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rtest_integrate2.mac
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/* GNU GENERAL PUBLIC LICENSE
Version 2, June 1991
This file is a modification of 'rtest_integrate' from the Maxima CAS project. I thank
the Maxima developers for the herculean effort made to create this test file.
To adapt this test file for `my_int`, I have replaced all calls of 'integrate' to 'my_int.'
And I am in the process of:
(a) removing the comments on the problem numbers
(b) commenting out only the tests that cause my_int to stall
(c) attempting to fix the cases that make my_int stall
(d) removing much of the simplification functions on the expected outputs
(e) fixing up the expected outputs
(f) replace nounform expected outputs with non-nounform outputs
(g) replace tests that look like
ratsimp(integrate(XXX,x) - YYY);
0$
with
integrate(XXX,x);
YYY$
(h) remove unneeded assumptions
As of 7 January 2025, this test runs OK with 123/758 tests passed. That seems terrible, but
there are tests in this file that are beyond the objectives of `my_int`. */
(kill(all),
load(incomplete_gamma_int),
declare(z,complex,n,integer),
assume(n>0),
assume(notequal(v,0)),
domain:complex,
radexpand:true,
/* This function checks if we get back the integrand.
*
* integral is the result of an integration,
* var the integration variable, and
* simplifier a simplifying function.
*/
diffcheck(integral,var,simplifier):=
if op(integral)=nounify(my_int) then
integral
else
simplifier(diff(integral,var)),
factor_radcan(expr):=factor(radcan(expr)),
done);
done;
/* ----- Involving only one direct function --------------------------------- */
block([gamma_expand : true], my_int(a^(c*z),z));
a^(c*z)/(log(a)*c);
block([gamma_expand : true], my_int(a^z,z));
a^z/log(a);
block([gamma_expand : true], my_int(%e^(a*z+b),z));
%e^(a*z+b)/a;
block([gamma_expand : true], my_int(%e^(a*z),z));
%e^(a*z)/a;
block([gamma_expand : true], my_int(%e^z,z));
%e^z;
/* ----- Power arguments and base a ----------------------------------------- */
my_int(a^(b*z^r),z);
z*gamma_incomplete(1/r,-b*z^r*log(a))*(-b*z^r*log(a))^(-1/r)/-r;
my_int(a^(b*z^2),z);
(gamma_incomplete(1/2,-(log(a)*b*z^2))*a^(b*z^2)*sqrt(-(log(a)*b*z^2))*%e^-(log(a)*b*z^2))$
my_int(a^(b*sqrt(z)),z);
2*(log(a)*b*sqrt(z)-1)*a^(b*sqrt(z))/(log(a)^2*b^2);
my_int(a^(b*(z^r)^p),z);
z*gamma_incomplete(1/(p*r),-log(a)*b*(z^r)^p)
/(-p*r*(-b*log(a)*(z^r)^p)^(1/(p*r)));
my_int(a^(b*(z^r)^(1/r)),z), domain:real, radcan;
a^(b*z)/(log(a)*b)$
my_int(a^(b*sqrt(z^2)),z), radcan;
a^(b*z)/(log(a)*b)$
/* ----- Power arguments and base %e ---------------------------------------- */
my_int(%e^(a*z^r),z);
-z*gamma_incomplete(1/r,-a*z^r)/(r*(-a*z^r)^(1/r));
my_int(%e^(a*z^2),z);
(gamma_incomplete(1/2,-(a*z^2))*sqrt(-(a*z^2)))/(2*a*z)$
my_int(%e^(a*sqrt(z)),z);
2*(a*sqrt(z)-1)*%e^(a*sqrt(z))/a^2;
my_int(%e^(a*(z^r)^p),z);
-z*gamma_incomplete(1/(p*r),-a*(z^r)^p)/(p*r*(-a*(z^r)^p)^(1/(p*r)));
my_int(%e^(a*(z^r)^(1/r)),z), domain:real;
%e^(a*z)/a$
my_int(%e^(a*sqrt(z^2)),z);
%e^(a*z)/a$
/* ----- Involving z^(v-1), arguments b*z and base a ------------------------ */
my_int(z^(v-1)*a^(b*z),z);
-((a^(b*z)*gamma_incomplete(v,-(log(a)*b*z))*z^v*%e^-(log(a)*b*z))/(-(log(a)*b*z))^v)$
my_int(a^(b*z)/z,z),gamma_expand:true,logcontract;
expintegral_ei(b*z*log(a));
my_int(z^(v-1)*a^z,z);
-((a^z*gamma_incomplete(v,-(log(a)*z))*z^v*%e^-(log(a)*z))/(-(log(a)*z))^v)$
my_int(a^z/z,z),gamma_expand:true,logcontract;
expintegral_ei(z*log(a));
/* ----- Involving z^(v-1), arguments a*z and base %e ----------------------- */
my_int(z^(v-1)*%e^(a*z),z);
-gamma_incomplete(v,-a*z)*z^v/(-a*z)^v;
my_int(z^(v-1)*%e^z,z);
-gamma_incomplete(v,-z)*z^v/(-z)^v;
my_int(z^n*%e^(a*z),z),factor;
a^(-n-1)*(-1)^n*gamma_incomplete(n+1,-a*z);
my_int(%e^(a*z)/z^n,z),factor;
a^(n-1)*(-1)^n*gamma_incomplete(1-n,-a*z);
my_int(z*%e^(a*z),z);
-(gamma_incomplete(2,-(a*z))/a^2)$
my_int(sqrt(z^2)*%e^(a*z),z);
-((gamma_incomplete(2,-(a*z))*sqrt(z^2))/(a^2*z))$
my_int(z^2*%e^(a*z),z);
gamma_incomplete(3,-(a*z))/a^3$
my_int(z^3*%e^(a*z),z);
-(gamma_incomplete(4,-(a*z))/a^4)$
my_int(z^4*%e^(a*z),z);
(gamma_incomplete(5,-(a*z))/a^5)$
my_int(z^5*%e^(a*z),z);
-(gamma_incomplete(6,-(a*z))/a^6)$
my_int(z^6*%e^(a*z),z);
gamma_incomplete(7,-(a*z))/a^7$
my_int(z^7*%e^(a*z),z);
-(gamma_incomplete(8,-(a*z))/a^8)$
my_int(z^8*%e^(a*z),z);
gamma_incomplete(9,-(a*z))/a^9$
my_int(%e^(a*z)/z,z);
expintegral_ei(a*z);
my_int(%e^(a*z)/z^2,z);
gamma_incomplete(-1, -a*z)*a$
my_int(%e^(a*z)/z^3,z);
-(gamma_incomplete(-2,-(a*z))*a^2)$
my_int(%e^(a*z)/z^4,z);
gamma_incomplete(-3,-(a*z))*a^3$
my_int(%e^(a*z)/z^5,z);
-(gamma_incomplete(-4,-(a*z))*a^4)$
my_int(z^(1/2+n)*%e^(a*z),z);
(a^(-n-1)*(-1)^n*gamma_incomplete((2*n+3)/2,-(a*z))*sqrt(z))/sqrt(-(a*z))$
my_int(sqrt(z)*%e^(a*z),z);
(gamma_incomplete(3/2,-(a*z))*sqrt(-(1/a)))/a$
my_int(z^(3/2)*%e^(a*z),z);
(gamma_incomplete(5/2,-(a*z))*sqrt(-(1/a^3)))/a$
my_int(z^(5/2)*%e^(a*z),z);
(gamma_incomplete(7/2,-(a*z))*sqrt(-(1/a^5)))/a$
my_int(z^(7/2)*%e^(a*z),z);
(gamma_incomplete(9/2,-(a*z))*sqrt(-(1/a^7)))/a$
my_int(z^(9/2)*%e^(a*z),z);
(gamma_incomplete(11/2,-(a*z))*sqrt(-(1/a^9)))/a$
my_int(%e^(a*z)/sqrt(z),z);
(gamma_incomplete(1/2,-(a*z))*sqrt(-a))/a$
my_int(%e^(a*z)/sqrt(z),z);
(gamma_incomplete(1/2,-(a*z))*sqrt(-a))/a$
my_int(%e^(a*z)/sqrt(z^3),z);
(gamma_incomplete(-(1/2),-(a*z))*sqrt(-a^3))/a$
my_int(%e^(a*z)/z^(5/2),z);
(gamma_incomplete(-(3/2),-(a*z))*sqrt(-a^5))/a$
my_int(%e^(a*z)/z^(7/2),z);
(gamma_incomplete(-(5/2),-(a*z))*sqrt(-a^7))/a$
my_int(%e^(a*z)/z^(9/2),z);
(gamma_incomplete(-(7/2),-(a*z))*sqrt(-a^9))/a$
/* ----- Involving z^(v-1), arguments b*z+d and base a ---------------------- */
my_int(z^(v-1)*a^(b*z+d),z);
-((a^(b*z+d)*gamma_incomplete(v,-(log(a)*b*z))*z^v*%e^-(log(a)*b*z))/(-(log(a)*b*z))^v)$
/* Problem 70: NEW 2 */
my_int(a^(b*z+d)/z,z);
a^d*expintegral_ei(b*z*log(a));
/* ----- Involving z^(v-1), arguments b*z+d and base %e --------------------- */
diffcheck(my_int(z^(v-1)*%e^(a*z+b),z),z,factor);
z^(v-1)*%e^(a*z+b);
my_int(z^n*%e^(a*z+b),z);
(%e^b*gamma_incomplete(n+1,-(a*z))*z^n)/(a*(-(a*z))^n)$
my_int(z*%e^(a*z+b),z);
-((gamma_incomplete(2,-(a*z))*%e^b)/a^2)$
my_int(%e^(a*z+b)/z,z),gamma_expand:true,logcontract;
%e^b*expintegral_ei(a*z);
my_int(%e^(a*z+b)/sqrt(z),z),erf_representation:erfi;
(gamma_incomplete(1/2,-(a*z))*sqrt(-a)*%e^b)/a$
/* ----- Involving z^(v-1), arguments b*z^r and base a ---------------------- */
my_int(z^(v-1)*a^(b*z^r),z);
-gamma_incomplete(v/r,-log(a)*b*z^r)*z^v/(r*(-log(a)*b*z^r)^(v/r))$
my_int(a^(b*z^r)/z,z),gamma_expand:true,logcontract;
expintegral_ei(log(a)*b*z^r)/r;
/* ----- Involving z^(v-1), arguments a*z^r and base %e --------------------- */
my_int(z^(v-1)*%e^(a*z^r),z);
-gamma_incomplete(v/r,-a*z^r)*z^v/(r*(-a*z^r)^(v/r))$
my_int(%e^(a*z^r)/z,z),gamma_expand:true,logcontract;
expintegral_ei(a*z^r)/r;
my_int(z^(2*n)*%e^(a*z^2),z);
(a^(-n-1)*(-1)^n*gamma_incomplete((2*n+1)/2,-(a*z^2))*sqrt(-(a*z^2)))/(2*z)$
my_int(z^(2*n-1)*%e^(a*z^2),z),factor;
-((gamma_incomplete(n,-(a*z^2))*z^(2*n))/(2*(-(a*z^2))^n))$
my_int(z*%e^(a*z^2),z);
gamma_incomplete(1,-(a*z^2))/(2*a)$
my_int(z^2*%e^(a*z^2),z);
(gamma_incomplete(3/2,-(a*z^2))*sqrt(-(1/(a*z^2)))*z)/(2*a)$
my_int(z^3*%e^(a*z^2),z);
-(gamma_incomplete(2,-(a*z^2))/(2*a^2))$
my_int(z^4*%e^(a*z^2),z),erf_representation:erfi;
(gamma_incomplete(5/2,-(a*z^2))*sqrt(-(1/(a^3*z^6)))*z^3)/(2*a)$
my_int(z^5*%e^(a*z^2),z);
gamma_incomplete(3,-(a*z^2))/(2*a^3)$
my_int(%e^(a*z^2)/z,z);
expintegral_ei(a*z^2)/2;
my_int(%e^(a*z^2)/z^2,z);
(gamma_incomplete(-(1/2),-(a*z^2))*sqrt(-a^3)*abs(z))/(2*a*z)$
my_int(%e^(a*z^2)/z^3,z);
(gamma_incomplete(-1,-(a*z^2))*a)/2$
my_int(%e^(a*z^2)/z^4,z)$
(gamma_incomplete(-(3/2),-(a*z^2))*sqrt(-a^5)*abs(z))/(2*a*z)$
my_int(%e^(a*z^2)/z^5,z),gamma_expand:true,logcontract;
-((gamma_incomplete(-2,-(a*z^2))*a^2)/2)$
my_int(z^n*%e^(a*sqrt(z)),z);
-2*gamma_incomplete(2*(n+1),-a*sqrt(z))/a^(2*(n+1));
my_int(z*%e^(a*sqrt(z)),z);
2*(a^3*z^(3/2)-3*a^2*z+6*a*sqrt(z)-6)*%e^(a*sqrt(z))/a^4;
my_int(z^2*%e^(a*sqrt(z)),z);
2*(a^5*z^(5/2)-5*a^4*z^2+20*a^3*z^(3/2)-60*a^2*z+120*a*sqrt(z)-120)
*%e^(a*sqrt(z))
/a^6;
my_int(z^3*%e^(a*sqrt(z)),z);
2*(a^7*z^(7/2)-7*a^6*z^3+42*a^5*z^(5/2)-210*a^4*z^2+840*a^3*z^(3/2)
-2520*a^2*z+5040*a*sqrt(z)-5040)*%e^(a*sqrt(z))
/a^8;
my_int(z^4*%e^(a*sqrt(z)),z);
2*(a^9*z^(9/2)-9*a^8*z^4+72*a^7*z^(7/2)-504*a^6*z^3+3024*a^5*z^(5/2)
-15120*a^4*z^2+60480*a^3*z^(3/2)-181440*a^2*z
+362880*a*sqrt(z)-362880)*%e^(a*sqrt(z))
/a^10;
my_int(z^5*%e^(a*sqrt(z)),z);
2*(a^11*z^(11/2)-11*a^10*z^5+110*a^9*z^(9/2)-990*a^8*z^4
+7920*a^7*z^(7/2)-55440*a^6*z^3+332640*a^5*z^(5/2)
-1663200*a^4*z^2+6652800*a^3*z^(3/2)-19958400*a^2*z
+39916800*a*sqrt(z)-39916800)*%e^(a*sqrt(z))
/a^12;
my_int(%e^(a*sqrt(z))/z,z);
2*expintegral_ei(a*sqrt(z));
my_int(%e^(a*sqrt(z))/z^2,z);
((2*a*z+2*sqrt(z))*%e^(a*sqrt(z))-2*a^2*expintegral_ei(a*sqrt(z))*z^(3/2))
/-(2*z^(3/2));
my_int(%e^(a*sqrt(z))/z^3,z);
((2*a^3*z^3+2*a^2*z^(5/2)+4*a*z^2+12*z^(3/2))*%e^(a*sqrt(z))
-2*a^4*expintegral_ei(a*sqrt(z))*z^(7/2))
/-(24*z^(7/2));
my_int(%e^(a*sqrt(z))/z^4,z);
((2*a^5*z^5+2*a^4*z^(9/2)+4*a^3*z^4+12*a^2*z^(7/2)+48*a*z^3+240*z^(5/2))
*%e^(a*sqrt(z))
-2*a^6*expintegral_ei(a*sqrt(z))*z^(11/2))
/-(720*z^(11/2));
my_int(%e^(a*sqrt(z))/z^5,z);
((2*a^7*z^7+2*a^6*z^(13/2)+4*a^5*z^6+12*a^4*z^(11/2)+48*a^3*z^5
+240*a^2*z^(9/2)+1440*a*z^4+10080*z^(7/2))
*%e^(a*sqrt(z))
-2*a^8*expintegral_ei(a*sqrt(z))*z^(15/2))
/-(40320*z^(15/2));
/* ----- Involving z^(v-1), arguments b*z^r and base a ---------------------- */
diffcheck(my_int(z^(v-1)*a^(d+b*z^r),z),z,radcan);
z^(v-1)*a^(d+b*z^r);
my_int(a^(d+b*z^r)/z,z),gamma_expand:true,logcontract;
a^d*expintegral_ei(log(a)*b*z^r)/r;
/* ----- Involving z^(v-1), arguments b*z^r and base %e --------------------- */
diffcheck(my_int(z^(v-1)*%e^(b+a*z^r),z),z,factor);
z^(v-1)*%e^(b+a*z^r);
my_int(%e^(b+a*z^r)/z,z);
%e^b*expintegral_ei(a*z^r)/r;
/* ----- Involving (a*z+b)^-n ----------------------------------------------- */
my_int(%e^(c*z)/(b+a*z),z),expintexpand:true,logcontract;
%e^-(b*c/a)*expintegral_ei((a*c*z+b*c)/a)/a;
my_int(%e^(c*z)/(b+a*z)^2,z);
(gamma_incomplete(-1,-((c*(a*z+b))/a))*c*%e^-((b*c)/a))/a^2$
my_int(%e^(c*z)/(b+a*z)^3,z);
-((gamma_incomplete(-2,-((c*(a*z+b))/a))*c^2*%e^-((b*c)/a))/a^3)$
my_int(%e^(c*z)/(b+a*z)^4,z);
(gamma_incomplete(-3,-((c*(a*z+b))/a))*c^3*%e^-((b*c)/a))/a^4$
my_int(%e^(c*z)/(b+a*z)^5,z);
-((gamma_incomplete(-4,-((c*(a*z+b))/a))*c^4*%e^-((b*c)/a))/a^5)$
my_int(%e^(c*z)/(b+a*z)^6,z);
(gamma_incomplete(-5,-((c*(a*z+b))/a))*c^5*%e^-((b*c)/a))/a^6$
my_int(z*%e^(c*z)/(b+a*z),z);
z*%e^(c*z)/(a*c*z+b*c)-b*'my_int(%e^(c*z)/(a^2*c*z^2+2*a*b*c*z+b^2*c),z);
my_int(z^2*%e^(c*z)/(b+a*z),z);
exp(c*x)/(a*c) - (b*exp(-b*c/a) * expintegral_ei(b*c/a + c*z))/a^2$
/* Problem 121: NOUN */
/* DOES NOT FINISH
my_int(z*%e^(c*z)/(b+a*z)^2,z);
'my_int((a*z-b)*%e^(c*z)/(a^3*c*z^3+3*a^2*b*c*z^2+3*a*b^2*c*z+b^3*c),z)
+z*%e^(c*z)/(a^2*c*z^2+2*a*b*c*z+b^2*c);
/* Problem 122: NOUN */
my_int(z^2*%e^(c*z)/(b+a*z)^2,z);
z^2*%e^(c*z)/(a^2*c*z^2+2*a*b*c*z+b^2*c)
-2*b*'my_int(z*%e^(c*z)/(a^3*c*z^3+3*a^2*b*c*z^2+3*a*b^2*c*z+b^3*c),z);
/* Problem 123: NOUN */
my_int(z^3*%e^(c*z)/(b+a*z)^2,z);
'my_int((2*b^2*c*z+2*b^2)*%e^(c*z)/(a^4*c^2*z^3+3*a^3*b*c^2*z^2
+3*a^2*b^2*c^2*z+a*b^3*c^2),z)
+(a*c*z^3-a*z^2-2*b*z)*%e^(c*z)/(a^3*c^2*z^2+2*a^2*b*c^2*z+a*b^2*c^2);
/* Problem 124: NOUN */
my_int(z^4*%e^(c*z)/(b+a*z)^2,z);
'my_int(((2*a*b^2*c-2*b^3*c^2)*z-2*b^3*c-4*a*b^2)*%e^(c*z)
/(a^5*c^3*z^3+3*a^4*b*c^3*z^2+3*a^3*b^2*c^3*z+a^2*b^3*c^3),z)
+(a^2*c^2*z^4-2*a^2*c*z^3+(2*a^2-2*a*b*c)*z^2+(2*b^2*c+4*a*b)*z)*%e^(c*z)
/(a^4*c^3*z^2+2*a^3*b*c^3*z+a^2*b^2*c^3);
/* ----- Involving (a*z^2+b)^-n and base %e --------------------------------- */
/* Problem 125: NOUN */
my_int(%e^(c*z)/(b+a*z^2),z);
'my_int(%e^(c*z)/(a*z^2+b),z);
/* Problem 126: NOUN */
my_int(z*%e^(c*z)/(b+a*z^2),z);
'my_int((a*z^2-b)*%e^(c*z)/(a^2*c*z^4+2*a*b*c*z^2+b^2*c),z)
+z*%e^(c*z)/(a*c*z^2+b*c);
/* Problem 127: NOUN */
my_int(%e^(c*z)/(b+a*z^2)^2,z);
'my_int(%e^(c*z)/(a*z^2+b)^2,z);
/* Problem 128: NOUN */
my_int(z*%e^(c*z)/(b+a*z^2)^2,z);
'my_int((3*a*z^2-b)*%e^(c*z)/(a^3*c*z^6+3*a^2*b*c*z^4+3*a*b^2*c*z^2+b^3*c),z)
+z*%e^(c*z)/(a^2*c*z^4+2*a*b*c*z^2+b^2*c);
/* ----- Involving (a*z^2+b*z+c)^-n ----------------------------------------- */
/* Problem 129: NOUN */
my_int(%e^(d*z)/(c+b*z+a*z^2),z);
'my_int(%e^(d*z)/(a*z^2+b*z+c),z);
/* Problem 130: NOUN */
my_int(z*%e^(d*z)/(c+b*z+a*z^2),z);
'my_int((a*z^2-c)*%e^(d*z)/(a^2*d*z^4+2*a*b*d*z^3+(2*a*c*d+b^2*d)*z^2
+2*b*c*d*z+c^2*d),z)
+z*%e^(d*z)/(a*d*z^2+b*d*z+c*d);
/* Problem 131: NOUN */
my_int(%e^(d*z)/(c+b*z+a*z^2)^2,z);
'my_int(%e^(d*z)/(a*z^2+b*z+c)^2,z);
/* Problem 132: NOUN */
my_int(z*%e^(d*z)/(c+b*z+a*z^2)^2,z);
'my_int((3*a*z^2+b*z-c)*%e^(d*z)/(a^3*d*z^6+3*a^2*b*d*z^5
+(3*a^2*c*d+3*a*b^2*d)*z^4
+(6*a*b*c*d+b^3*d)*z^3
+(3*a*c^2*d+3*b^2*c*d)*z^2
+3*b*c^2*d*z+c^3*d),z)
+z*%e^(d*z)/(a^2*d*z^4+2*a*b*d*z^3+(2*a*c*d+b^2*d)*z^2+2*b*c*d*z+c^2*d);
*/
/* ----- Involving (a*z*b)^w and base %e ------------------------------------ */
my_int((a*z+b)^w*%e^(c*z+d),z);
(%e^(d-(b*c)/a)*gamma_incomplete(w+1,-((c*(a*z+b))/a))*(a*z+b)^w)
/(c*(-((c*(a*z+b))/a))^w)$
my_int((a*z+b)^w*%e^(c*z),z);
(%e^-((b*c)/a)*gamma_incomplete(w+1,-((c*(a*z+b))/a))*(a*z+b)^w)
/(c*(-((c*(a*z+b))/a))^w)$
/* Problem 135: OK */
multthru(my_int((a*z+b)^(3/2)*%e^(c*z),z));
3*sqrt(%pi)*a*%e^-(b*c/a)*erf(sqrt(-c/a)*sqrt(a*z+b))/(4*c^2*sqrt(-c/a))
+(2*a*c*(a*z+b)^(3/2)-3*a^2*sqrt(a*z+b))*%e^(c*(a*z+b)/a-b*c/a)/(2*a*c^2);
/* Problem 136: OK */
multthru(my_int(sqrt(a*z+b)*%e^(c*z),z));
sqrt(a*z+b)*%e^(c*(a*z+b)/a-b*c/a)/c-sqrt(%pi)*%e^-(b*c/a)
*erf(sqrt(-c/a)*sqrt(a*z+b))
/(2*c*sqrt(-c/a));
/* Problem 137: OK */
my_int(%e^(c*z)/sqrt(a*z+b),z);
sqrt(%pi)*%e^-(b*c/a)*erf(sqrt(-c/a)*sqrt(a*z+b))/(a*sqrt(-c/a));
/* Problem 138: NEW 2
* This example is first transformed to an example of type 2.
*/
diffcheck(my_int(%e^(c*z)/(a*z+b)^(3/2),z),z,factor);
%e^(c*z)/(a*z+b)^(3/2);
/* ----- Involving a*z^2+b*z+c ---------------------------------------------- */
/* Problem 139: OK */
my_int(d^(c+b*z+a*z^2),z),erf_representation:erfi,radexpand:all,factor;
sqrt(%pi)*d^c*%e^-(b^2*log(d)/(4*a))*erfi(sqrt(log(d))*(2*a*z+b)/(2*sqrt(a)))
/(2*sqrt(a)*sqrt(log(d)));
/* Problem 140: OK */
my_int(%e^(c+b*z+a*z^2),z),erf_representation:erfi,radexpand:all,factor;
sqrt(%pi)*%e^(c-b^2/(4*a))*erfi((2*a*z+b)/(2*sqrt(a)))/(2*sqrt(a));
/* ----- Involving a*z^2+b -------------------------------------------------- */
/* Problem 141: OK */
my_int(d^(b*z+a*z^2),z),erf_representation:erfi,radexpand:all,factor;
sqrt(%pi)*%e^-(b^2*log(d)/(4*a))*erfi(sqrt(log(d))*(2*a*z+b)/(2*sqrt(a)))
/(2*sqrt(a)*sqrt(log(d)));
/* Problem 142: OK */
my_int(%e^(b*z+a*z^2),z),erf_representation:erfi,radexpand:all,factor;
sqrt(%pi)*%e^-(b^2/(4*a))*erfi((2*a*z+b)/(2*sqrt(a)))/(2*sqrt(a))$
/* Involving a*z^2+c -------------------------------------------------------- */
/* Problem 143: OK */
my_int(d^(c+a*z^2),z),erf_representation:erfi,radexpand:all;
sqrt(%pi)*d^c*erfi(sqrt(a)*sqrt(log(d))*z)/(2*sqrt(a)*sqrt(log(d)))$
/* Problem 144: OK */
my_int(%e^(c+a*z^2),z),erf_representation:erfi,radexpand:all;
sqrt(%pi)*%e^c*erfi(sqrt(a)*z)/(2*sqrt(a));
/* Involving a*z^2+b/z^2 ---------------------------------------------------- */
/* Problem 145: NEW 4 */
diffcheck(my_int(d^(b/z^2+a*z^2),z),z,factor_radcan);
d^(b/z^2+a*z^2);
/* Problem 146: NEW 4 */
diffcheck(my_int(%e^(b/z^2+a*z^2),z),z,factor);
%e^(b/z^2+a*z^2);
/* ----- Involving az^2+b/z^2+c --------------------------------------------- */
/* Problem 147: NEW 4 */
diffcheck(my_int(d^(c+b/z^2+a*z^2),z),z,factor_radcan);
d^(c+b/z^2+a*z^2);
/* Problem 148: NEW 4 */
diffcheck(my_int(%e^(c+b/z^2+a*z^2),z),z,factor);
%e^(c+b/z^2+a*z^2);
/* ----- Involving a*z+b*sqrt(z)+c ------------------------------------------ */
/* Problem 149: NEW 5
* This integral is first transformed to an integral of type 5.
*/
diffcheck(my_int(d^(c+a*z+sqrt(z)*b),z),z,radcan);
d^(c+a*z+sqrt(z)*b);
/* Problem 150: NEW 5
* This integral is first transformed to an integral of type 5.
*/
diffcheck(my_int(%e^(c+a*z+sqrt(z)*b),z),z,factor), domain:real;
%e^(c+a*z+sqrt(z)*b);
/* ----- Involving a*z+b*sqrt(z) -------------------------------------------- */
/* Problem 151: NEW 5 */
diffcheck(my_int(d^(a*z+sqrt(z)*b),z),z,radcan);
d^(a*z+sqrt(z)*b);
/* Problem 152: NEW 5 */
diffcheck(my_int(%e^(a*z+sqrt(z)*b),z),z,factor), domain:real;
%e^(a*z+sqrt(z)*b);
/* ----- Involving a*z^r+c -------------------------------------------------- */
/* Problem 153: NEW 1 */
diffcheck(my_int(d^(c+a*z^r),z),z,radcan);
d^(c+a*z^r);
/* Problem 154: NEW 1 */
diffcheck(my_int(%e^(c+a*z^r),z),z,factor);
%e^(c+a*z^r);
/* Problem 155: OK */
my_int(%e^(c+a*z^2),z),erf_representation:erfi,radexpand:all;
sqrt(%pi)*%e^c*erfi(sqrt(a)*z)/(2*sqrt(a))$
/* Problem 156: OK */
my_int(%e^(c+a*sqrt(z)),z);
2*(a*%e^c*sqrt(z)-%e^c)*%e^(a*sqrt(z))/a^2;
/* ----- Arguments involving exponential functions -------------------------- */
/* Problem 157: NEW 2 */
my_int(a^b^z,z),gamma_expand:true,logcontract;
expintegral_ei(log(a)*b^z)/log(b);
/* Problem 158: NEW 2 */
my_int(%e^a^z,z),gamma_expand:true,logcontract,domain:complex;
(expintegral_ei(a^z)+log(-1))/log(a);
/* Problem 159: NEW 2 */
my_int(%e^%e^z,z),gamma_expand:true,logexpand:all;
expintegral_ei(%e^z)+log(-1);
/* ----- Involving trigonometric functions ---------------------------------- */
/* Problem 160: NOUN */
my_int(%e^tan(z),z);
'my_int(%e^tan(z),z);
/* Problem 161: NOUN */
my_int(%e^(a*tan(z)),z);
'my_int(%e^(a*tan(z)),z);
/* Problem 162: NOUN */
my_int(%e^cot(z),z);
'my_int(%e^cot(z),z);
/* Problem 163: NOUN */
my_int(%e^(a*cot(z)),z);
'my_int(%e^(a*cot(z)),z);
/* ----- Involving hyperbolic functions ------------------------------------- */
/* Problem 164: NOUN */
my_int(%e^tanh(z),z);
'my_int(%e^tanh(z),z);
/* Problem 165: NOUN */
my_int(%e^(a*tanh(z)),z);
'my_int(%e^(a*tanh(z)),z);
/* Problem 166: NOUN */
my_int(%e^coth(z),z);
'my_int(%e^coth(z),z);
/* Problem 167: NOUN */
my_int(%e^(a*coth(z)),z);
'my_int(%e^(a*coth(z)),z);
/* ----- Involving inverse trigonometric functions -------------------------- */
/* Problem 168: NOUN */
my_int(%e^asin(z),z);
'my_int(%e^asin(z),z);
/* Problem 169: NOUN */
my_int(%e^(a*asin(z)),z);
'my_int(%e^(a*asin(z)),z);
/* Problem 170: NOUN */
my_int(%e^acos(z),z);
'my_int(%e^acos(z),z);
/* Problem 171: NOUN */
my_int(%e^(a*acos(z)),z);
'my_int(%e^(a*acos(z)),z);
/* DOES NOT FINISH
/* Problem 172: NOUN */
my_int(%e^atan(z),z);
'my_int(%e^atan(z),z);
/* Problem 173: NOUN */
my_int(%e^(a*atan(z)),z);
'my_int(%e^(a*atan(z)),z);
/* Problem 174: NOUN */
my_int(%e^acot(z),z);
'my_int(%e^acot(z),z);
/* Problem 175: NOUN */
my_int(%e^(a*acot(z)),z);
'my_int(%e^(a*acot(z)),z);
/* Problem 176: NOUN */
my_int(%e^acsc(z),z);
'my_int(%e^acsc(z),z);
/* Problem 177: NOUN */
my_int(%e^(a*acsc(z)),z);
'my_int(%e^(a*acsc(z)),z);
/* Problem 178: NOUN */
my_int(%e^asec(z),z);
'my_int(%e^asec(z),z);
/* Problem 179: NOUN */
my_int(%e^(a*asec(z)),z);
'my_int(%e^(a*asec(z)),z);
*/
/* Arguments invovling inverse hyperbolic functions ------------------------- */
/* Problem 180: NEW 12/2009 */
my_int(%e^asinh(z),z);
log(sqrt(z^2+1)+z)/2+z*sqrt(z^2+1)/2+z^2/2;
/* Problem 181: NOUN */
my_int(%e^(a*asinh(z)),z);
'my_int(%e^(a*asinh(z)),z);
/* Problem 182: NEW 12/2009 */
my_int(%e^acosh(z),z);
-log(2*sqrt(z^2-1)+2*z)/2+z*sqrt(z^2-1)/2+z^2/2;
/* Problem 183: NOUN */
my_int(%e^(a*acosh(z)),z);
'my_int(%e^(a*acosh(z)),z);
/* DOES NOT FINISH
/* Problem 184: NOUN */
my_int(%e^atan(z),z);
'my_int(%e^atan(z),z);
/* Problem 185: NOUN */
my_int(%e^(a*atan(z)),z);
'my_int(%e^(a*atan(z)),z);
/* Problem 186: NOUN */
my_int(%e^acot(z),z);
'my_int(%e^acot(z),z);
/* Problem 187: NOUN */
my_int(%e^(a*acot(z)),z);
'my_int(%e^(a*acot(z)),z);
*/
/* Problem 189: NEW 08/2010
* This example works nicely for x>0. For z complex we get abs(z) expressions.
*/
assume(x>0);
[x>0];
my_int(%e^acsch(x),x),radcan;
-log(sqrt(x^2+1)+1)+2*log(x)+sqrt(x^2+1);
forget(x>0);
[x>0];
/* Problem 191: NOUN */
my_int(%e^(a*acsch(z)),z);
'my_int(%e^(a*acsch(z)),z);
/* Problem 193: NEW 12/2009 */
assume(x>0);
[x>0];
my_int(%e^asech(x),x),ratsimp;
-log((2*sqrt(1-x^2)+2)/x)+log(x)+sqrt(1-x^2)$
forget(x>0);
[x>0];
/* Problem 195: NOUN */
my_int(%e^(a*asech(z)),z);
'my_int(%e^(a*asech(z)),z);
/* ----- Involving z^n*d^(a*z^r+b*z) ---------------------------------------- */
/* Problem 196: NEW 5 */
niceindices(my_int(z^n*d^(b*z+a*z^2),z));
-a^(-n-1)*('sum(b^(n-i)*gamma_incomplete((i+1)/2,-log(d)*(2*a*z+b)^2/(4*a))
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*a*z+b)^(i+1)
/(-log(d)*(2*a*z+b)^2/a)^((i+1)/2),i,0,n))
/(2*d^(b^2/(4*a)));
/* ----- Prove algorithm type 5 for some special case ----- */
diffcheck(my_int(z*d^(b*z+a*z^2),z),z,radcan);
z*d^(b*z+a*z^2);
diffcheck(my_int(z^2*d^(b*z+a*z^2),z),z,radcan);
z^2*d^(b*z+a*z^2);
diffcheck(my_int(z^3*d^(b*z+a*z^2),z),z,radcan);
z^3*d^(b*z+a*z^2);
/* Problem 200: NEW 5 */
niceindices(my_int(z^n*%e^(b*z+a*z^2),z));
-a^(-n-1)*%e^-(b^2/(4*a))
*('sum(b^(n-i)*gamma_incomplete((i+1)/2,-(2*a*z+b)^2/(4*a))
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*a*z+b)^(i+1)
/(-(2*a*z+b)^2/a)^((i+1)/2),i,0,n))
/2;
/* ----- Prove algorithm type 5 for some special case ----- */
diffcheck(my_int(z*%e^(b*z+a*z^2),z),z,factor), domain:real;
z*%e^(a*z^2+b*z)$
diffcheck(my_int(z^2*%e^(b*z+a*z^2),z),z,factor), domain:real;
z^2*%e^(a*z^2+b*z)$
diffcheck(my_int(z^3*%e^(b*z+a*z^2),z),z,factor), domain:real;
z^3*%e^(a*z^2+b*z)$
/* Problem 204: NEW 6 */
niceindices(niceindices(my_int(z^n*d^(b*z+a*sqrt(z)),z)));
log(d)^(-2*(n+1)+2*n+1)*2^(-n-1)
*('sum(4^j*binomial(n,j)
*'sum(a^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
*(2*b*sqrt(z)+a)^(j+i)
*(a*log(d)
*gamma_incomplete(
(j+i+1)/2,
-log(d)
*(2*b*sqrt(z)+a)^2
/(4*b))*(2*b*sqrt(z)+a)
+2*b
*gamma_incomplete(
(j+i+2)/2,
-log(d)
*(2*b*sqrt(z)+a)^2
/(4*b))
*sqrt(
-log(d)
*(2*b*sqrt(z)+a)^2
/b))
/(-log(d)*(2*b*sqrt(z)+a)^2/b)
^((j+i+1)/2),i,0,j),j,0,n))
/(b^(2*(n+1))*d^(a^2/(4*b)));
/* --- For special cases n the integral is transformed to type 5 ------------ */
diffcheck(my_int(z*d^(b*z+a*sqrt(z)),z),z,radcan);
z*d^(b*z+a*sqrt(z));
diffcheck(my_int(z^2*d^(b*z+a*sqrt(z)),z),z,radcan);
z^2*d^(b*z+a*sqrt(z));
diffcheck(my_int(z^3*d^(b*z+a*sqrt(z)),z),z,radcan);
z^3*d^(b*z+a*sqrt(z));
/* Problem 208: NEW 6 */
niceindices(niceindices(my_int(z^n*%e^(b*z+a*sqrt(z)),z)));
%e^-(a^2/(4*b))*2^(-n-1)
*('sum(4^j*binomial(n,j)
*'sum(a^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
*(2*b*sqrt(z)+a)^(j+i)
*(a
*gamma_incomplete(
(j+i+1)/2,
-(2*b*sqrt(z)+a)^2/(4*b))
*(2*b*sqrt(z)+a)
+2*b
*gamma_incomplete(
(j+i+2)/2,
-(2*b*sqrt(z)+a)^2/(4*b))
*sqrt(-(2*b*sqrt(z)+a)^2/b))
/(-(2*b*sqrt(z)+a)^2/b)^((j+i+1)/2),i,0,j),j,
0,n))
/b^(2*(n+1));
/* --- Again the problem is my_intd with type 5 --- */
diffcheck(my_int(z*%e^(b*z+a*sqrt(z)),z),z,factor), domain:real;
z*%e^(b*z+a*sqrt(z));
diffcheck(my_int(z^2*%e^(b*z+a*sqrt(z)),z),z,factor), domain:real;
z^2*%e^(b*z+a*sqrt(z));
diffcheck(my_int(z^3*%e^(b*z+a*sqrt(z)),z),z,factor), domain:real;
z^3*%e^(b*z+a*sqrt(z));
/* ----- Involving z^n*d^(a*z^r+b*z+c) -------------------------------------- */
/* Problem 212: NEW 5 */
niceindices(my_int(z^n*d^(c+b*z+a*z^2),z));
-a^(-n-1)*d^(c-b^2/(4*a))
*('sum(b^(n-i)*gamma_incomplete((i+1)/2,-log(d)*(2*a*z+b)^2/(4*a))
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*a*z+b)^(i+1)
/(-log(d)*(2*a*z+b)^2/a)^((i+1)/2),i,0,n))
/2;
/* Problem 213: NEW 5 */
niceindices(my_int(z^n*%e^(c+b*z+a*z^2),z));
-a^(-n-1)*%e^(c-b^2/(4*a))
*('sum(b^(n-i)*gamma_incomplete((i+1)/2,-(2*a*z+b)^2/(4*a))
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*a*z+b)^(i+1)
/(-(2*a*z+b)^2/a)^((i+1)/2),i,0,n))
/2;
/* Problem 214: NEW 6 */
niceindices(niceindices(my_int(z^n*d^(c+b*z+a*sqrt(z)),z)));
d^(c-a^2/(4*b))*log(d)^(-2*(n+1)+2*n+1)*2^(-n-1)
*('sum(4^j*binomial(n,j)
*'sum(a^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
*(2*b*sqrt(z)+a)^(j+i)
*(a*log(d)
*gamma_incomplete(
(j+i+1)/2,
-log(d)*(2*b*sqrt(z)+a)^2/(4*b))
*(2*b*sqrt(z)+a)
+2*b
*gamma_incomplete(
(j+i+2)/2,
-log(d)*(2*b*sqrt(z)+a)^2/(4*b))
*sqrt(
-log(d)*(2*b*sqrt(z)+a)^2/b))
/(-log(d)*(2*b*sqrt(z)+a)^2/b)^((j+i+1)/2),i,
0,j),j,0,n))
/b^(2*(n+1));
/* Problem 215: NEW 6 */
niceindices(niceindices(my_int(z^n*%e^(c+b*z+a*sqrt(z)),z)));
%e^(c-a^2/(4*b))*2^(-n-1)
*('sum(4^j*binomial(n,j)
*'sum(a^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
*(2*b*sqrt(z)+a)^(j+i)
*(a
*gamma_incomplete(
(j+i+1)/2,
-(2*b*sqrt(z)+a)^2/(4*b))
*(2*b*sqrt(z)+a)
+2*b
*gamma_incomplete(
(j+i+2)/2,
-(2*b*sqrt(z)+a)^2/(4*b))
*sqrt(-(2*b*sqrt(z)+a)^2/b))
/(-(2*b*sqrt(z)+a)^2/b)^((j+i+1)/2),i,0,j),j,
0,n))
/b^(2*(n+1));
/* ----- Involving z^(v-1)*d^(c+a*z^r) -------------------------------------- */
/* Problem 216: NEW 2 */
my_int(z^(v-1)*d^(c+a*z^r),z);
-d^c*gamma_incomplete(v/r,-a*log(d)*z^r)*z^v/(r*(-a*log(d)*z^r)^(v/r));
/* Problem 217: NEW 2 */
my_int(z^(v-1)*%e^(c+a*z^r),z);
-%e^c*gamma_incomplete(v/r,-a*z^r)*z^v/(r*(-a*z^r)^(v/r));
/* Problem 218: NEW 2 */
my_int(z^(v-1)*d^(c+a*z^2),z);
-d^c*gamma_incomplete(v/2,-a*log(d)*z^2)*z^v/(2*(-a*log(d)*z^2)^(v/2));
/* Problem 219: NEW 2 */
my_int(z^(v-1)*%e^(c+a*z^2),z);
-%e^c*gamma_incomplete(v/2,-a*z^2)*z^v/(2*(-a*z^2)^(v/2));
/* Problem 220: NEW 2 */
my_int(z^(v-1)*d^(c+a*sqrt(z)),z);
-2*d^c*gamma_incomplete(2*v,-a*log(d)*sqrt(z))*z^v/(-a*log(d)*sqrt(z))^(2*v);
/* Problem 221: NEW 2 */
my_int(z^(v-1)*%e^(c+a*sqrt(z)),z);
-2*%e^c*gamma_incomplete(2*v,-a*sqrt(z))*z^v/(-a*sqrt(z))^(2*v);
/* ----- Involving z^(2*n)*d^(a*z^2+b/Z^2) ---------------------------------- */
/* Problem 222: NOUN */
my_int(z^(2*n)*d^(b/z^2+a*z^2),z);
'my_int(d^(a*z^2+b/z^2)*z^(2*n),z);
/* Problem 223: NOUN */
my_int(z^(2*n)*%e^(b/z^2+a*z^2),z);
'my_int(z^(2*n)*%e^(a*z^2+b/z^2),z);
/* ----- Involving z^(2*n)*d^(a*z^2+b/Z^2+c) -------------------------------- */
/* Problem 224: NOUN */
my_int(z^(2*n)*d^(c+b/z^2+a*z^2),z);
'my_int(d^(a*z^2+b/z^2+c)*z^(2*n),z);
/* Problem 225: NOUN */
my_int(z^(2*n)*%e^(c+b/z^2+a*z^2),z);
'my_int(z^(2*n)*%e^(a*z^2+b/z^2+c),z);
/* ----- Involving powers of exp with arguments c*z and base a -------------- */
/* Problem 226: OK */
my_int((a^(c*z))^v,z);
(a^(c*z))^v/(c*v*log(a));
/* Problem 227: OK */
my_int((a^z)^v,z);
(a^z)^v/(log(a)*v);
/* ----- with arguments c*z and base %e ------------------------------------- */
/* The following examples 217 - 224 are not really correct, because Maxima
* simplifies (%e^(c*z))^v -> %e^(v*c*z), which is not correct in general.
*/
/* Problem 228: OK */
my_int((%e^(c*z))^v,z);
(%e^(c*z))^v/(c*v);
/* Problem 229: OK */
my_int((%e^z)^v,z);
(%e^z)^v/v;
/* Problem 230: OK */
my_int(sqrt(%e^(c*z)),z);
2*sqrt(%e^(c*z))/c;
/* Problem 231: OK */
my_int((%e^(c*z))^(3/2),z);
2*(%e^(c*z))^(3/2)/(3*c);
/* Problem 232: OK */
my_int((%e^(c*z))^(5/2),z);
2*(%e^(c*z))^(5/2)/(5*c);
/* Problem 233: OK */
my_int(1/sqrt(%e^(c*z)),z);
-2/(c*sqrt(%e^(c*z)));
/* Problem 234: OK */
my_int(1/(%e^(c*z))^(3/2),z);
-2*(%e^(c*z))^(-3/2)/(3*c);
/* Problem 235: OK */
my_int(1/(%e^(c*z))^(5/2),z);
-2*(%e^(c*z))^(-5/2)/(5*c);
/* ----- with arguments c*z+b and base a ------------------------------------ */
/* Problem 236: OK */
my_int((a^(c*z+b))^v,z);
(a^(c*z+b))^v/(log(a)*c*v);