Chapter 4
Listing of integrals solved by CAS which has no known antiderivatives

4.1 Test file Number [5]

4.1.1 Maxima

Integral number [145] $$\int x\cos (k\csc (x))\cot (x)\csc (x)dx$$

[C]   time = 0.222839 (sec), size = 240 ,normalized size = 21.82 $$-\frac{(x{e}^{(\frac{4k\cos \left(2x\right)\cos \left(x\right)}{\cos {\left(2x\right)}^2+\sin {\left(2x\right)}^2-2\cos \left(2x\right)+1}+\frac{4k\sin \left(2x\right)\sin \left(x\right)}{\cos {\left(2x\right)}^2+\sin {\left(2x\right)}^2-2\cos \left(2x\right)+1})}+x{e}^{\left(\frac{4k\cos \left(x\right)}{\cos {\left(2x\right)}^2+\sin {\left(2x\right)}^2-2\cos \left(2x\right)+1}\right)})e^{(-\frac{2k\cos \left(2x\right)\cos \left(x\right)}{\cos {\left(2x\right)}^2+\sin {\left(2x\right)}^2-2\cos \left(2x\right)+1}-\frac{2k\sin \left(2x\right)\sin \left(x\right)}{\cos {\left(2x\right)}^2+\sin {\left(2x\right)}^2-2\cos \left(2x\right)+1}-\frac{2k\cos \left(x\right)}{\cos {\left(2x\right)}^2+\sin {\left(2x\right)}^2-2\cos \left(2x\right)+1})}\sin \left(\frac{2(k\cos \left(x\right)\sin \left(2x\right)-k\cos \left(2x\right)\sin \left(x\right)+k\sin \left(x\right))}{\cos {\left(2x\right)}^2+\sin {\left(2x\right)}^2-2\cos \left(2x\right)+1}\right)}{2k}$$

[In]

integrate(x*cos(x)*cos(k/sin(x))/sin(x)^2,x, algorithm=""maxima"")

output

-1/2*(x*e^(4*k*cos(2*x)*cos(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1) + 4* 
k*sin(2*x)*sin(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)) + x*e^(4*k*cos(x 
)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1)))*e^(-2*k*cos(2*x)*cos(x)/(cos(2* 
x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1) - 2*k*sin(2*x)*sin(x)/(cos(2*x)^2 + sin(2* 
x)^2 - 2*cos(2*x) + 1) - 2*k*cos(x)/(cos(2*x)^2 + sin(2*x)^2 - 2*cos(2*x) + 1) 
)*sin(2*(k*cos(x)*sin(2*x) - k*cos(2*x)*sin(x) + k*sin(x))/(cos(2*x)^2 + sin(2 
*x)^2 - 2*cos(2*x) + 1))/k
 

4.2 Test file Number [57]

4.2.1 Mathematica

Integral number [166] $$\int \frac{(fx{)}^m(a+b\log \left(c{x}^n\right))}{d+ex}dx$$

[B]   time = 0.128514 (sec), size = 72 ,normalized size = 3.13 $$\frac{x(fx{)}^m(-bn{}_3{F}_2(1,1+m,1+m;2+m,2+m;-\frac{ex}{d})+(1+m)\mathrm{Hypergeometric}2\mathrm{F}1(1,1+m,2+m,-\frac{ex}{d})(a+b\log \left(c{x}^n\right)))}{d(1+m{)}^2}$$

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x),x]

output

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{1, 1 + m, 1 + m}, {2 + m, 2 + m}, -((e*x) 
/d)]) + (1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((e*x)/d)]*(a + b*Log[c*x^ 
n])))/(d*(1 + m)^2)
 

Integral number [167] $$\int \frac{(fx{)}^m(a+b\log \left(c{x}^n\right))}{(d+ex{)}^2}dx$$

[B]   time = 0.117558 (sec), size = 72 ,normalized size = 3.13 $$\frac{x(fx{)}^m(-bn{}_3{F}_2(2,1+m,1+m;2+m,2+m;-\frac{ex}{d})+(1+m)\mathrm{Hypergeometric}2\mathrm{F}1(2,1+m,2+m,-\frac{ex}{d})(a+b\log \left(c{x}^n\right)))}{d^2(1+m{)}^2}$$

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x)^2,x]

output

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{2, 1 + m, 1 + m}, {2 + m, 2 + m}, -((e*x) 
/d)]) + (1 + m)*Hypergeometric2F1[2, 1 + m, 2 + m, -((e*x)/d)]*(a + b*Log[c*x^ 
n])))/(d^2*(1 + m)^2)
 

Integral number [168] $$\int x(a+bx{)}^m\log \left(c{x}^n\right)dx$$

[B]   time = 0.16101 (sec), size = 173 ,normalized size = 11.53 $$\frac{(a+bx{)}^m{(1+\frac{bx}{a})}^{-m}(-n(2abx{(1+\frac{bx}{a})}^m+b^2{x}^2{(1+\frac{bx}{a})}^m+a^2(-1+{(1+\frac{bx}{a})}^m))+ab(2+m)nx{}_3{F}_2(1,1,-1-m;2,2;-\frac{bx}{a})+(abmx{(1+\frac{bx}{a})}^m+b^2(1+m)x^2{(1+\frac{bx}{a})}^m-a^2(-1+{(1+\frac{bx}{a})}^m))\log \left(c{x}^n\right))}{b^2(1+m)(2+m)}$$

[In]

Integrate[x*(a + b*x)^m*Log[c*x^n],x]

output

((a + b*x)^m*(-(n*(2*a*b*x*(1 + (b*x)/a)^m + b^2*x^2*(1 + (b*x)/a)^m + a^2*(-1 
 + (1 + (b*x)/a)^m))) + a*b*(2 + m)*n*x*HypergeometricPFQ[{1, 1, -1 - m}, {2, 
2}, -((b*x)/a)] + (a*b*m*x*(1 + (b*x)/a)^m + b^2*(1 + m)*x^2*(1 + (b*x)/a)^m - 
 a^2*(-1 + (1 + (b*x)/a)^m))*Log[c*x^n]))/(b^2*(1 + m)*(2 + m)*(1 + (b*x)/a)^m 
)
 

Integral number [170] $$\int \frac{(a+bx{)}^m\log \left(c{x}^n\right)}{x}dx$$

[B]   time = 0.0419056 (sec), size = 89 ,normalized size = 5.24 $$\frac{{(1+\frac{a}{bx})}^{-m}(a+bx{)}^m(-n{}_3{F}_2(-m,-m,-m;1-m,1-m;-\frac{a}{bx})+m\mathrm{Hypergeometric}2\mathrm{F}1(-m,-m,1-m,-\frac{a}{bx})\log \left(c{x}^n\right))}{m^2}$$

[In]

Integrate[((a + b*x)^m*Log[c*x^n])/x,x]

output

((a + b*x)^m*(-(n*HypergeometricPFQ[{-m, -m, -m}, {1 - m, 1 - m}, -(a/(b*x))]) 
 + m*Hypergeometric2F1[-m, -m, 1 - m, -(a/(b*x))]*Log[c*x^n]))/(m^2*(1 + a/(b* 
x))^m)
 

Integral number [322] $$\int \frac{(fx{)}^m(a+b\log \left(c{x}^n\right))}{d+e{x}^2}dx$$

[B]   time = 0.703872 (sec), size = 108 ,normalized size = 4.32 $$\frac{x(fx{)}^m(-bn{}_3{F}_2(1,\frac{1}{2}+\frac{m}{2},\frac{1}{2}+\frac{m}{2};\frac{3}{2}+\frac{m}{2},\frac{3}{2}+\frac{m}{2};-\frac{e{x}^2}{d})+(1+m)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{1+m}{2},\frac{3+m}{2},-\frac{e{x}^2}{d})(a+b\log \left(c{x}^n\right)))}{d(1+m{)}^2}$$

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^2),x]

output

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{1, 1/2 + m/2, 1/2 + m/2}, {3/2 + m/2, 3/2 
 + m/2}, -((e*x^2)/d)]) + (1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, - 
((e*x^2)/d)]*(a + b*Log[c*x^n])))/(d*(1 + m)^2)
 

Integral number [323] $$\int \frac{(fx{)}^m(a+b\log \left(c{x}^n\right))}{{(d+e{x}^2)}^2}dx$$

[B]   time = 0.136591 (sec), size = 108 ,normalized size = 4.32 $$\frac{x(fx{)}^m(-bn{}_3{F}_2(2,\frac{1}{2}+\frac{m}{2},\frac{1}{2}+\frac{m}{2};\frac{3}{2}+\frac{m}{2},\frac{3}{2}+\frac{m}{2};-\frac{e{x}^2}{d})+(1+m)\mathrm{Hypergeometric}2\mathrm{F}1(2,\frac{1+m}{2},\frac{3+m}{2},-\frac{e{x}^2}{d})(a+b\log \left(c{x}^n\right)))}{d^2(1+m{)}^2}$$

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^2)^2,x]

output

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{2, 1/2 + m/2, 1/2 + m/2}, {3/2 + m/2, 3/2 
 + m/2}, -((e*x^2)/d)]) + (1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, - 
((e*x^2)/d)]*(a + b*Log[c*x^n])))/(d^2*(1 + m)^2)
 

Integral number [406] $$\int \frac{x^3(a+b\log \left(c{x}^n\right))}{d+e{x}^r}dx$$

[B]   time = 0.0916891 (sec), size = 87 ,normalized size = 3.78 $$\frac{x^4(-bn{}_3{F}_2(1,\frac{4}{r},\frac{4}{r};1+\frac{4}{r},1+\frac{4}{r};-\frac{e{x}^r}{d})+4\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{4}{r},\frac{4+r}{r},-\frac{e{x}^r}{d})(a+b\log \left(c{x}^n\right)))}{16d}$$

[In]

Integrate[(x^3*(a + b*Log[c*x^n]))/(d + e*x^r),x]

output

(x^4*(-(b*n*HypergeometricPFQ[{1, 4/r, 4/r}, {1 + 4/r, 1 + 4/r}, -((e*x^r)/d)] 
) + 4*Hypergeometric2F1[1, 4/r, (4 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n])))/ 
(16*d)
 

Integral number [407] $$\int \frac{x(a+b\log \left(c{x}^n\right))}{d+e{x}^r}dx$$

[B]   time = 0.0764756 (sec), size = 87 ,normalized size = 4.14 $$\frac{x^2(-bn{}_3{F}_2(1,\frac{2}{r},\frac{2}{r};1+\frac{2}{r},1+\frac{2}{r};-\frac{e{x}^r}{d})+2\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{2}{r},\frac{2+r}{r},-\frac{e{x}^r}{d})(a+b\log \left(c{x}^n\right)))}{4d}$$

[In]

Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x^r),x]

output

(x^2*(-(b*n*HypergeometricPFQ[{1, 2/r, 2/r}, {1 + 2/r, 1 + 2/r}, -((e*x^r)/d)] 
) + 2*Hypergeometric2F1[1, 2/r, (2 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n])))/ 
(4*d)
 

Integral number [409] $$\int \frac{a+b\log \left(c{x}^n\right)}{x^3(d+e{x}^r)}dx$$

[B]   time = 0.0886791 (sec), size = 86 ,normalized size = 3.74 $$-\frac{bn{}_3{F}_2(1,-\frac{2}{r},-\frac{2}{r};1-\frac{2}{r},1-\frac{2}{r};-\frac{e{x}^r}{d})+2\mathrm{Hypergeometric}2\mathrm{F}1(1,-\frac{2}{r},\frac{-2+r}{r},-\frac{e{x}^r}{d})(a+b\log \left(c{x}^n\right))}{4d{x}^2}$$

[In]

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)),x]

output

-1/4*(b*n*HypergeometricPFQ[{1, -2/r, -2/r}, {1 - 2/r, 1 - 2/r}, -((e*x^r)/d)] 
 + 2*Hypergeometric2F1[1, -2/r, (-2 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n]))/ 
(d*x^2)
 

Integral number [410] $$\int \frac{x^2(a+b\log \left(c{x}^n\right))}{d+e{x}^r}dx$$

[B]   time = 0.0840796 (sec), size = 87 ,normalized size = 3.78 $$\frac{x^3(-bn{}_3{F}_2(1,\frac{3}{r},\frac{3}{r};1+\frac{3}{r},1+\frac{3}{r};-\frac{e{x}^r}{d})+3\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{3}{r},\frac{3+r}{r},-\frac{e{x}^r}{d})(a+b\log \left(c{x}^n\right)))}{9d}$$

[In]

Integrate[(x^2*(a + b*Log[c*x^n]))/(d + e*x^r),x]

output

(x^3*(-(b*n*HypergeometricPFQ[{1, 3/r, 3/r}, {1 + 3/r, 1 + 3/r}, -((e*x^r)/d)] 
) + 3*Hypergeometric2F1[1, 3/r, (3 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n])))/ 
(9*d)
 

Integral number [411] $$\int \frac{a+b\log \left(c{x}^n\right)}{d+e{x}^r}dx$$

[B]   time = 0.0650704 (sec), size = 69 ,normalized size = 3.45 $$\frac{x(-bn{}_3{F}_2(1,\frac{1}{r},\frac{1}{r};1+\frac{1}{r},1+\frac{1}{r};-\frac{e{x}^r}{d})+\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{1}{r},1+\frac{1}{r},-\frac{e{x}^r}{d})(a+b\log \left(c{x}^n\right)))}{d}$$

[In]

Integrate[(a + b*Log[c*x^n])/(d + e*x^r),x]

output

(x*(-(b*n*HypergeometricPFQ[{1, r^(-1), r^(-1)}, {1 + r^(-1), 1 + r^(-1)}, -(( 
e*x^r)/d)]) + Hypergeometric2F1[1, r^(-1), 1 + r^(-1), -((e*x^r)/d)]*(a + b*Lo 
g[c*x^n])))/d
 

Integral number [412] $$\int \frac{a+b\log \left(c{x}^n\right)}{x^2(d+e{x}^r)}dx$$

[B]   time = 0.0852945 (sec), size = 83 ,normalized size = 3.61 $$-\frac{bn{}_3{F}_2(1,-\frac{1}{r},-\frac{1}{r};1-\frac{1}{r},1-\frac{1}{r};-\frac{e{x}^r}{d})+\mathrm{Hypergeometric}2\mathrm{F}1(1,-\frac{1}{r},\frac{-1+r}{r},-\frac{e{x}^r}{d})(a+b\log \left(c{x}^n\right))}{dx}$$

[In]

Integrate[(a + b*Log[c*x^n])/(x^2*(d + e*x^r)),x]

output

-((b*n*HypergeometricPFQ[{1, -r^(-1), -r^(-1)}, {1 - r^(-1), 1 - r^(-1)}, -((e 
*x^r)/d)] + Hypergeometric2F1[1, -r^(-1), (-1 + r)/r, -((e*x^r)/d)]*(a + b*Log 
[c*x^n]))/(d*x))
 

Integral number [413] $$\int \frac{x^3(a+b\log \left(c{x}^n\right))}{{(d+e{x}^r)}^2}dx$$

[B]   time = 0.164696 (sec), size = 140 ,normalized size = 6.09 $$\frac{x^4(-bn(-4+r)(d+e{x}^r){}_3{F}_2(1,\frac{4}{r},\frac{4}{r};1+\frac{4}{r},1+\frac{4}{r};-\frac{e{x}^r}{d})+16d(a+b\log \left(c{x}^n\right))+4(d+e{x}^r)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{4}{r},\frac{4+r}{r},-\frac{e{x}^r}{d})(-bn+a(-4+r)+b(-4+r)\log \left(c{x}^n\right)))}{16d^{2}r(d+e{x}^r)}$$

[In]

Integrate[(x^3*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

output

(x^4*(-(b*n*(-4 + r)*(d + e*x^r)*HypergeometricPFQ[{1, 4/r, 4/r}, {1 + 4/r, 1 
+ 4/r}, -((e*x^r)/d)]) + 16*d*(a + b*Log[c*x^n]) + 4*(d + e*x^r)*Hypergeometri 
c2F1[1, 4/r, (4 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(-4 + r) + b*(-4 + r)*Log[c* 
x^n])))/(16*d^2*r*(d + e*x^r))
 

Integral number [414] $$\int \frac{x(a+b\log \left(c{x}^n\right))}{{(d+e{x}^r)}^2}dx$$

[B]   time = 0.161746 (sec), size = 140 ,normalized size = 6.67 $$\frac{x^2(-bn(-2+r)(d+e{x}^r){}_3{F}_2(1,\frac{2}{r},\frac{2}{r};1+\frac{2}{r},1+\frac{2}{r};-\frac{e{x}^r}{d})+4d(a+b\log \left(c{x}^n\right))+2(d+e{x}^r)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{2}{r},\frac{2+r}{r},-\frac{e{x}^r}{d})(-bn+a(-2+r)+b(-2+r)\log \left(c{x}^n\right)))}{4d^{2}r(d+e{x}^r)}$$

[In]

Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

output

(x^2*(-(b*n*(-2 + r)*(d + e*x^r)*HypergeometricPFQ[{1, 2/r, 2/r}, {1 + 2/r, 1 
+ 2/r}, -((e*x^r)/d)]) + 4*d*(a + b*Log[c*x^n]) + 2*(d + e*x^r)*Hypergeometric 
2F1[1, 2/r, (2 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(-2 + r) + b*(-2 + r)*Log[c*x 
^n])))/(4*d^2*r*(d + e*x^r))
 

Integral number [416] $$\int \frac{a+b\log \left(c{x}^n\right)}{x^3{(d+e{x}^r)}^2}dx$$

[B]   time = 0.163581 (sec), size = 139 ,normalized size = 6.04 $$-\frac{bn(2+r)(d+e{x}^r){}_3{F}_2(1,-\frac{2}{r},-\frac{2}{r};1-\frac{2}{r},1-\frac{2}{r};-\frac{e{x}^r}{d})-4d(a+b\log \left(c{x}^n\right))+2(d+e{x}^r)\mathrm{Hypergeometric}2\mathrm{F}1(1,-\frac{2}{r},\frac{-2+r}{r},-\frac{e{x}^r}{d})(-bn+a(2+r)+b(2+r)\log \left(c{x}^n\right))}{4d^{2}r{x}^2(d+e{x}^r)}$$

[In]

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)^2),x]

output

-1/4*(b*n*(2 + r)*(d + e*x^r)*HypergeometricPFQ[{1, -2/r, -2/r}, {1 - 2/r, 1 - 
 2/r}, -((e*x^r)/d)] - 4*d*(a + b*Log[c*x^n]) + 2*(d + e*x^r)*Hypergeometric2F 
1[1, -2/r, (-2 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(2 + r) + b*(2 + r)*Log[c*x^n 
]))/(d^2*r*x^2*(d + e*x^r))
 

Integral number [417] $$\int \frac{x^2(a+b\log \left(c{x}^n\right))}{{(d+e{x}^r)}^2}dx$$

[B]   time = 0.156319 (sec), size = 140 ,normalized size = 6.09 $$\frac{x^3(-bn(-3+r)(d+e{x}^r){}_3{F}_2(1,\frac{3}{r},\frac{3}{r};1+\frac{3}{r},1+\frac{3}{r};-\frac{e{x}^r}{d})+9d(a+b\log \left(c{x}^n\right))+3(d+e{x}^r)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{3}{r},\frac{3+r}{r},-\frac{e{x}^r}{d})(-bn+a(-3+r)+b(-3+r)\log \left(c{x}^n\right)))}{9d^{2}r(d+e{x}^r)}$$

[In]

Integrate[(x^2*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

output

(x^3*(-(b*n*(-3 + r)*(d + e*x^r)*HypergeometricPFQ[{1, 3/r, 3/r}, {1 + 3/r, 1 
+ 3/r}, -((e*x^r)/d)]) + 9*d*(a + b*Log[c*x^n]) + 3*(d + e*x^r)*Hypergeometric 
2F1[1, 3/r, (3 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(-3 + r) + b*(-3 + r)*Log[c*x 
^n])))/(9*d^2*r*(d + e*x^r))
 

Integral number [418] $$\int \frac{a+b\log \left(c{x}^n\right)}{{(d+e{x}^r)}^2}dx$$

[B]   time = 1.70687 (sec), size = 161 ,normalized size = 8.05 $$\frac{x(adr\mathrm{Hypergeometric}2\mathrm{F}1(2,\frac{1}{r},1+\frac{1}{r},-\frac{e{x}^r}{d})+aer{x}^r\mathrm{Hypergeometric}2\mathrm{F}1(2,\frac{1}{r},1+\frac{1}{r},-\frac{e{x}^r}{d})-bn(-1+r)(d+e{x}^r){}_3{F}_2(1,\frac{1}{r},\frac{1}{r};1+\frac{1}{r},1+\frac{1}{r};-\frac{e{x}^r}{d})+bd\log \left(c{x}^n\right)-b(d+e{x}^r)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{1}{r},1+\frac{1}{r},-\frac{e{x}^r}{d})(n-(-1+r)\log \left(c{x}^n\right)))}{d^{2}r(d+e{x}^r)}$$

[In]

Integrate[(a + b*Log[c*x^n])/(d + e*x^r)^2,x]

output

(x*(a*d*r*Hypergeometric2F1[2, r^(-1), 1 + r^(-1), -((e*x^r)/d)] + a*e*r*x^r*H 
ypergeometric2F1[2, r^(-1), 1 + r^(-1), -((e*x^r)/d)] - b*n*(-1 + r)*(d + e*x^ 
r)*HypergeometricPFQ[{1, r^(-1), r^(-1)}, {1 + r^(-1), 1 + r^(-1)}, -((e*x^r)/ 
d)] + b*d*Log[c*x^n] - b*(d + e*x^r)*Hypergeometric2F1[1, r^(-1), 1 + r^(-1), 
-((e*x^r)/d)]*(n - (-1 + r)*Log[c*x^n])))/(d^2*r*(d + e*x^r))
 

Integral number [419] $$\int \frac{a+b\log \left(c{x}^n\right)}{x^2{(d+e{x}^r)}^2}dx$$

[B]   time = 0.138563 (sec), size = 135 ,normalized size = 5.87 $$\frac{-bn(1+r)(d+e{x}^r){}_3{F}_2(1,-\frac{1}{r},-\frac{1}{r};1-\frac{1}{r},1-\frac{1}{r};-\frac{e{x}^r}{d})+d(a+b\log \left(c{x}^n\right))-(d+e{x}^r)\mathrm{Hypergeometric}2\mathrm{F}1(1,-\frac{1}{r},\frac{-1+r}{r},-\frac{e{x}^r}{d})(a-bn+ar+b(1+r)\log \left(c{x}^n\right))}{d^{2}rx(d+e{x}^r)}$$

[In]

Integrate[(a + b*Log[c*x^n])/(x^2*(d + e*x^r)^2),x]

output

(-(b*n*(1 + r)*(d + e*x^r)*HypergeometricPFQ[{1, -r^(-1), -r^(-1)}, {1 - r^(-1 
), 1 - r^(-1)}, -((e*x^r)/d)]) + d*(a + b*Log[c*x^n]) - (d + e*x^r)*Hypergeome 
tric2F1[1, -r^(-1), (-1 + r)/r, -((e*x^r)/d)]*(a - b*n + a*r + b*(1 + r)*Log[c 
*x^n]))/(d^2*r*x*(d + e*x^r))
 

Integral number [444] $$\int \frac{(fx{)}^m(a+b\log \left(c{x}^n\right))}{d+e{x}^r}dx$$

[B]   time = 0.157935 (sec), size = 111 ,normalized size = 4.44 $$\frac{x(fx{)}^m(-bn{}_3{F}_2(1,\frac{1}{r}+\frac{m}{r},\frac{1}{r}+\frac{m}{r};1+\frac{1}{r}+\frac{m}{r},1+\frac{1}{r}+\frac{m}{r};-\frac{e{x}^r}{d})+(1+m)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{1+m}{r},\frac{1+m+r}{r},-\frac{e{x}^r}{d})(a+b\log \left(c{x}^n\right)))}{d(1+m{)}^2}$$

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^r),x]

output

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{1, r^(-1) + m/r, r^(-1) + m/r}, {1 + r^(- 
1) + m/r, 1 + r^(-1) + m/r}, -((e*x^r)/d)]) + (1 + m)*Hypergeometric2F1[1, (1 
+ m)/r, (1 + m + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n])))/(d*(1 + m)^2)
 

Integral number [445] $$\int \frac{(fx{)}^m(a+b\log \left(c{x}^n\right))}{{(d+e{x}^r)}^2}dx$$

[B]   time = 0.296047 (sec), size = 177 ,normalized size = 7.08 $$\frac{x(fx{)}^m(bn(1+m-r)(d+e{x}^r){}_3{F}_2(1,\frac{1}{r}+\frac{m}{r},\frac{1}{r}+\frac{m}{r};1+\frac{1}{r}+\frac{m}{r},1+\frac{1}{r}+\frac{m}{r};-\frac{e{x}^r}{d})-(1+m)(-d(1+m)(a+b\log \left(c{x}^n\right))+(d+e{x}^r)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{1+m}{r},\frac{1+m+r}{r},-\frac{e{x}^r}{d})(bn+a(1+m-r)+b(1+m-r)\log \left(c{x}^n\right))))}{d^2(1+m{)}^{2}r(d+e{x}^r)}$$

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

output

(x*(f*x)^m*(b*n*(1 + m - r)*(d + e*x^r)*HypergeometricPFQ[{1, r^(-1) + m/r, r^ 
(-1) + m/r}, {1 + r^(-1) + m/r, 1 + r^(-1) + m/r}, -((e*x^r)/d)] - (1 + m)*(-( 
d*(1 + m)*(a + b*Log[c*x^n])) + (d + e*x^r)*Hypergeometric2F1[1, (1 + m)/r, (1 
 + m + r)/r, -((e*x^r)/d)]*(b*n + a*(1 + m - r) + b*(1 + m - r)*Log[c*x^n])))) 
/(d^2*(1 + m)^2*r*(d + e*x^r))
 

4.3 Test file Number [58]

4.3.1 Mathematica

Integral number [138] $$\int (gx{)}^q(a+b\log \left(c{x}^n\right))\log \left(d{(e+f{x}^m)}^k\right)dx$$

[B]   time = 0.33242 (sec), size = 304 ,normalized size = 10.86 $$\frac{x(gx{)}^q(-akm+2bkmn-akmq-bkmn{}_3{F}_2(1,\frac{1}{m}+\frac{q}{m},\frac{1}{m}+\frac{q}{m};1+\frac{1}{m}+\frac{q}{m},1+\frac{1}{m}+\frac{q}{m};-\frac{f{x}^m}{e})-bkm\log \left(c{x}^n\right)-bkmq\log \left(c{x}^n\right)+km\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{1+q}{m},\frac{1+m+q}{m},-\frac{f{x}^m}{e})(a-bn+aq+b(1+q)\log \left(c{x}^n\right))+a\log \left(d{(e+f{x}^m)}^k\right)-bn\log \left(d{(e+f{x}^m)}^k\right)+2aq\log \left(d{(e+f{x}^m)}^k\right)-bnq\log \left(d{(e+f{x}^m)}^k\right)+a{q}^2\log \left(d{(e+f{x}^m)}^k\right)+b\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)+2bq\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)+b{q}^2\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right))}{(1+q{)}^3}$$

[In]

Integrate[(g*x)^q*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

output

(x*(g*x)^q*(-(a*k*m) + 2*b*k*m*n - a*k*m*q - b*k*m*n*HypergeometricPFQ[{1, m^( 
-1) + q/m, m^(-1) + q/m}, {1 + m^(-1) + q/m, 1 + m^(-1) + q/m}, -((f*x^m)/e)] 
- b*k*m*Log[c*x^n] - b*k*m*q*Log[c*x^n] + k*m*Hypergeometric2F1[1, (1 + q)/m, 
(1 + m + q)/m, -((f*x^m)/e)]*(a - b*n + a*q + b*(1 + q)*Log[c*x^n]) + a*Log[d* 
(e + f*x^m)^k] - b*n*Log[d*(e + f*x^m)^k] + 2*a*q*Log[d*(e + f*x^m)^k] - b*n*q 
*Log[d*(e + f*x^m)^k] + a*q^2*Log[d*(e + f*x^m)^k] + b*Log[c*x^n]*Log[d*(e + f 
*x^m)^k] + 2*b*q*Log[c*x^n]*Log[d*(e + f*x^m)^k] + b*q^2*Log[c*x^n]*Log[d*(e + 
 f*x^m)^k]))/(1 + q)^3
 

Integral number [144] $$\int x^2(a+b\log \left(c{x}^n\right))\log \left(d{(e+f{x}^m)}^k\right)dx$$

[B]   time = 0.152851 (sec), size = 292 ,normalized size = 11.23 $$-\frac{x^3(-6bekmn-2bek{m}^{2}n+9afkm{x}^m\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{3+m}{m},2+\frac{3}{m},-\frac{f{x}^m}{e})+bekm(3+m)n{}_3{F}_2(1,\frac{3}{m},\frac{3}{m};1+\frac{3}{m},1+\frac{3}{m};-\frac{f{x}^m}{e})+bekm(3+m)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{3}{m},\frac{3+m}{m},-\frac{f{x}^m}{e})(n-3\log \left(c{x}^n\right))+9bekm\log \left(c{x}^n\right)+3bek{m}^2\log \left(c{x}^n\right)-27ae\log \left(d{(e+f{x}^m)}^k\right)-9aem\log \left(d{(e+f{x}^m)}^k\right)+9ben\log \left(d{(e+f{x}^m)}^k\right)+3bemn\log \left(d{(e+f{x}^m)}^k\right)-27be\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)-9bem\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right))}{27e(3+m)}$$

[In]

Integrate[x^2*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

output

-1/27*(x^3*(-6*b*e*k*m*n - 2*b*e*k*m^2*n + 9*a*f*k*m*x^m*Hypergeometric2F1[1, 
(3 + m)/m, 2 + 3/m, -((f*x^m)/e)] + b*e*k*m*(3 + m)*n*HypergeometricPFQ[{1, 3/ 
m, 3/m}, {1 + 3/m, 1 + 3/m}, -((f*x^m)/e)] + b*e*k*m*(3 + m)*Hypergeometric2F1 
[1, 3/m, (3 + m)/m, -((f*x^m)/e)]*(n - 3*Log[c*x^n]) + 9*b*e*k*m*Log[c*x^n] + 
3*b*e*k*m^2*Log[c*x^n] - 27*a*e*Log[d*(e + f*x^m)^k] - 9*a*e*m*Log[d*(e + f*x^ 
m)^k] + 9*b*e*n*Log[d*(e + f*x^m)^k] + 3*b*e*m*n*Log[d*(e + f*x^m)^k] - 27*b*e 
*Log[c*x^n]*Log[d*(e + f*x^m)^k] - 9*b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k]))/( 
e*(3 + m))
 

Integral number [145] $$\int x(a+b\log \left(c{x}^n\right))\log \left(d{(e+f{x}^m)}^k\right)dx$$

[B]   time = 0.136601 (sec), size = 292 ,normalized size = 12.17 $$-\frac{x^2(-4bekmn-2bek{m}^{2}n+4afkm{x}^m\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{2+m}{m},2+\frac{2}{m},-\frac{f{x}^m}{e})+bekm(2+m)n{}_3{F}_2(1,\frac{2}{m},\frac{2}{m};1+\frac{2}{m},1+\frac{2}{m};-\frac{f{x}^m}{e})+bekm(2+m)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{2}{m},\frac{2+m}{m},-\frac{f{x}^m}{e})(n-2\log \left(c{x}^n\right))+4bekm\log \left(c{x}^n\right)+2bek{m}^2\log \left(c{x}^n\right)-8ae\log \left(d{(e+f{x}^m)}^k\right)-4aem\log \left(d{(e+f{x}^m)}^k\right)+4ben\log \left(d{(e+f{x}^m)}^k\right)+2bemn\log \left(d{(e+f{x}^m)}^k\right)-8be\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)-4bem\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right))}{8e(2+m)}$$

[In]

Integrate[x*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

output

-1/8*(x^2*(-4*b*e*k*m*n - 2*b*e*k*m^2*n + 4*a*f*k*m*x^m*Hypergeometric2F1[1, ( 
2 + m)/m, 2 + 2/m, -((f*x^m)/e)] + b*e*k*m*(2 + m)*n*HypergeometricPFQ[{1, 2/m 
, 2/m}, {1 + 2/m, 1 + 2/m}, -((f*x^m)/e)] + b*e*k*m*(2 + m)*Hypergeometric2F1[ 
1, 2/m, (2 + m)/m, -((f*x^m)/e)]*(n - 2*Log[c*x^n]) + 4*b*e*k*m*Log[c*x^n] + 2 
*b*e*k*m^2*Log[c*x^n] - 8*a*e*Log[d*(e + f*x^m)^k] - 4*a*e*m*Log[d*(e + f*x^m) 
^k] + 4*b*e*n*Log[d*(e + f*x^m)^k] + 2*b*e*m*n*Log[d*(e + f*x^m)^k] - 8*b*e*Lo 
g[c*x^n]*Log[d*(e + f*x^m)^k] - 4*b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k]))/(e*( 
2 + m))
 

Integral number [146] $$\int (a+b\log \left(c{x}^n\right))\log \left(d{(e+f{x}^m)}^k\right)dx$$

[B]   time = 0.152115 (sec), size = 165 ,normalized size = 7.17 $$bkmnx-kmx(a+b(-n\log (x)+\log \left(c{x}^n\right)))+x(bkmn-bkmn{}_3{F}_2(1,\frac{1}{m},\frac{1}{m};1+\frac{1}{m},1+\frac{1}{m};-\frac{f{x}^m}{e})-bkmn\log (x)+km\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{1}{m},1+\frac{1}{m},-\frac{f{x}^m}{e})(a-bn+b\log \left(c{x}^n\right))+a\log \left(d{(e+f{x}^m)}^k\right)-bn\log \left(d{(e+f{x}^m)}^k\right)+b\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right))$$

[In]

Integrate[(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

output

b*k*m*n*x - k*m*x*(a + b*(-(n*Log[x]) + Log[c*x^n])) + x*(b*k*m*n - b*k*m*n*Hy 
pergeometricPFQ[{1, m^(-1), m^(-1)}, {1 + m^(-1), 1 + m^(-1)}, -((f*x^m)/e)] - 
 b*k*m*n*Log[x] + k*m*Hypergeometric2F1[1, m^(-1), 1 + m^(-1), -((f*x^m)/e)]*( 
a - b*n + b*Log[c*x^n]) + a*Log[d*(e + f*x^m)^k] - b*n*Log[d*(e + f*x^m)^k] + 
b*Log[c*x^n]*Log[d*(e + f*x^m)^k])
 

Integral number [148] $$\int \frac{(a+b\log \left(c{x}^n\right))\log \left(d{(e+f{x}^m)}^k\right)}{x^2}dx$$

[B]   time = 0.125142 (sec), size = 282 ,normalized size = 10.85 $$\frac{2bekmn-2bek{m}^{2}n+afkm{x}^m\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{-1+m}{m},2-\frac{1}{m},-\frac{f{x}^m}{e})+bek(-1+m)mn{}_3{F}_2(1,-\frac{1}{m},-\frac{1}{m};1-\frac{1}{m},1-\frac{1}{m};-\frac{f{x}^m}{e})+bekm\log \left(c{x}^n\right)-bek{m}^2\log \left(c{x}^n\right)+bek(-1+m)m\mathrm{Hypergeometric}2\mathrm{F}1(1,-\frac{1}{m},\frac{-1+m}{m},-\frac{f{x}^m}{e})(n+\log \left(c{x}^n\right))+ae\log \left(d{(e+f{x}^m)}^k\right)-aem\log \left(d{(e+f{x}^m)}^k\right)+ben\log \left(d{(e+f{x}^m)}^k\right)-bemn\log \left(d{(e+f{x}^m)}^k\right)+be\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)-bem\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)}{e(-1+m)x}$$

[In]

Integrate[((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^2,x]

output

(2*b*e*k*m*n - 2*b*e*k*m^2*n + a*f*k*m*x^m*Hypergeometric2F1[1, (-1 + m)/m, 2 
- m^(-1), -((f*x^m)/e)] + b*e*k*(-1 + m)*m*n*HypergeometricPFQ[{1, -m^(-1), -m 
^(-1)}, {1 - m^(-1), 1 - m^(-1)}, -((f*x^m)/e)] + b*e*k*m*Log[c*x^n] - b*e*k*m 
^2*Log[c*x^n] + b*e*k*(-1 + m)*m*Hypergeometric2F1[1, -m^(-1), (-1 + m)/m, -(( 
f*x^m)/e)]*(n + Log[c*x^n]) + a*e*Log[d*(e + f*x^m)^k] - a*e*m*Log[d*(e + f*x^ 
m)^k] + b*e*n*Log[d*(e + f*x^m)^k] - b*e*m*n*Log[d*(e + f*x^m)^k] + b*e*Log[c* 
x^n]*Log[d*(e + f*x^m)^k] - b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k])/(e*(-1 + m) 
*x)
 

Integral number [149] $$\int \frac{(a+b\log \left(c{x}^n\right))\log \left(d{(e+f{x}^m)}^k\right)}{x^3}dx$$

[B]   time = 0.128627 (sec), size = 292 ,normalized size = 11.23 $$\frac{4bekmn-2bek{m}^{2}n+4afkm{x}^m\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{-2+m}{m},2-\frac{2}{m},-\frac{f{x}^m}{e})+bek(-2+m)mn{}_3{F}_2(1,-\frac{2}{m},-\frac{2}{m};1-\frac{2}{m},1-\frac{2}{m};-\frac{f{x}^m}{e})+4bekm\log \left(c{x}^n\right)-2bek{m}^2\log \left(c{x}^n\right)+bek(-2+m)m\mathrm{Hypergeometric}2\mathrm{F}1(1,-\frac{2}{m},\frac{-2+m}{m},-\frac{f{x}^m}{e})(n+2\log \left(c{x}^n\right))+8ae\log \left(d{(e+f{x}^m)}^k\right)-4aem\log \left(d{(e+f{x}^m)}^k\right)+4ben\log \left(d{(e+f{x}^m)}^k\right)-2bemn\log \left(d{(e+f{x}^m)}^k\right)+8be\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)-4bem\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)}{8e(-2+m)x^2}$$

[In]

Integrate[((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^3,x]

output

(4*b*e*k*m*n - 2*b*e*k*m^2*n + 4*a*f*k*m*x^m*Hypergeometric2F1[1, (-2 + m)/m, 
2 - 2/m, -((f*x^m)/e)] + b*e*k*(-2 + m)*m*n*HypergeometricPFQ[{1, -2/m, -2/m}, 
 {1 - 2/m, 1 - 2/m}, -((f*x^m)/e)] + 4*b*e*k*m*Log[c*x^n] - 2*b*e*k*m^2*Log[c* 
x^n] + b*e*k*(-2 + m)*m*Hypergeometric2F1[1, -2/m, (-2 + m)/m, -((f*x^m)/e)]*( 
n + 2*Log[c*x^n]) + 8*a*e*Log[d*(e + f*x^m)^k] - 4*a*e*m*Log[d*(e + f*x^m)^k] 
+ 4*b*e*n*Log[d*(e + f*x^m)^k] - 2*b*e*m*n*Log[d*(e + f*x^m)^k] + 8*b*e*Log[c* 
x^n]*Log[d*(e + f*x^m)^k] - 4*b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k])/(8*e*(-2 
+ m)*x^2)
 

Integral number [220] $$\int -(dx{)}^m(a+b\log \left(c{x}^n\right))\log (1-e{x}^q)dx$$

[B]   time = 0.296332 (sec), size = 266 ,normalized size = 10.23 $$-\frac{x(dx{)}^m(-aq-amq+2bnq-bnq{}_3{F}_2(1,\frac{1}{q}+\frac{m}{q},\frac{1}{q}+\frac{m}{q};1+\frac{1}{q}+\frac{m}{q},1+\frac{1}{q}+\frac{m}{q};e{x}^q)-bq\log \left(c{x}^n\right)-bmq\log \left(c{x}^n\right)+q\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{1+m}{q},\frac{1+m+q}{q},e{x}^q)(a+am-bn+b(1+m)\log \left(c{x}^n\right))+a\log (1-e{x}^q)+2am\log (1-e{x}^q)+a{m}^2\log (1-e{x}^q)-bn\log (1-e{x}^q)-bmn\log (1-e{x}^q)+b\log \left(c{x}^n\right)\log (1-e{x}^q)+2bm\log \left(c{x}^n\right)\log (1-e{x}^q)+b{m}^2\log \left(c{x}^n\right)\log (1-e{x}^q))}{(1+m{)}^3}$$

[In]

Integrate[-((d*x)^m*(a + b*Log[c*x^n])*Log[1 - e*x^q]),x]

output

-((x*(d*x)^m*(-(a*q) - a*m*q + 2*b*n*q - b*n*q*HypergeometricPFQ[{1, q^(-1) + 
m/q, q^(-1) + m/q}, {1 + q^(-1) + m/q, 1 + q^(-1) + m/q}, e*x^q] - b*q*Log[c*x 
^n] - b*m*q*Log[c*x^n] + q*Hypergeometric2F1[1, (1 + m)/q, (1 + m + q)/q, e*x^ 
q]*(a + a*m - b*n + b*(1 + m)*Log[c*x^n]) + a*Log[1 - e*x^q] + 2*a*m*Log[1 - e 
*x^q] + a*m^2*Log[1 - e*x^q] - b*n*Log[1 - e*x^q] - b*m*n*Log[1 - e*x^q] + b*L 
og[c*x^n]*Log[1 - e*x^q] + 2*b*m*Log[c*x^n]*Log[1 - e*x^q] + b*m^2*Log[c*x^n]* 
Log[1 - e*x^q]))/(1 + m)^3)
 

4.3.2 Maple

Integral number [221] $$\int (dx{)}^m(a+b\log \left(c{x}^n\right))\mathrm{PolyLog}(2,e{x}^q)dx$$

[B]   time = 0.089 (sec), size = 867 ,normalized size = 37.7 $$-\frac{{\left(dx\right)}^m{x}^{-m}{(-e)}^{-\frac{m}{q}-\frac{1}{q}}a(-\frac{q^2{x}^{1+m}{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln (1-e{x}^q)}{{(1+m)}^2}-\frac{q{x}^{1+m}{(-e)}^{\frac{m}{q}+\frac{1}{q}}{\mathrm{Li}}_2\left(e{x}^q\right)}{1+m}-\frac{q^2{x}^{1+m+q}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{{(1+m)}^2})}{q}-\frac{{\left(dx\right)}^m{x}^{-m}{(-e)}^{-\frac{m}{q}-\frac{1}{q}}b\ln \left(c\right)(-\frac{q^2{x}^{1+m}{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln (1-e{x}^q)}{{(1+m)}^2}-\frac{q{x}^{1+m}{(-e)}^{\frac{m}{q}+\frac{1}{q}}{\mathrm{Li}}_2\left(e{x}^q\right)}{1+m}-\frac{q^2{x}^{1+m+q}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{{(1+m)}^2})}{q}+(\frac{{(-e)}^{-\frac{m}{q}-\frac{1}{q}}\ln (-e){\left(dx\right)}^m{x}^{-m}bn(-\frac{q^2{x}^m{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln (1-e{x}^q)}{{(1+m)}^2}-\frac{q{x}^m{(-e)}^{\frac{m}{q}+\frac{1}{q}}{\mathrm{Li}}_2\left(e{x}^q\right)}{1+m}-\frac{q^2{x}^{q+m}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{{(1+m)}^2})}{q^2}-\frac{{(-e)}^{-\frac{m}{q}-\frac{1}{q}}{\left(dx\right)}^m{x}^{-m}bn(-\frac{q^2{x}^m{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln \left(x\right)\ln (1-e{x}^q)}{{(1+m)}^2}-\frac{q{x}^m{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln (-e)\ln (1-e{x}^q)}{{(1+m)}^2}+\frac{2q^2{x}^m{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln (1-e{x}^q)}{{(1+m)}^3}-\frac{q{x}^m{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln \left(x\right){\mathrm{Li}}_2\left(e{x}^q\right)}{1+m}-\frac{x^m{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln (-e){\mathrm{Li}}_2\left(e{x}^q\right)}{1+m}+\frac{q{x}^m{(-e)}^{\frac{m}{q}+\frac{1}{q}}{\mathrm{Li}}_2\left(e{x}^q\right)}{{(1+m)}^2}-\frac{q^2{x}^{q+m}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln \left(x\right)\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{{(1+m)}^2}-\frac{q{x}^{q+m}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln (-e)\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{{(1+m)}^2}+\frac{2q^2{x}^{q+m}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{{(1+m)}^3}+\frac{q{x}^{q+m}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}\mathrm{\Phi }(e{x}^q,2,\frac{1+m+q}{q})}{{(1+m)}^2})}{q})x$$

[In]

int((d*x)^m*(a+b*ln(c*x^n))*polylog(2,e*x^q),x)

output

-(d*x)^m*x^(-m)*(-e)^(-m/q-1/q)*a/q*(-q^2*x^(1+m)*(-e)^(m/q+1/q)/(1+m)^2*ln(1- 
e*x^q)-q*x^(1+m)*(-e)^(m/q+1/q)/(1+m)*polylog(2,e*x^q)-q^2*x^(1+m+q)*e*(-e)^(m 
/q+1/q)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q))-(d*x)^m*x^(-m)*(-e)^(-m/q-1/q)*b* 
ln(c)/q*(-q^2*x^(1+m)*(-e)^(m/q+1/q)/(1+m)^2*ln(1-e*x^q)-q*x^(1+m)*(-e)^(m/q+1 
/q)/(1+m)*polylog(2,e*x^q)-q^2*x^(1+m+q)*e*(-e)^(m/q+1/q)/(1+m)^2*LerchPhi(e*x 
^q,1,(1+m+q)/q))+((-e)^(-m/q-1/q)*ln(-e)/q^2*(d*x)^m*x^(-m)*b*n*(-q^2*x^m*(-e) 
^(m/q+1/q)/(1+m)^2*ln(1-e*x^q)-q*x^m*(-e)^(m/q+1/q)/(1+m)*polylog(2,e*x^q)-q^2 
*x^(q+m)*e*(-e)^(m/q+1/q)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q))-(-e)^(-m/q-1/q) 
*(d*x)^m*x^(-m)*b*n/q*(-q^2*x^m*(-e)^(m/q+1/q)*ln(x)/(1+m)^2*ln(1-e*x^q)-q*x^m 
*(-e)^(m/q+1/q)*ln(-e)/(1+m)^2*ln(1-e*x^q)+2*q^2*x^m*(-e)^(m/q+1/q)/(1+m)^3*ln 
(1-e*x^q)-q*x^m*(-e)^(m/q+1/q)*ln(x)/(1+m)*polylog(2,e*x^q)-x^m*(-e)^(m/q+1/q) 
*ln(-e)/(1+m)*polylog(2,e*x^q)+q*x^m*(-e)^(m/q+1/q)/(1+m)^2*polylog(2,e*x^q)-q 
^2*x^(q+m)*e*(-e)^(m/q+1/q)*ln(x)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q)-q*x^(q+m 
)*e*(-e)^(m/q+1/q)*ln(-e)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q)+2*q^2*x^(q+m)*e* 
(-e)^(m/q+1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q)+q*x^(q+m)*e*(-e)^(m/q+1/q)/ 
(1+m)^2*LerchPhi(e*x^q,2,(1+m+q)/q)))*x
 

4.4 Test file Number [63]

4.4.1 Mathematica

Integral number [98] $$\int x^2{\log }^3\left(c{(a+b{x}^2)}^p\right)dx$$

[B]   time = 3.12679 (sec), size = 909 ,normalized size = 50.5 $$\frac{2apx{(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))}^2}{b}-\frac{2a^{3/2}p\arctan \left(\frac{\sqrt{b}x}{\sqrt{a}}\right){(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))}^2}{b^{3/2}}+p{x}^3\log (a+b{x}^2){(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))}^2+\frac{1}{3}{x}^3{(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))}^2(-2p-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))+3p^2(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))(\frac{1}{3}{x}^3{\log }^2(a+b{x}^2)-\frac{4(9i{a}^{3/2}\arctan {\left(\frac{\sqrt{b}x}{\sqrt{a}}\right)}^2+3a^{3/2}\arctan \left(\frac{\sqrt{b}x}{\sqrt{a}}\right)(-8+6\log \left(\frac{2\sqrt{a}}{\sqrt{a}+i\sqrt{b}x}\right)+3\log (a+b{x}^2))+\sqrt{b}x(24a-2b{x}^2+(-9a+3b{x}^2)\log (a+b{x}^2))+9i{a}^{3/2}\mathrm{PolyLog}(2,\frac{i\sqrt{a}+\sqrt{b}x}{-i\sqrt{a}+\sqrt{b}x}))}{27b^{3/2}})+\frac{p^3(416\sqrt{-a}{a}^{3/2}\sqrt{\frac{b{x}^2}{a+b{x}^2}}\sqrt{a+b{x}^2}\arcsin \left(\frac{\sqrt{a}}{\sqrt{a+b{x}^2}}\right)+\frac{2}{3}\sqrt{-a}b{x}^2(624a-16b{x}^2+(-288a+24b{x}^2)\log (a+b{x}^2)+18(3a-b{x}^2){\log }^2(a+b{x}^2)+9b{x}^2{\log }^3(a+b{x}^2))+36\sqrt{-a}{a}^{3/2}\sqrt{\frac{b{x}^2}{a+b{x}^2}}(8\sqrt{a}{}_4{F}_3(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{a}{a+b{x}^2})+\log (a+b{x}^2)(4\sqrt{a}{}_3{F}_2(\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{a}{a+b{x}^2})+\sqrt{a+b{x}^2}\arcsin \left(\frac{\sqrt{a}}{\sqrt{a+b{x}^2}}\right)\log (a+b{x}^2)))-48a^2(4\sqrt{b{x}^2}\text{arctanh}\left(\frac{\sqrt{b{x}^2}}{\sqrt{-a}}\right)(\log (a+b{x}^2)-\log (1+\frac{b{x}^2}{a}))-\sqrt{-a}\sqrt{-\frac{b{x}^2}{a}}({\log }^2(1+\frac{b{x}^2}{a})-4\log (1+\frac{b{x}^2}{a})\log \left(\frac{1}{2}(1+\sqrt{-\frac{b{x}^2}{a}})\right)+2{\log }^2\left(\frac{1}{2}(1+\sqrt{-\frac{b{x}^2}{a}})\right)-4\mathrm{PolyLog}(2,\frac{1}{2}-\frac{1}{2}\sqrt{-\frac{b{x}^2}{a}}))))}{18\sqrt{-a}{b}^{2}x}$$

[In]

Integrate[x^2*Log[c*(a + b*x^2)^p]^3,x]

output

(2*a*p*x*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/b - (2*a^(3/2)*p*ArcT 
an[(Sqrt[b]*x)/Sqrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/b^(3/2 
) + p*x^3*Log[a + b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2 + (x^3 
*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2*(-2*p - p*Log[a + b*x^2] + Log 
[c*(a + b*x^2)^p]))/3 + 3*p^2*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])*((x 
^3*Log[a + b*x^2]^2)/3 - (4*((9*I)*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2 + 3*a 
^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-8 + 6*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b 
]*x)] + 3*Log[a + b*x^2]) + Sqrt[b]*x*(24*a - 2*b*x^2 + (-9*a + 3*b*x^2)*Log[a 
 + b*x^2]) + (9*I)*a^(3/2)*PolyLog[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + 
Sqrt[b]*x)]))/(27*b^(3/2))) + (p^3*(416*Sqrt[-a]*a^(3/2)*Sqrt[(b*x^2)/(a + b*x 
^2)]*Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]] + (2*Sqrt[-a]*b*x^2*(624* 
a - 16*b*x^2 + (-288*a + 24*b*x^2)*Log[a + b*x^2] + 18*(3*a - b*x^2)*Log[a + b 
*x^2]^2 + 9*b*x^2*Log[a + b*x^2]^3))/3 + 36*Sqrt[-a]*a^(3/2)*Sqrt[(b*x^2)/(a + 
 b*x^2)]*(8*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, a 
/(a + b*x^2)] + Log[a + b*x^2]*(4*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2}, { 
3/2, 3/2}, a/(a + b*x^2)] + Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Lo 
g[a + b*x^2])) - 48*a^2*(4*Sqrt[b*x^2]*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*(Log[a + 
b*x^2] - Log[1 + (b*x^2)/a]) - Sqrt[-a]*Sqrt[-((b*x^2)/a)]*(Log[1 + (b*x^2)/a] 
^2 - 4*Log[1 + (b*x^2)/a]*Log[(1 + Sqrt[-((b*x^2)/a)])/2] + 2*Log[(1 + Sqrt[-( 
(b*x^2)/a)])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[-((b*x^2)/a)]/2]))))/(18*Sqrt[-...
 

Integral number [99] $$\int {\log }^3\left(c{(a+b{x}^2)}^p\right)dx$$

[B]   time = 2.66678 (sec), size = 789 ,normalized size = 56.36 $$\frac{6\sqrt{a}p\arctan \left(\frac{\sqrt{b}x}{\sqrt{a}}\right){(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))}^2}{\sqrt{b}}+3px\log (a+b{x}^2){(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))}^2+x{(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))}^2(-6p-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))-\frac{3p^2(p\log (a+b{x}^2)-\log \left(c{(a+b{x}^2)}^p\right))(4i\sqrt{a}\arctan {\left(\frac{\sqrt{b}x}{\sqrt{a}}\right)}^2+4\sqrt{a}\arctan \left(\frac{\sqrt{b}x}{\sqrt{a}}\right)(-2+2\log \left(\frac{2\sqrt{a}}{\sqrt{a}+i\sqrt{b}x}\right)+\log (a+b{x}^2))+\sqrt{b}x(8-4\log (a+b{x}^2)+{\log }^2(a+b{x}^2))+4i\sqrt{a}\mathrm{PolyLog}(2,\frac{i\sqrt{a}+\sqrt{b}x}{-i\sqrt{a}+\sqrt{b}x}))}{\sqrt{b}}+\frac{p^3(-48\sqrt{-a^2}\sqrt{\frac{b{x}^2}{a+b{x}^2}}\sqrt{a+b{x}^2}\arcsin \left(\frac{\sqrt{a}}{\sqrt{a+b{x}^2}}\right)+\sqrt{-a}b{x}^2(-48+24\log (a+b{x}^2)-6{\log }^2(a+b{x}^2)+{\log }^3(a+b{x}^2))-6\sqrt{-a^2}\sqrt{\frac{b{x}^2}{a+b{x}^2}}(8\sqrt{a}{}_4{F}_3(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{a}{a+b{x}^2})+\log (a+b{x}^2)(4\sqrt{a}{}_3{F}_2(\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{a}{a+b{x}^2})+\sqrt{a+b{x}^2}\arcsin \left(\frac{\sqrt{a}}{\sqrt{a+b{x}^2}}\right)\log (a+b{x}^2)))+24a\sqrt{b{x}^2}\text{arctanh}\left(\frac{\sqrt{b{x}^2}}{\sqrt{-a}}\right)(\log (a+b{x}^2)-\log (1+\frac{b{x}^2}{a}))+6(-a{)}^{3/2}\sqrt{-\frac{b{x}^2}{a}}({\log }^2(1+\frac{b{x}^2}{a})-4\log (1+\frac{b{x}^2}{a})\log \left(\frac{1}{2}(1+\sqrt{-\frac{b{x}^2}{a}})\right)+2{\log }^2\left(\frac{1}{2}(1+\sqrt{-\frac{b{x}^2}{a}})\right)-4\mathrm{PolyLog}(2,\frac{1}{2}-\frac{1}{2}\sqrt{-\frac{b{x}^2}{a}})))}{\sqrt{-a}bx}$$

[In]

Integrate[Log[c*(a + b*x^2)^p]^3,x]

output

(6*Sqrt[a]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x 
^2)^p])^2)/Sqrt[b] + 3*p*x*Log[a + b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b* 
x^2)^p])^2 + x*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2*(-6*p - p*Log[a 
+ b*x^2] + Log[c*(a + b*x^2)^p]) - (3*p^2*(p*Log[a + b*x^2] - Log[c*(a + b*x^2 
)^p])*((4*I)*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2 + 4*Sqrt[a]*ArcTan[(Sqrt[b] 
*x)/Sqrt[a]]*(-2 + 2*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)] + Log[a + b*x^2] 
) + Sqrt[b]*x*(8 - 4*Log[a + b*x^2] + Log[a + b*x^2]^2) + (4*I)*Sqrt[a]*PolyLo 
g[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b]*x)]))/Sqrt[b] + (p^3*(-48 
*Sqrt[-a^2]*Sqrt[(b*x^2)/(a + b*x^2)]*Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + 
b*x^2]] + Sqrt[-a]*b*x^2*(-48 + 24*Log[a + b*x^2] - 6*Log[a + b*x^2]^2 + Log[a 
 + b*x^2]^3) - 6*Sqrt[-a^2]*Sqrt[(b*x^2)/(a + b*x^2)]*(8*Sqrt[a]*Hypergeometri 
cPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, a/(a + b*x^2)] + Log[a + b*x^2]*(4 
*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, a/(a + b*x^2)] + Sqrt[ 
a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Log[a + b*x^2])) + 24*a*Sqrt[b*x^2] 
*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*(Log[a + b*x^2] - Log[1 + (b*x^2)/a]) + 6*(-a)^ 
(3/2)*Sqrt[-((b*x^2)/a)]*(Log[1 + (b*x^2)/a]^2 - 4*Log[1 + (b*x^2)/a]*Log[(1 + 
 Sqrt[-((b*x^2)/a)])/2] + 2*Log[(1 + Sqrt[-((b*x^2)/a)])/2]^2 - 4*PolyLog[2, 1 
/2 - Sqrt[-((b*x^2)/a)]/2])))/(Sqrt[-a]*b*x)
 

Integral number [100] $$\int \frac{{\log }^3\left(c{(a+b{x}^2)}^p\right)}{x^2}dx$$

[C]   time = 1.00436 (sec), size = 505 ,normalized size = 28.06 $$\frac{p^3(-96\sqrt{a}\sqrt{1-\frac{a}{a+b{x}^2}}{}_4{F}_3(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{a}{a+b{x}^2})-48\sqrt{a}\sqrt{1-\frac{a}{a+b{x}^2}}{}_3{F}_2(\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{a}{a+b{x}^2})\log (a+b{x}^2)-2{\log }^2(a+b{x}^2)(6\sqrt{a+b{x}^2}\sqrt{1-\frac{a}{a+b{x}^2}}\arcsin \left(\frac{\sqrt{a}}{\sqrt{a+b{x}^2}}\right)+\sqrt{a}\log (a+b{x}^2)))}{2\sqrt{a}x}+\frac{6\sqrt{b}p\arctan \left(\frac{\sqrt{b}x}{\sqrt{a}}\right){(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))}^2}{\sqrt{a}}-\frac{3p\log (a+b{x}^2){(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))}^2}{x}-\frac{{(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))}^3}{x}+3p^2(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))(-\frac{{\log }^2(a+b{x}^2)}{x}+\frac{4\sqrt{b}(\arctan \left(\frac{\sqrt{b}x}{\sqrt{a}}\right)(i\arctan \left(\frac{\sqrt{b}x}{\sqrt{a}}\right)+2\log \left(\frac{2i}{i-\frac{\sqrt{b}x}{\sqrt{a}}}\right)+\log (a+b{x}^2))+i\mathrm{PolyLog}(2,\frac{i\sqrt{a}+\sqrt{b}x}{-i\sqrt{a}+\sqrt{b}x}))}{\sqrt{a}})$$

[In]

Integrate[Log[c*(a + b*x^2)^p]^3/x^2,x]

output

(p^3*(-96*Sqrt[a]*Sqrt[1 - a/(a + b*x^2)]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/ 
2}, {3/2, 3/2, 3/2}, a/(a + b*x^2)] - 48*Sqrt[a]*Sqrt[1 - a/(a + b*x^2)]*Hyper 
geometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, a/(a + b*x^2)]*Log[a + b*x^2] - 2*Lo 
g[a + b*x^2]^2*(6*Sqrt[a + b*x^2]*Sqrt[1 - a/(a + b*x^2)]*ArcSin[Sqrt[a]/Sqrt[ 
a + b*x^2]] + Sqrt[a]*Log[a + b*x^2])))/(2*Sqrt[a]*x) + (6*Sqrt[b]*p*ArcTan[(S 
qrt[b]*x)/Sqrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/Sqrt[a] - ( 
3*p*Log[a + b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/x - (-(p*Lo 
g[a + b*x^2]) + Log[c*(a + b*x^2)^p])^3/x + 3*p^2*(-(p*Log[a + b*x^2]) + Log[c 
*(a + b*x^2)^p])*(-(Log[a + b*x^2]^2/x) + (4*Sqrt[b]*(ArcTan[(Sqrt[b]*x)/Sqrt[ 
a]]*(I*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + 2*Log[(2*I)/(I - (Sqrt[b]*x)/Sqrt[a])] + 
Log[a + b*x^2]) + I*PolyLog[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b] 
*x)]))/Sqrt[a])
 

Integral number [101] $$\int \frac{{\log }^3\left(c{(a+b{x}^2)}^p\right)}{x^4}dx$$

[B]   time = 2.19457 (sec), size = 851 ,normalized size = 47.28 $$\frac{a^2{(p\log (a+b{x}^2)-\log \left(c{(a+b{x}^2)}^p\right))}^3-6abp{x}^2{(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))}^2-6\sqrt{a}{b}^{3/2}p{x}^3\arctan \left(\frac{\sqrt{b}x}{\sqrt{a}}\right){(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))}^2-3a^{2}p\log (a+b{x}^2){(-p\log (a+b{x}^2)+\log \left(c{(a+b{x}^2)}^p\right))}^2+3\sqrt{a}{p}^2(p\log (a+b{x}^2)-\log \left(c{(a+b{x}^2)}^p\right))(a^{3/2}{\log }^2(a+b{x}^2)+4b{x}^2(i\sqrt{b}x\arctan {\left(\frac{\sqrt{b}x}{\sqrt{a}}\right)}^2+\sqrt{a}\log (a+b{x}^2)+\sqrt{b}x\arctan \left(\frac{\sqrt{b}x}{\sqrt{a}}\right)(-2+2\log \left(\frac{2\sqrt{a}}{\sqrt{a}+i\sqrt{b}x}\right)+\log (a+b{x}^2))+i\sqrt{b}x\mathrm{PolyLog}(2,\frac{i\sqrt{a}+\sqrt{b}x}{-i\sqrt{a}+\sqrt{b}x})))+p^3(48ab{x}^2\sqrt{\frac{b{x}^2}{a+b{x}^2}}{}_4{F}_3(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{a}{a+b{x}^2})+24\sqrt{-a}{\left(b{x}^2\right)}^{3/2}\text{arctanh}\left(\frac{\sqrt{b{x}^2}}{\sqrt{-a}}\right)\log (a+b{x}^2)+24ab{x}^2\sqrt{\frac{b{x}^2}{a+b{x}^2}}{}_3{F}_2(\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{a}{a+b{x}^2})\log (a+b{x}^2)-6ab{x}^2{\log }^2(a+b{x}^2)+6\sqrt{a}{\left(\frac{b{x}^2}{a+b{x}^2}\right)}^{3/2}{(a+b{x}^2)}^{3/2}\arcsin \left(\frac{\sqrt{a}}{\sqrt{a+b{x}^2}}\right){\log }^2(a+b{x}^2)-a^2{\log }^3(a+b{x}^2)-24\sqrt{-a}{\left(b{x}^2\right)}^{3/2}\text{arctanh}\left(\frac{\sqrt{b{x}^2}}{\sqrt{-a}}\right)\log (1+\frac{b{x}^2}{a})-6a^2{(-\frac{b{x}^2}{a})}^{3/2}{\log }^2(1+\frac{b{x}^2}{a})+24a^2{(-\frac{b{x}^2}{a})}^{3/2}\log (1+\frac{b{x}^2}{a})\log \left(\frac{1}{2}(1+\sqrt{-\frac{b{x}^2}{a}})\right)-12a^2{(-\frac{b{x}^2}{a})}^{3/2}{\log }^2\left(\frac{1}{2}(1+\sqrt{-\frac{b{x}^2}{a}})\right)+24a^2{(-\frac{b{x}^2}{a})}^{3/2}\mathrm{PolyLog}(2,\frac{1}{2}-\frac{1}{2}\sqrt{-\frac{b{x}^2}{a}}))}{3a^2{x}^3}$$

[In]

Integrate[Log[c*(a + b*x^2)^p]^3/x^4,x]

output

(a^2*(p*Log[a + b*x^2] - Log[c*(a + b*x^2)^p])^3 - 6*a*b*p*x^2*(-(p*Log[a + b* 
x^2]) + Log[c*(a + b*x^2)^p])^2 - 6*Sqrt[a]*b^(3/2)*p*x^3*ArcTan[(Sqrt[b]*x)/S 
qrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2 - 3*a^2*p*Log[a + b*x^2 
]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2 + 3*Sqrt[a]*p^2*(p*Log[a + b* 
x^2] - Log[c*(a + b*x^2)^p])*(a^(3/2)*Log[a + b*x^2]^2 + 4*b*x^2*(I*Sqrt[b]*x* 
ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2 + Sqrt[a]*Log[a + b*x^2] + Sqrt[b]*x*ArcTan[(Sqr 
t[b]*x)/Sqrt[a]]*(-2 + 2*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)] + Log[a + b* 
x^2]) + I*Sqrt[b]*x*PolyLog[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b] 
*x)])) + p^3*(48*a*b*x^2*Sqrt[(b*x^2)/(a + b*x^2)]*HypergeometricPFQ[{1/2, 1/2 
, 1/2, 1/2}, {3/2, 3/2, 3/2}, a/(a + b*x^2)] + 24*Sqrt[-a]*(b*x^2)^(3/2)*ArcTa 
nh[Sqrt[b*x^2]/Sqrt[-a]]*Log[a + b*x^2] + 24*a*b*x^2*Sqrt[(b*x^2)/(a + b*x^2)] 
*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, a/(a + b*x^2)]*Log[a + b*x^2] 
- 6*a*b*x^2*Log[a + b*x^2]^2 + 6*Sqrt[a]*((b*x^2)/(a + b*x^2))^(3/2)*(a + b*x^ 
2)^(3/2)*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Log[a + b*x^2]^2 - a^2*Log[a + b*x^2] 
^3 - 24*Sqrt[-a]*(b*x^2)^(3/2)*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*Log[1 + (b*x^2)/a 
] - 6*a^2*(-((b*x^2)/a))^(3/2)*Log[1 + (b*x^2)/a]^2 + 24*a^2*(-((b*x^2)/a))^(3 
/2)*Log[1 + (b*x^2)/a]*Log[(1 + Sqrt[-((b*x^2)/a)])/2] - 12*a^2*(-((b*x^2)/a)) 
^(3/2)*Log[(1 + Sqrt[-((b*x^2)/a)])/2]^2 + 24*a^2*(-((b*x^2)/a))^(3/2)*PolyLog 
[2, 1/2 - Sqrt[-((b*x^2)/a)]/2]))/(3*a^2*x^3)
 

Integral number [158] $$\int (fx{)}^m{\log }^3\left(c{(d+e{x}^2)}^p\right)dx$$

[B]   time = 1.8218 (sec), size = 994 ,normalized size = 49.7 $$\frac{(fx{)}^m((1+m)p^3{x}^2{\log }^3(d+e{x}^2)+\frac{6p^3{(-\frac{e{x}^2}{d})}^{\frac{1-m}{2}}(-((1+m)(d+e{x}^2){}_4{F}_3(1,1,1,\frac{1}{2}-\frac{m}{2};2,2,2;1+\frac{e{x}^2}{d}))+(1+m)(d+e{x}^2){}_3{F}_2(1,1,\frac{1}{2}-\frac{m}{2};2,2;1+\frac{e{x}^2}{d})\log (d+e{x}^2)+d(-1+{(-\frac{e{x}^2}{d})}^{\frac{1+m}{2}}){\log }^2(d+e{x}^2))}{e}+\frac{6d(1+m)p^3{\left(\frac{e{x}^2}{d+e{x}^2}\right)}^{\frac{1}{2}-\frac{m}{2}}(8{}_4{F}_3(\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2};\frac{3}{2}-\frac{m}{2},\frac{3}{2}-\frac{m}{2},\frac{3}{2}-\frac{m}{2};\frac{d}{d+e{x}^2})+(-1+m)\log (d+e{x}^2)(-4{}_3{F}_2(\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2};\frac{3}{2}-\frac{m}{2},\frac{3}{2}-\frac{m}{2};\frac{d}{d+e{x}^2})+(-1+m)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2},\frac{3}{2}-\frac{m}{2},\frac{d}{d+e{x}^2})\log (d+e{x}^2)))}{e(-1+m{)}^3}-\frac{3p^2{(-\frac{e{x}^2}{d})}^{\frac{1-m}{2}}(-((1+m)(d+e{x}^2){}_4{F}_3(1,1,1,\frac{1}{2}-\frac{m}{2};2,2,2;1+\frac{e{x}^2}{d}))+(1+m)(d+e{x}^2){}_3{F}_2(1,1,\frac{1}{2}-\frac{m}{2};2,2;1+\frac{e{x}^2}{d})\log (d+e{x}^2)+d(-1+{(-\frac{e{x}^2}{d})}^{\frac{1+m}{2}}){\log }^2(d+e{x}^2))(-p\log (d+e{x}^2)+\log \left(c{(d+e{x}^2)}^p\right))}{e}-\frac{3m{p}^2{(-\frac{e{x}^2}{d})}^{\frac{1-m}{2}}(-((1+m)(d+e{x}^2){}_4{F}_3(1,1,1,\frac{1}{2}-\frac{m}{2};2,2,2;1+\frac{e{x}^2}{d}))+(1+m)(d+e{x}^2){}_3{F}_2(1,1,\frac{1}{2}-\frac{m}{2};2,2;1+\frac{e{x}^2}{d})\log (d+e{x}^2)+d(-1+{(-\frac{e{x}^2}{d})}^{\frac{1+m}{2}}){\log }^2(d+e{x}^2))(-p\log (d+e{x}^2)+\log \left(c{(d+e{x}^2)}^p\right))}{e}+\frac{3p{x}^2(-2e{x}^2\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{3+m}{2},\frac{5+m}{2},-\frac{e{x}^2}{d})+d(3+m)\log (d+e{x}^2)){(-p\log (d+e{x}^2)+\log \left(c{(d+e{x}^2)}^p\right))}^2}{d(3+m)}+\frac{3mp{x}^2(-2e{x}^2\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{3+m}{2},\frac{5+m}{2},-\frac{e{x}^2}{d})+d(3+m)\log (d+e{x}^2)){(-p\log (d+e{x}^2)+\log \left(c{(d+e{x}^2)}^p\right))}^2}{d(3+m)}+x^2{(-p\log (d+e{x}^2)+\log \left(c{(d+e{x}^2)}^p\right))}^3+m{x}^2{(-p\log (d+e{x}^2)+\log \left(c{(d+e{x}^2)}^p\right))}^3)}{(1+m{)}^{2}x}$$

[In]

Integrate[(f*x)^m*Log[c*(d + e*x^2)^p]^3,x]

output

((f*x)^m*((1 + m)*p^3*x^2*Log[d + e*x^2]^3 + (6*p^3*(-((e*x^2)/d))^((1 - m)/2) 
*(-((1 + m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1, 1/2 - m/2}, {2, 2, 2}, 1 + 
 (e*x^2)/d]) + (1 + m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1/2 - m/2}, {2, 2} 
, 1 + (e*x^2)/d]*Log[d + e*x^2] + d*(-1 + (-((e*x^2)/d))^((1 + m)/2))*Log[d + 
e*x^2]^2))/e + (6*d*(1 + m)*p^3*((e*x^2)/(d + e*x^2))^(1/2 - m/2)*(8*Hypergeom 
etricPFQ[{1/2 - m/2, 1/2 - m/2, 1/2 - m/2, 1/2 - m/2}, {3/2 - m/2, 3/2 - m/2, 
3/2 - m/2}, d/(d + e*x^2)] + (-1 + m)*Log[d + e*x^2]*(-4*HypergeometricPFQ[{1/ 
2 - m/2, 1/2 - m/2, 1/2 - m/2}, {3/2 - m/2, 3/2 - m/2}, d/(d + e*x^2)] + (-1 + 
 m)*Hypergeometric2F1[1/2 - m/2, 1/2 - m/2, 3/2 - m/2, d/(d + e*x^2)]*Log[d + 
e*x^2])))/(e*(-1 + m)^3) - (3*p^2*(-((e*x^2)/d))^((1 - m)/2)*(-((1 + m)*(d + e 
*x^2)*HypergeometricPFQ[{1, 1, 1, 1/2 - m/2}, {2, 2, 2}, 1 + (e*x^2)/d]) + (1 
+ m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1/2 - m/2}, {2, 2}, 1 + (e*x^2)/d]*L 
og[d + e*x^2] + d*(-1 + (-((e*x^2)/d))^((1 + m)/2))*Log[d + e*x^2]^2)*(-(p*Log 
[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/e - (3*m*p^2*(-((e*x^2)/d))^((1 - m)/2)* 
(-((1 + m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1, 1/2 - m/2}, {2, 2, 2}, 1 + 
(e*x^2)/d]) + (1 + m)*(d + e*x^2)*HypergeometricPFQ[{1, 1, 1/2 - m/2}, {2, 2}, 
 1 + (e*x^2)/d]*Log[d + e*x^2] + d*(-1 + (-((e*x^2)/d))^((1 + m)/2))*Log[d + e 
*x^2]^2)*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/e + (3*p*x^2*(-2*e*x^2* 
Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)] + d*(3 + m)*Log[d + e 
*x^2])*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(d*(3 + m)) + (3*m*p...
 

Integral number [159] $$\int (fx{)}^m{\log }^2\left(c{(d+e{x}^2)}^p\right)dx$$

[B]   time = 0.528109 (sec), size = 466 ,normalized size = 23.3 $$\frac{(fx{)}^m(4p^{2}x(\frac{2e{x}^2\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{3+m}{2},\frac{5+m}{2},-\frac{e{x}^2}{d})}{d(3+m)}-\log (d+e{x}^2))+(1+m)p^{2}x{\log }^2(d+e{x}^2)+\frac{4d(1+m)p^2{\left(\frac{e{x}^2}{d+e{x}^2}\right)}^{\frac{1}{2}-\frac{m}{2}}(-2{}_3{F}_2(\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2};\frac{3}{2}-\frac{m}{2},\frac{3}{2}-\frac{m}{2};\frac{d}{d+e{x}^2})+(-1+m)\mathrm{Hypergeometric}2\mathrm{F}1(\frac{1}{2}-\frac{m}{2},\frac{1}{2}-\frac{m}{2},\frac{3}{2}-\frac{m}{2},\frac{d}{d+e{x}^2})\log (d+e{x}^2))}{e(-1+m{)}^{2}x}+\frac{2p(2e{x}^3\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{3+m}{2},\frac{5+m}{2},-\frac{e{x}^2}{d})-d(3+m)x\log (d+e{x}^2))(p\log (d+e{x}^2)-\log \left(c{(d+e{x}^2)}^p\right))}{d(3+m)}-\frac{2mp(-2e{x}^3\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{3+m}{2},\frac{5+m}{2},-\frac{e{x}^2}{d})+d(3+m)x\log (d+e{x}^2))(p\log (d+e{x}^2)-\log \left(c{(d+e{x}^2)}^p\right))}{d(3+m)}+x{(-p\log (d+e{x}^2)+\log \left(c{(d+e{x}^2)}^p\right))}^2+mx{(-p\log (d+e{x}^2)+\log \left(c{(d+e{x}^2)}^p\right))}^2)}{(1+m{)}^2}$$

[In]

Integrate[(f*x)^m*Log[c*(d + e*x^2)^p]^2,x]

output

((f*x)^m*(4*p^2*x*((2*e*x^2*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^ 
2)/d)])/(d*(3 + m)) - Log[d + e*x^2]) + (1 + m)*p^2*x*Log[d + e*x^2]^2 + (4*d* 
(1 + m)*p^2*((e*x^2)/(d + e*x^2))^(1/2 - m/2)*(-2*HypergeometricPFQ[{1/2 - m/2 
, 1/2 - m/2, 1/2 - m/2}, {3/2 - m/2, 3/2 - m/2}, d/(d + e*x^2)] + (-1 + m)*Hyp 
ergeometric2F1[1/2 - m/2, 1/2 - m/2, 3/2 - m/2, d/(d + e*x^2)]*Log[d + e*x^2]) 
)/(e*(-1 + m)^2*x) + (2*p*(2*e*x^3*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, 
-((e*x^2)/d)] - d*(3 + m)*x*Log[d + e*x^2])*(p*Log[d + e*x^2] - Log[c*(d + e*x 
^2)^p]))/(d*(3 + m)) - (2*m*p*(-2*e*x^3*Hypergeometric2F1[1, (3 + m)/2, (5 + m 
)/2, -((e*x^2)/d)] + d*(3 + m)*x*Log[d + e*x^2])*(p*Log[d + e*x^2] - Log[c*(d 
+ e*x^2)^p]))/(d*(3 + m)) + x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2 + 
 m*x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2))/(1 + m)^2
 

Integral number [277] $$\int (f+g{x}^2){\log }^3\left(c{(d+e{x}^2)}^p\right)dx$$

[B]   time = 9.0714 (sec), size = 1772 ,normalized size = 80.55 $$\text{result too large to display}$$

[In]

Integrate[(f + g*x^2)*Log[c*(d + e*x^2)^p]^3,x]

output

(2*d*g*p*x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/e + (6*Sqrt[d]*f*p* 
ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/Sq 
rt[e] - (2*d^(3/2)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-(p*Log[d + e*x^2]) + Log[ 
c*(d + e*x^2)^p])^2)/e^(3/2) + 3*f*p*x*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + L 
og[c*(d + e*x^2)^p])^2 + g*p*x^3*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + Log[c*( 
d + e*x^2)^p])^2 + f*x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2*(-6*p - 
p*Log[d + e*x^2] + Log[c*(d + e*x^2)^p]) + (g*x^3*(-(p*Log[d + e*x^2]) + Log[c 
*(d + e*x^2)^p])^2*(-2*p - p*Log[d + e*x^2] + Log[c*(d + e*x^2)^p]))/3 + 3*f*p 
^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])*(x*Log[d + e*x^2]^2 - (4*((-I) 
*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + Sqrt[e]*x*(-2 + Log[d + e*x^2]) - Sqr 
t[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-2 + 2*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]* 
x)] + Log[d + e*x^2]) - I*Sqrt[d]*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqr 
t[d] + Sqrt[e]*x)]))/Sqrt[e]) + 3*g*p^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^ 
2)^p])*((x^3*Log[d + e*x^2]^2)/3 - (4*((9*I)*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d 
]]^2 + 3*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-8 + 6*Log[(2*Sqrt[d])/(Sqrt[d] 
+ I*Sqrt[e]*x)] + 3*Log[d + e*x^2]) + Sqrt[e]*x*(24*d - 2*e*x^2 + (-9*d + 3*e* 
x^2)*Log[d + e*x^2]) + (9*I)*d^(3/2)*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)* 
Sqrt[d] + Sqrt[e]*x)]))/(27*e^(3/2))) + (g*p^3*(416*Sqrt[-d]*d^(3/2)*Sqrt[d + 
e*x^2]*Sqrt[1 - d/(d + e*x^2)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]] + 36*Sqrt[-d]*d 
^(3/2)*Sqrt[1 - d/(d + e*x^2)]*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2,...
 

Integral number [298] $$\int {(f+g{x}^3)}^2{\log }^3\left(c{(d+e{x}^2)}^p\right)dx$$

[B]   time = 8.70025 (sec), size = 2385 ,normalized size = 99.38 $$\text{Result too large to show}$$

[In]

Integrate[(f + g*x^3)^2*Log[c*(d + e*x^2)^p]^3,x]

output

(f*g*p^3*(d + e*x^2)*(-8*d*(-6 + 6*Log[d + e*x^2] - 3*Log[d + e*x^2]^2 + Log[d 
 + e*x^2]^3) + (d + e*x^2)*(-3 + 6*Log[d + e*x^2] - 6*Log[d + e*x^2]^2 + 4*Log 
[d + e*x^2]^3)))/(8*e^2) + 6*f*g*p^2*((x^4*Log[d + e*x^2]^2)/4 - e*((3*d*x^2)/ 
(4*e^2) - x^4/(8*e) - (3*d^2*Log[d + e*x^2])/(4*e^3) - (d*x^2*Log[d + e*x^2])/ 
(2*e^2) + (x^4*Log[d + e*x^2])/(4*e) + (d^2*Log[d + e*x^2]^2)/(4*e^3)))*(-(p*L 
og[d + e*x^2]) + Log[c*(d + e*x^2)^p]) + (3*d*f*g*p*x^2*(-(p*Log[d + e*x^2]) + 
 Log[c*(d + e*x^2)^p])^2)/(2*e) - (2*d^2*g^2*p*x^3*(-(p*Log[d + e*x^2]) + Log[ 
c*(d + e*x^2)^p])^2)/(7*e^2) + (6*d*g^2*p*x^5*(-(p*Log[d + e*x^2]) + Log[c*(d 
+ e*x^2)^p])^2)/(35*e) - (3*d^2*f*g*p*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + Lo 
g[c*(d + e*x^2)^p])^2)/(2*e^2) + (3*p*x*(14*f^2 + 7*f*g*x^3 + 2*g^2*x^6)*Log[d 
 + e*x^2]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/14 + (f*g*x^4*(-(p*L 
og[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2*(-3*p + 2*(-(p*Log[d + e*x^2]) + Log[ 
c*(d + e*x^2)^p])))/4 + (g^2*x^7*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^ 
2*(-6*p + 7*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])))/49 + (x*(-(p*Log[d 
+ e*x^2]) + Log[c*(d + e*x^2)^p])^2*(-42*e^3*f^2*p + 6*d^3*g^2*p + 7*e^3*f^2*( 
-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])))/(7*e^3) - (6*ArcTan[(Sqrt[e]*x)/ 
Sqrt[d]]*(-7*d*e^3*f^2*p*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2 + d^4* 
g^2*p*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2))/(7*Sqrt[d]*e^(7/2)) + 3 
*f^2*p^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])*(x*Log[d + e*x^2]^2 - (4 
*((-I)*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + Sqrt[e]*x*(-2 + Log[d + e*x^...
 

Integral number [299] $$\int (f+g{x}^3){\log }^3\left(c{(d+e{x}^2)}^p\right)dx$$

[B]   time = 1.29533 (sec), size = 1051 ,normalized size = 47.77 $$\frac{1}{4}g{x}^4{\log }^3\left(c{(d+e{x}^2)}^p\right)+\frac{6\sqrt{d}fp\arctan \left(\frac{\sqrt{e}x}{\sqrt{d}}\right){(-p\log (d+e{x}^2)+\log \left(c{(d+e{x}^2)}^p\right))}^2}{\sqrt{e}}+3fpx\log (d+e{x}^2){(-p\log (d+e{x}^2)+\log \left(c{(d+e{x}^2)}^p\right))}^2+fx{(-p\log (d+e{x}^2)+\log \left(c{(d+e{x}^2)}^p\right))}^2(-6p-p\log (d+e{x}^2)+\log \left(c{(d+e{x}^2)}^p\right))-\frac{3}{4}gp(-\frac{7d{p}^2{x}^2}{2e}+\frac{p^2{x}^4}{4}+\frac{d^2{p}^2\log (d+e{x}^2)}{2e^2}+\frac{3d^{2}p\log \left(c{(d+e{x}^2)}^p\right)}{e^2}+\frac{3dp{x}^2\log \left(c{(d+e{x}^2)}^p\right)}{e}-\frac{1}{2}p{x}^4\log \left(c{(d+e{x}^2)}^p\right)-\frac{3d^2{\log }^2\left(c{(d+e{x}^2)}^p\right)}{2e^2}-\frac{d{x}^2{\log }^2\left(c{(d+e{x}^2)}^p\right)}{e}+\frac{1}{2}{x}^4{\log }^2\left(c{(d+e{x}^2)}^p\right)+\frac{d^2{\log }^3\left(c{(d+e{x}^2)}^p\right)}{3e^{2}p})+3f{p}^2(-p\log (d+e{x}^2)+\log \left(c{(d+e{x}^2)}^p\right))(x{\log }^2(d+e{x}^2)-\frac{4(-i\sqrt{d}\arctan {\left(\frac{\sqrt{e}x}{\sqrt{d}}\right)}^2+\sqrt{e}x(-2+\log (d+e{x}^2))-\sqrt{d}\arctan \left(\frac{\sqrt{e}x}{\sqrt{d}}\right)(-2+2\log \left(\frac{2\sqrt{d}}{\sqrt{d}+i\sqrt{e}x}\right)+\log (d+e{x}^2))-i\sqrt{d}\mathrm{PolyLog}(2,\frac{i\sqrt{d}+\sqrt{e}x}{-i\sqrt{d}+\sqrt{e}x}))}{\sqrt{e}})+\frac{f{p}^3(-48\sqrt{-d^2}\sqrt{d+e{x}^2}\sqrt{1-\frac{d}{d+e{x}^2}}\arcsin \left(\frac{\sqrt{d}}{\sqrt{d+e{x}^2}}\right)-6\sqrt{-d^2}\sqrt{1-\frac{d}{d+e{x}^2}}(8\sqrt{d}{}_4{F}_3(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{d}{d+e{x}^2})+4\sqrt{d}{}_3{F}_2(\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{d}{d+e{x}^2})\log (d+e{x}^2)+\sqrt{d+e{x}^2}\arcsin \left(\frac{\sqrt{d}}{\sqrt{d+e{x}^2}}\right){\log }^2(d+e{x}^2))+\sqrt{-d}e{x}^2(-48+24\log (d+e{x}^2)-6{\log }^2(d+e{x}^2)+{\log }^3(d+e{x}^2))+24d\sqrt{e{x}^2}\text{arctanh}\left(\frac{\sqrt{e{x}^2}}{\sqrt{-d}}\right)(\log (d+e{x}^2)-\log \left(\frac{d+e{x}^2}{d}\right))+6(-d{)}^{3/2}\sqrt{1-\frac{d+e{x}^2}{d}}({\log }^2\left(\frac{d+e{x}^2}{d}\right)-4\log \left(\frac{d+e{x}^2}{d}\right)\log \left(\frac{1}{2}(1+\sqrt{1-\frac{d+e{x}^2}{d}})\right)+2{\log }^2\left(\frac{1}{2}(1+\sqrt{1-\frac{d+e{x}^2}{d}})\right)-4\mathrm{PolyLog}(2,\frac{1}{2}-\frac{1}{2}\sqrt{1-\frac{d+e{x}^2}{d}})))}{\sqrt{-d}ex}$$

[In]

Integrate[(f + g*x^3)*Log[c*(d + e*x^2)^p]^3,x]

output

(g*x^4*Log[c*(d + e*x^2)^p]^3)/4 + (6*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]* 
(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/Sqrt[e] + 3*f*p*x*Log[d + e*x^ 
2]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2 + f*x*(-(p*Log[d + e*x^2]) + 
 Log[c*(d + e*x^2)^p])^2*(-6*p - p*Log[d + e*x^2] + Log[c*(d + e*x^2)^p]) - (3 
*g*p*((-7*d*p^2*x^2)/(2*e) + (p^2*x^4)/4 + (d^2*p^2*Log[d + e*x^2])/(2*e^2) + 
(3*d^2*p*Log[c*(d + e*x^2)^p])/e^2 + (3*d*p*x^2*Log[c*(d + e*x^2)^p])/e - (p*x 
^4*Log[c*(d + e*x^2)^p])/2 - (3*d^2*Log[c*(d + e*x^2)^p]^2)/(2*e^2) - (d*x^2*L 
og[c*(d + e*x^2)^p]^2)/e + (x^4*Log[c*(d + e*x^2)^p]^2)/2 + (d^2*Log[c*(d + e* 
x^2)^p]^3)/(3*e^2*p)))/4 + 3*f*p^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p] 
)*(x*Log[d + e*x^2]^2 - (4*((-I)*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + Sqrt[ 
e]*x*(-2 + Log[d + e*x^2]) - Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-2 + 2*Log[( 
2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + Log[d + e*x^2]) - I*Sqrt[d]*PolyLog[2, ( 
I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)]))/Sqrt[e]) + (f*p^3*(-48*Sq 
rt[-d^2]*Sqrt[d + e*x^2]*Sqrt[1 - d/(d + e*x^2)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2 
]] - 6*Sqrt[-d^2]*Sqrt[1 - d/(d + e*x^2)]*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1 
/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^2)] + 4*Sqrt[d]*HypergeometricPFQ[{ 
1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^2)]*Log[d + e*x^2] + Sqrt[d + e*x^2]*Ar 
cSin[Sqrt[d]/Sqrt[d + e*x^2]]*Log[d + e*x^2]^2) + Sqrt[-d]*e*x^2*(-48 + 24*Log 
[d + e*x^2] - 6*Log[d + e*x^2]^2 + Log[d + e*x^2]^3) + 24*d*Sqrt[e*x^2]*ArcTan 
h[Sqrt[e*x^2]/Sqrt[-d]]*(Log[d + e*x^2] - Log[(d + e*x^2)/d]) + 6*(-d)^(3/2...
 

Integral number [485] $$\int x^2{(a+b\log \left(c{(d+e{x}^{2/3})}^n\right))}^{3}dx$$

[A]   time = 7.79559 (sec), size = 1552 ,normalized size = 64.67 $$\text{result too large to display}$$

[In]

Integrate[x^2*(a + b*Log[c*(d + e*x^(2/3))^n])^3,x]

output

(-2*b*d^4*n*x^(1/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^ 
2)/e^4 + (2*b*d^3*n*x*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n] 
)^2)/(3*e^3) - (2*b*d^2*n*x^(5/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e 
*x^(2/3))^n])^2)/(5*e^2) + (2*b*d*n*x^(7/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Lo 
g[c*(d + e*x^(2/3))^n])^2)/(7*e) + (2*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqr 
t[d]]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/e^(9/2) + b 
*n*x^3*Log[d + e*x^(2/3)]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3) 
)^n])^2 + (x^3*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2*(3* 
a - 2*b*n - 3*b*n*Log[d + e*x^(2/3)] + 3*b*Log[c*(d + e*x^(2/3))^n]))/9 - (b^3 
*n^3*(1094783760*d^(9/2)*Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))] 
*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]] - e*x^(2/3)*(-16*(68423985*d^4 - 4186770* 
d^3*e*x^(2/3) + 871542*d^2*e^2*x^(4/3) - 217125*d*e^3*x^2 + 42875*e^4*x^(8/3)) 
 + 2520*(177345*d^4 - 26040*d^3*e*x^(2/3) + 9009*d^2*e^2*x^(4/3) - 3600*d*e^3* 
x^2 + 1225*e^4*x^(8/3))*Log[d + e*x^(2/3)] - 198450*(315*d^4 - 105*d^3*e*x^(2/ 
3) + 63*d^2*e^2*x^(4/3) - 45*d*e^3*x^2 + 35*e^4*x^(8/3))*Log[d + e*x^(2/3)]^2 
+ 10418625*e^4*x^(8/3)*Log[d + e*x^(2/3)]^3) + 62511750*d^(9/2)*Sqrt[(e*x^(2/3 
))/(d + e*x^(2/3))]*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3 
/2, 3/2}, d/(d + e*x^(2/3))] + Log[d + e*x^(2/3)]*(4*Sqrt[d]*HypergeometricPFQ 
[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))] + Sqrt[d + e*x^(2/3)]*ArcSin[ 
Sqrt[d]/Sqrt[d + e*x^(2/3)]]*Log[d + e*x^(2/3)])) + 111727350*(-d)^(9/2)*(4...
 

Integral number [486] $$\int {(a+b\log \left(c{(d+e{x}^{2/3})}^n\right))}^{3}dx$$

[B]   time = 5.89109 (sec), size = 1299 ,normalized size = 64.95 $$\text{result too large to display}$$

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^3,x]

output

(6*b*d*n*x^(1/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/ 
e - (6*b*d^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a - b*n*Log[d + e*x^(2/3 
)] + b*Log[c*(d + e*x^(2/3))^n])^2)/e^(3/2) + 3*b*n*x*Log[d + e*x^(2/3)]*(a - 
b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 + x*(a - b*n*Log[d + e* 
x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2*(a - 2*b*n - b*n*Log[d + e*x^(2/3)] + 
 b*Log[c*(d + e*x^(2/3))^n]) + (b^2*n^2*x^(1/3)*(a - b*n*Log[d + e*x^(2/3)] + 
b*Log[c*(d + e*x^(2/3))^n])*((-96*d^(3/2)*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]]) 
/(Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]) - d*(104 - 48*Log[d + 
 e*x^(2/3)] + 9*Log[d + e*x^(2/3)]^2) + (d + e*x^(2/3))*(8 - 12*Log[d + e*x^(2 
/3)] + 9*Log[d + e*x^(2/3)]^2) + (36*(-d)^(3/2)*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[- 
d]]*(Log[d + e*x^(2/3)] - Log[1 + (e*x^(2/3))/d]))/Sqrt[e*x^(2/3)] + (9*d*(2*L 
og[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 - 4*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]*L 
og[1 + (e*x^(2/3))/d] + Log[1 + (e*x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[-(( 
e*x^(2/3))/d)]/2]))/Sqrt[-((e*x^(2/3))/d)]))/(3*e) + (b^3*n^3*(624*d*e*x^(2/3) 
 - 16*e^2*x^(4/3) + 624*d^(3/2)*Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e*x^ 
(2/3))]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]] + 432*d^2*Sqrt[(e*x^(2/3))/(d + e* 
x^(2/3))]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^ 
(2/3))] + 144*d^2*Sqrt[-((e*x^(2/3))/d)]*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 
 - 288*d*e*x^(2/3)*Log[d + e*x^(2/3)] + 24*e^2*x^(4/3)*Log[d + e*x^(2/3)] + 28 
8*Sqrt[-d]*d*Sqrt[e*x^(2/3)]*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*Log[d + e*x^...
 

Integral number [487] $$\int \frac{{(a+b\log \left(c{(d+e{x}^{2/3})}^n\right))}^3}{x^2}dx$$

[B]   time = 6.62541 (sec), size = 1158 ,normalized size = 48.25 $$\text{result too large to display}$$

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^2,x]

output

(-6*b*e*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/(d*x^(1 
/3)) - (6*b*e^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a - b*n*Log[d + e*x^( 
2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/d^(3/2) - (3*b*n*Log[d + e*x^(2/3)]*(a 
- b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/x - (a - b*n*Log[d + 
 e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^3/x + (3*b^2*e*n^2*(a - b*n*Log[d + 
e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])*((-16*Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^( 
2/3))/(d + e*x^(2/3))]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]])/d^(3/2) - (8*Log[d 
 + e*x^(2/3)])/d - (2*Log[d + e*x^(2/3)]^2)/(e*x^(2/3)) - (8*Sqrt[e*x^(2/3)]*A 
rcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[1 + (e*x^(2/3))/d]) 
)/(-d)^(3/2) - (2*Sqrt[-((e*x^(2/3))/d)]*(2*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2 
]^2 - 4*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]*Log[1 + (e*x^(2/3))/d] + Log[1 + ( 
e*x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[-((e*x^(2/3))/d)]/2]))/d))/(2*x^(1/3 
)) + (b^3*n^3*(48*Sqrt[-d^2]*e*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*x^(2/3)*Hyper 
geometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^(2/3))] - 12*d* 
Sqrt[-d^2]*(-((e*x^(2/3))/d))^(3/2)*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 - 24 
*Sqrt[d]*(e*x^(2/3))^(3/2)*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*Log[d + e*x^(2/3) 
] + 24*Sqrt[-d^2]*e*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*x^(2/3)*HypergeometricPF 
Q[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))]*Log[d + e*x^(2/3)] - 6*Sqrt[ 
-d^2]*e*x^(2/3)*Log[d + e*x^(2/3)]^2 + 6*Sqrt[-d]*(d + e*x^(2/3))^(3/2)*((e*x^ 
(2/3))/(d + e*x^(2/3)))^(3/2)*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]]*Log[d + e...
 

Integral number [488] $$\int \frac{{(a+b\log \left(c{(d+e{x}^{2/3})}^n\right))}^3}{x^4}dx$$

[B]   time = 7.58333 (sec), size = 1385 ,normalized size = 57.71 $$\text{result too large to display}$$

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^4,x]

output

((-60*b*e*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/(d*x^ 
(7/3)) + (84*b*e^2*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n]) 
^2)/(d^2*x^(5/3)) - (140*b*e^3*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e* 
x^(2/3))^n])^2)/(d^3*x) + (420*b*e^4*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*( 
d + e*x^(2/3))^n])^2)/(d^4*x^(1/3)) + (420*b*e^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3) 
)/Sqrt[d]]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/d^(9/2 
) - (210*b*n*Log[d + e*x^(2/3)]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x 
^(2/3))^n])^2)/x^3 - (70*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3)) 
^n])^3)/x^3 - (2*b^3*n^3*(1376*e^3*(d + e*x^(2/3))^(3/2)*((e*x^(2/3))/(d + e*x 
^(2/3)))^(3/2)*x^2*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]] + Sqrt[d]*(16*e^3*(d - 
15*e*x^(2/3))*x^2 + 8*(3*d^2*e^2*x^(4/3) - 12*d*e^3*x^2 + 71*e^4*x^(8/3))*Log[ 
d + e*x^(2/3)] + (30*d^3*e*x^(2/3) - 42*d^2*e^2*x^(4/3) + 70*d*e^3*x^2 - 210*e 
^4*x^(8/3))*Log[d + e*x^(2/3)]^2 + 35*d^4*Log[d + e*x^(2/3)]^3) + 210*e^4*Sqrt 
[(e*x^(2/3))/(d + e*x^(2/3))]*x^(8/3)*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 
1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^(2/3))] + Log[d + e*x^(2/3)]*(4*Sqrt[d] 
*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))] + Sqrt[d + 
e*x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]]*Log[d + e*x^(2/3)])) + (352*d^( 
3/2)*e^4*x^(8/3)*(4*Sqrt[e*x^(2/3)]*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + 
 e*x^(2/3)] - Log[1 + (e*x^(2/3))/d]) - Sqrt[-d]*Sqrt[-((e*x^(2/3))/d)]*(2*Log 
[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 - 4*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]*...
 

Integral number [528] $$\int x^2{(a+b\log \left(c{(d+\frac{e}{x^{2/3}})}^n\right))}^{3}dx$$

[B]   time = 23.0946 (sec), size = 5975 ,normalized size = 248.96 $$\text{Result too large to show}$$

[In]

Integrate[x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^3,x]

output

Result too large to show
 

Integral number [530] $$\int \frac{{(a+b\log \left(c{(d+\frac{e}{x^{2/3}})}^n\right))}^3}{x^2}dx$$

[B]   time = 13.157 (sec), size = 5504 ,normalized size = 229.33 $$\text{Result too large to show}$$

[In]

Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])^3/x^2,x]

output

Result too large to show
 

Integral number [531] $$\int \frac{{(a+b\log \left(c{(d+\frac{e}{x^{2/3}})}^n\right))}^3}{x^4}dx$$

[B]   time = 21.2985 (sec), size = 6328 ,normalized size = 263.67 $$\text{Result too large to show}$$

[In]

Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])^3/x^4,x]

output

Result too large to show
 

4.5 Test file Number [79]

4.5.1 Mathematica

Integral number [399] $$\int \frac{{\cos }^4(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 0.895414 (sec), size = 394 ,normalized size = 17.13 $$\frac{-i\text{RootSum}[-ib+3ib{\mathrm{\#}\text{1}}^2+8a{\mathrm{\#}\text{1}}^3-3ib{\mathrm{\#}\text{1}}^4+ib{\mathrm{\#}\text{1}}^6\mathrm{\&},\frac{2b\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)-ib\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2)+4ia\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)\mathrm{\#}\text{1}+2a\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2)\mathrm{\#}\text{1}+12b\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^2-6ib\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^2-4ia\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^3-2a\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^3+2b\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^4-ib\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^4}{b\mathrm{\#}\text{1}-4ia{\mathrm{\#}\text{1}}^2-2b{\mathrm{\#}\text{1}}^3+b{\mathrm{\#}\text{1}}^5}\mathrm{\&}]+\frac{24\cos (c+dx)(a+b\sin (c+dx))}{4a+3b\sin (c+dx)-b\sin (3(c+dx))}}{18abd}$$

[In]

Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]

output

((-I)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , ( 
2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*b*Log[1 - 2*Cos[c + d*x]*#1 + 
 #1^2] + (4*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 2*a*Log[1 - 2*C 
os[c + d*x]*#1 + #1^2]*#1 + 12*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 
 - (6*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (4*I)*a*ArcTan[Sin[c + d*x 
]/(Cos[c + d*x] - #1)]*#1^3 - 2*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*b 
*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*b*Log[1 - 2*Cos[c + d*x]*#1 
 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + (24*Cos[c + d*x 
]*(a + b*Sin[c + d*x]))/(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)]))/(18*a*b 
*d)
 

Integral number [400] $$\int \frac{{\cos }^2(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 0.765131 (sec), size = 273 ,normalized size = 11.87 $$\frac{-i\text{RootSum}[-ib+3ib{\mathrm{\#}\text{1}}^2+8a{\mathrm{\#}\text{1}}^3-3ib{\mathrm{\#}\text{1}}^4+ib{\mathrm{\#}\text{1}}^6\mathrm{\&},\frac{2\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)-i\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2)+12\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^2-6i\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^2+2\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^4-i\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^4}{b\mathrm{\#}\text{1}-4ia{\mathrm{\#}\text{1}}^2-2b{\mathrm{\#}\text{1}}^3+b{\mathrm{\#}\text{1}}^5}\mathrm{\&}]+\frac{12\sin (2(c+dx))}{4a+3b\sin (c+dx)-b\sin (3(c+dx))}}{18ad}$$

[In]

Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]

output

((-I)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , ( 
2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^ 
2] + 12*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (6*I)*Log[1 - 2*Cos[c 
+ d*x]*#1 + #1^2]*#1^2 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*L 
og[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^ 
5) & ] + (12*Sin[2*(c + d*x)])/(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)]))/ 
(18*a*d)
 

Integral number [401] $$\int \frac{1}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 0.608929 (sec), size = 502 ,normalized size = 35.86 $$\frac{\frac{i\text{RootSum}[-ib+3ib{\mathrm{\#}\text{1}}^2+8a{\mathrm{\#}\text{1}}^3-3ib{\mathrm{\#}\text{1}}^4+ib{\mathrm{\#}\text{1}}^6\mathrm{\&},\frac{2b^2\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)-i{b}^2\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2)+4iab\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)\mathrm{\#}\text{1}+2ab\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2)\mathrm{\#}\text{1}-24a^2\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^2+12b^2\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^2+12i{a}^2\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^2-6i{b}^2\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^2-4iab\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^3-2ab\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^3+2b^2\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^4-i{b}^2\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^4}{b\mathrm{\#}\text{1}-4ia{\mathrm{\#}\text{1}}^2-2b{\mathrm{\#}\text{1}}^3+b{\mathrm{\#}\text{1}}^5}\mathrm{\&}]}{a^2-b^2}-\frac{12b\cos (c+dx)(-3a+a\cos (2(c+dx))+2b\sin (c+dx))}{(a-b)(a+b)(4a+3b\sin (c+dx)-b\sin (3(c+dx)))}}{18ad}$$

[In]

Integrate[(a + b*Sin[c + d*x]^3)^(-2),x]

output

((I*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2* 
b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*b^2*Log[1 - 2*Cos[c + d*x]*#1 
 + #1^2] + (4*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 2*a*b*Log[1 
 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 24*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - # 
1)]*#1^2 + 12*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (12*I)*a^2*L 
og[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + 
#1^2]*#1^2 - (4*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 2*a*b*L 
og[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x 
] - #1)]*#1^4 - I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a* 
#1^2 - 2*b*#1^3 + b*#1^5) & ])/(a^2 - b^2) - (12*b*Cos[c + d*x]*(-3*a + a*Cos[ 
2*(c + d*x)] + 2*b*Sin[c + d*x]))/((a - b)*(a + b)*(4*a + 3*b*Sin[c + d*x] - b 
*Sin[3*(c + d*x)])))/(18*a*d)
 

Integral number [402] $$\int \frac{{\sec }^2(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 2.21428 (sec), size = 845 ,normalized size = 36.74 $$\frac{-\frac{ib\text{RootSum}[-ib+3ib{\mathrm{\#}\text{1}}^2+8a{\mathrm{\#}\text{1}}^3-3ib{\mathrm{\#}\text{1}}^4+ib{\mathrm{\#}\text{1}}^6\mathrm{\&},\frac{16a^{2}b\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)+2b^3\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)-8i{a}^{2}b\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2)-i{b}^3\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2)+20i{a}^3\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)\mathrm{\#}\text{1}+16ia{b}^2\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)\mathrm{\#}\text{1}+10a^3\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2)\mathrm{\#}\text{1}+8a{b}^2\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2)\mathrm{\#}\text{1}-120a^{2}b\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^2+12b^3\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^2+60i{a}^{2}b\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^2-6i{b}^3\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^2-20i{a}^3\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^3-16ia{b}^2\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^3-10a^3\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^3-8a{b}^2\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^3+16a^{2}b\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^4+2b^3\arctan \left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^4-8i{a}^{2}b\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^4-i{b}^3\log (1-2\cos (c+dx)\mathrm{\#}\text{1}+{\mathrm{\#}\text{1}}^2){\mathrm{\#}\text{1}}^4}{b\mathrm{\#}\text{1}-4ia{\mathrm{\#}\text{1}}^2-2b{\mathrm{\#}\text{1}}^3+b{\mathrm{\#}\text{1}}^5}\mathrm{\&}]}{a{(a^2-b^2)}^2}+\frac{18\sin (\frac{1}{2}(c+dx))}{(a+b{)}^2(\cos (\frac{1}{2}(c+dx))-\sin (\frac{1}{2}(c+dx)))}+\frac{18\sin (\frac{1}{2}(c+dx))}{(a-b{)}^2(\cos (\frac{1}{2}(c+dx))+\sin (\frac{1}{2}(c+dx)))}+\frac{12b\cos (c+dx)(-2a^3-7a{b}^2+3a{b}^2\cos (2(c+dx))+2b(2a^2+b^2)\sin (c+dx))}{a(a-b{)}^2(a+b{)}^2(4a+3b\sin (c+dx)-b\sin (3(c+dx)))}}{18d}$$

[In]

Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]

output

(((-I)*b*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & 
, (16*a^2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 2*b^3*ArcTan[Sin[c + d* 
x]/(Cos[c + d*x] - #1)] - (8*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - I*b^ 
3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (20*I)*a^3*ArcTan[Sin[c + d*x]/(Cos[c + 
d*x] - #1)]*#1 + (16*I)*a*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 10 
*a^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 8*a*b^2*Log[1 - 2*Cos[c + d*x]*#1 
+ #1^2]*#1 - 120*a^2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^3* 
ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (60*I)*a^2*b*Log[1 - 2*Cos[c + 
 d*x]*#1 + #1^2]*#1^2 - (6*I)*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (20 
*I)*a^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - (16*I)*a*b^2*ArcTan[Si 
n[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 10*a^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2 
]*#1^3 - 8*a*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 16*a^2*b*ArcTan[Sin[ 
c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 2*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - 
 #1)]*#1^4 - (8*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - I*b^3*Log[1 
- 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & 
])/(a*(a^2 - b^2)^2) + (18*Sin[(c + d*x)/2])/((a + b)^2*(Cos[(c + d*x)/2] - Si 
n[(c + d*x)/2])) + (18*Sin[(c + d*x)/2])/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c 
 + d*x)/2])) + (12*b*Cos[c + d*x]*(-2*a^3 - 7*a*b^2 + 3*a*b^2*Cos[2*(c + d*x)] 
 + 2*b*(2*a^2 + b^2)*Sin[c + d*x]))/(a*(a - b)^2*(a + b)^2*(4*a + 3*b*Sin[c + 
d*x] - b*Sin[3*(c + d*x)])))/(18*d)
 

Integral number [403] $$\int \frac{{\sec }^4(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 2.2548 (sec), size = 1158 ,normalized size = 50.35 $$\text{result too large to display}$$

[In]

Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]

output

((4*I)*b^2*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 
& , (14*a^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 74*a^2*b^2*ArcTan[Sin[c 
 + d*x]/(Cos[c + d*x] - #1)] + 2*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] 
- (7*I)*a^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (37*I)*a^2*b^2*Log[1 - 2*Cos[c 
 + d*x]*#1 + #1^2] - I*b^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (144*I)*a^3*b*A 
rcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + (36*I)*a*b^3*ArcTan[Sin[c + d*x]/ 
(Cos[c + d*x] - #1)]*#1 + 72*a^3*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 18*a 
*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 180*a^4*ArcTan[Sin[c + d*x]/(Cos[c 
 + d*x] - #1)]*#1^2 - 372*a^2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^ 
2 + 12*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (90*I)*a^4*Log[1 - 
2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (186*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + # 
1^2]*#1^2 - (6*I)*b^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (144*I)*a^3*b*A 
rcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - (36*I)*a*b^3*ArcTan[Sin[c + d*x 
]/(Cos[c + d*x] - #1)]*#1^3 - 72*a^3*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 
- 18*a*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 14*a^4*ArcTan[Sin[c + d*x] 
/(Cos[c + d*x] - #1)]*#1^4 + 74*a^2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1 
)]*#1^4 + 2*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (7*I)*a^4*Log[ 
1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (37*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 
+ #1^2]*#1^4 - I*b^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*# 
1^2 - 2*b*#1^3 + b*#1^5) & ] + (3*Sec[c + d*x]^3*(48*a^5*b + 568*a^3*b^3 + ...
 

4.5.2 Fricas

Integral number [399] $$\int \frac{{\cos }^4(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 9.39875 (sec), size = 9984 ,normalized size = 434.09 $$\text{Too large to display}$$

[In]

integrate(cos(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

output

-1/36*(sqrt(2)*sqrt(1/2)*(a^2*b*d - (a*b^2*d*cos(d*x + c)^2 - a*b^2*d)*sin(d*x 
 + c))*sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*(27/(a^6*b^6*d^6) - (a^2 - 4*b^2)^3/ 
(a^10*b^8*d^6) - (a^6 + 15*a^4*b^2 + 48*a^2*b^4 - 64*b^6)/(a^10*b^8*d^6))^(1/3 
) + 6/(a^2*b^2*d^2))*a^2*b^2*d^2 + 3*sqrt(1/3)*a^2*b^2*d^2*sqrt(-(((1/2)^(1/3) 
*(I*sqrt(3) + 1)*(27/(a^6*b^6*d^6) - (a^2 - 4*b^2)^3/(a^10*b^8*d^6) - (a^6 + 1 
5*a^4*b^2 + 48*a^2*b^4 - 64*b^6)/(a^10*b^8*d^6))^(1/3) + 6/(a^2*b^2*d^2))^2*a^ 
4*b^4*d^4 - 12*((1/2)^(1/3)*(I*sqrt(3) + 1)*(27/(a^6*b^6*d^6) - (a^2 - 4*b^2)^ 
3/(a^10*b^8*d^6) - (a^6 + 15*a^4*b^2 + 48*a^2*b^4 - 64*b^6)/(a^10*b^8*d^6))^(1 
/3) + 6/(a^2*b^2*d^2))*a^2*b^2*d^2 + 36)/(a^4*b^4*d^4)) - 18)/(a^2*b^2*d^2))*l 
og(1/4*(a^9*b^5 + 8*a^7*b^7)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(27/(a^6*b^6*d^6) - 
(a^2 - 4*b^2)^3/(a^10*b^8*d^6) - (a^6 + 15*a^4*b^2 + 48*a^2*b^4 - 64*b^6)/(a^1 
0*b^8*d^6))^(1/3) + 6/(a^2*b^2*d^2))^2*d^4*sin(d*x + c) - 2*a^6 + 96*a^2*b^4 - 
 256*b^6 - 3*(a^7*b^3 + 8*a^5*b^5)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(27/(a^6*b^6*d 
^6) - (a^2 - 4*b^2)^3/(a^10*b^8*d^6) - (a^6 + 15*a^4*b^2 + 48*a^2*b^4 - 64*b^6 
)/(a^10*b^8*d^6))^(1/3) + 6/(a^2*b^2*d^2))*d^2*sin(d*x + c) - 1/4*sqrt(2)*sqrt 
(1/2)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*(27/(a^6*b^6*d^6) - (a^2 - 4*b^2)^3/(a^10* 
b^8*d^6) - (a^6 + 15*a^4*b^2 + 48*a^2*b^4 - 64*b^6)/(a^10*b^8*d^6))^(1/3) + 6/ 
(a^2*b^2*d^2))^2*a^9*b^7*d^5*cos(d*x + c) - 16*(a^7*b^5 - a^5*b^7)*((1/2)^(1/3 
)*(I*sqrt(3) + 1)*(27/(a^6*b^6*d^6) - (a^2 - 4*b^2)^3/(a^10*b^8*d^6) - (a^6 + 
15*a^4*b^2 + 48*a^2*b^4 - 64*b^6)/(a^10*b^8*d^6))^(1/3) + 6/(a^2*b^2*d^2))*...
 

Integral number [400] $$\int \frac{{\cos }^2(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 3.24086 (sec), size = 36403 ,normalized size = 1582.74 $$\text{Too large to display}$$

[In]

integrate(cos(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

output

1/324*(3*sqrt(2/3)*sqrt(1/6)*(a^2*d - (a*b*d*cos(d*x + c)^2 - a*b*d)*sin(d*x + 
 c))*sqrt(-((a^4 - a^2*b^2)*((-I*sqrt(3) + 1)*(3/(a^6*b^2*d^4 - a^4*b^4*d^4) - 
 1/(a^4*d^2 - a^2*b^2*d^2)^2)/(-1/1062882*(a^4 - 16*a^2*b^2 + 64*b^4)/(a^12*b^ 
4*d^6 - a^10*b^6*d^6) + 1/118098/((a^6*b^2*d^4 - a^4*b^4*d^4)*(a^4*d^2 - a^2*b 
^2*d^2)) - 1/531441/(a^4*d^2 - a^2*b^2*d^2)^3 + 1/1062882*(a^6 + 28*a^4*b^2 - 
80*a^2*b^4 + 64*b^6)/((a^2 - b^2)^2*a^10*b^4*d^6))^(1/3) - 6561*(I*sqrt(3) + 1 
)*(-1/1062882*(a^4 - 16*a^2*b^2 + 64*b^4)/(a^12*b^4*d^6 - a^10*b^6*d^6) + 1/11 
8098/((a^6*b^2*d^4 - a^4*b^4*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/531441/(a^4*d^2 
 - a^2*b^2*d^2)^3 + 1/1062882*(a^6 + 28*a^4*b^2 - 80*a^2*b^4 + 64*b^6)/((a^2 - 
 b^2)^2*a^10*b^4*d^6))^(1/3) - 162/(a^4*d^2 - a^2*b^2*d^2))*d^2 + 3*sqrt(1/3)* 
(a^4 - a^2*b^2)*d^2*sqrt(-((a^8*b^2 - 2*a^6*b^4 + a^4*b^6)*((-I*sqrt(3) + 1)*( 
3/(a^6*b^2*d^4 - a^4*b^4*d^4) - 1/(a^4*d^2 - a^2*b^2*d^2)^2)/(-1/1062882*(a^4 
- 16*a^2*b^2 + 64*b^4)/(a^12*b^4*d^6 - a^10*b^6*d^6) + 1/118098/((a^6*b^2*d^4 
- a^4*b^4*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/531441/(a^4*d^2 - a^2*b^2*d^2)^3 + 
 1/1062882*(a^6 + 28*a^4*b^2 - 80*a^2*b^4 + 64*b^6)/((a^2 - b^2)^2*a^10*b^4*d^ 
6))^(1/3) - 6561*(I*sqrt(3) + 1)*(-1/1062882*(a^4 - 16*a^2*b^2 + 64*b^4)/(a^12 
*b^4*d^6 - a^10*b^6*d^6) + 1/118098/((a^6*b^2*d^4 - a^4*b^4*d^4)*(a^4*d^2 - a^ 
2*b^2*d^2)) - 1/531441/(a^4*d^2 - a^2*b^2*d^2)^3 + 1/1062882*(a^6 + 28*a^4*b^2 
 - 80*a^2*b^4 + 64*b^6)/((a^2 - b^2)^2*a^10*b^4*d^6))^(1/3) - 162/(a^4*d^2 - a 
^2*b^2*d^2))^2*d^4 + 324*(a^4*b^2 - a^2*b^4)*((-I*sqrt(3) + 1)*(3/(a^6*b^2*...
 

Integral number [401] $$\int \frac{1}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 8.47099 (sec), size = 70185 ,normalized size = 5013.21 $$\text{Too large to display}$$

[In]

integrate(1/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

output

-1/108*(36*a*b*cos(d*x + c)^3 + 36*b^2*cos(d*x + c)*sin(d*x + c) - sqrt(2/3)*s 
qrt(1/2)*((a^4 - a^2*b^2)*d - ((a^3*b - a*b^3)*d*cos(d*x + c)^2 - (a^3*b - a*b 
^3)*d)*sin(d*x + c))*sqrt(-(1458*a^4 + 486*a^2*b^2 - 486*b^4 - (a^8 - 3*a^6*b^ 
2 + 3*a^4*b^4 - a^2*b^6)*((-I*sqrt(3) + 1)*(3*(3*a^4 + a^2*b^2 - b^4)^2/(a^8*d 
^2 - 3*a^6*b^2*d^2 + 3*a^4*b^4*d^2 - a^2*b^6*d^2)^2 - (27*a^2 - 11*b^2)/(a^10* 
d^4 - 3*a^8*b^2*d^4 + 3*a^6*b^4*d^4 - a^4*b^6*d^4))/(-1/1062882*(729*a^4 - 432 
*a^2*b^2 + 64*b^4)/(a^16*d^6 - 3*a^14*b^2*d^6 + 3*a^12*b^4*d^6 - a^10*b^6*d^6) 
 - 1/19683*(3*a^4 + a^2*b^2 - b^4)^3/(a^8*d^2 - 3*a^6*b^2*d^2 + 3*a^4*b^4*d^2 
- a^2*b^6*d^2)^3 + 1/39366*(3*a^4 + a^2*b^2 - b^4)*(27*a^2 - 11*b^2)/((a^10*d^ 
4 - 3*a^8*b^2*d^4 + 3*a^6*b^4*d^4 - a^4*b^6*d^4)*(a^8*d^2 - 3*a^6*b^2*d^2 + 3* 
a^4*b^4*d^2 - a^2*b^6*d^2)) + 1/1062882*(3375*a^8 - 4573*a^6*b^2 + 2460*a^4*b^ 
4 - 624*a^2*b^6 + 64*b^8)*b^2/((a^2 - b^2)^6*a^10*d^6))^(1/3) + 2187*(I*sqrt(3 
) + 1)*(-1/1062882*(729*a^4 - 432*a^2*b^2 + 64*b^4)/(a^16*d^6 - 3*a^14*b^2*d^6 
 + 3*a^12*b^4*d^6 - a^10*b^6*d^6) - 1/19683*(3*a^4 + a^2*b^2 - b^4)^3/(a^8*d^2 
 - 3*a^6*b^2*d^2 + 3*a^4*b^4*d^2 - a^2*b^6*d^2)^3 + 1/39366*(3*a^4 + a^2*b^2 - 
 b^4)*(27*a^2 - 11*b^2)/((a^10*d^4 - 3*a^8*b^2*d^4 + 3*a^6*b^4*d^4 - a^4*b^6*d 
^4)*(a^8*d^2 - 3*a^6*b^2*d^2 + 3*a^4*b^4*d^2 - a^2*b^6*d^2)) + 1/1062882*(3375 
*a^8 - 4573*a^6*b^2 + 2460*a^4*b^4 - 624*a^2*b^6 + 64*b^8)*b^2/((a^2 - b^2)^6* 
a^10*d^6))^(1/3) + 162*(3*a^4 + a^2*b^2 - b^4)/(a^8*d^2 - 3*a^6*b^2*d^2 + 3*a^ 
4*b^4*d^2 - a^2*b^6*d^2))*d^2 + 3*sqrt(1/3)*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - ...
 

Integral number [402] $$\int \frac{{\sec }^2(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 43.884 (sec), size = 102913 ,normalized size = 4474.48 $$\text{Too large to display}$$

[In]

integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

output

1/108*(108*(a^3*b + 2*a*b^3)*cos(d*x + c)^4 - 108*a^3*b + 108*a*b^3 - sqrt(2)* 
sqrt(1/2)*((a^6 - 2*a^4*b^2 + a^2*b^4)*d*cos(d*x + c) - ((a^5*b - 2*a^3*b^3 + 
a*b^5)*d*cos(d*x + c)^3 - (a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*x + c))*sin(d*x 
+ c))*sqrt(-(5670*a^6*b^2 + 31590*a^4*b^4 + 2916*a^2*b^6 - 810*b^8 - (a^12 - 5 
*a^10*b^2 + 10*a^8*b^4 - 10*a^6*b^6 + 5*a^4*b^8 - a^2*b^10)*((-I*sqrt(3) + 1)* 
((35*a^6*b^2 + 195*a^4*b^4 + 18*a^2*b^6 - 5*b^8)^2/(a^12*d^2 - 5*a^10*b^2*d^2 
+ 10*a^8*b^4*d^2 - 10*a^6*b^6*d^2 + 5*a^4*b^8*d^2 - a^2*b^10*d^2)^2 - 45*(10*a 
^2*b^4 - b^6)/(a^14*d^4 - 5*a^12*b^2*d^4 + 10*a^10*b^4*d^4 - 10*a^8*b^6*d^4 + 
5*a^6*b^8*d^4 - a^4*b^10*d^4))/(-1/19683*(35*a^6*b^2 + 195*a^4*b^4 + 18*a^2*b^ 
6 - 5*b^8)^3/(a^12*d^2 - 5*a^10*b^2*d^2 + 10*a^8*b^4*d^2 - 10*a^6*b^6*d^2 + 5* 
a^4*b^8*d^2 - a^2*b^10*d^2)^3 - 1/1062882*(15625*a^4*b^4 - 2000*a^2*b^6 + 64*b 
^8)/(a^20*d^6 - 5*a^18*b^2*d^6 + 10*a^16*b^4*d^6 - 10*a^14*b^6*d^6 + 5*a^12*b^ 
8*d^6 - a^10*b^10*d^6) + 5/1458*(35*a^6*b^2 + 195*a^4*b^4 + 18*a^2*b^6 - 5*b^8 
)*(10*a^2*b^4 - b^6)/((a^14*d^4 - 5*a^12*b^2*d^4 + 10*a^10*b^4*d^4 - 10*a^8*b^ 
6*d^4 + 5*a^6*b^8*d^4 - a^4*b^10*d^4)*(a^12*d^2 - 5*a^10*b^2*d^2 + 10*a^8*b^4* 
d^2 - 10*a^6*b^6*d^2 + 5*a^4*b^8*d^2 - a^2*b^10*d^2)) - 1/1062882*(15625*a^14 
+ 959375*a^12*b^2 + 24861*a^10*b^4 - 1094705*a^8*b^6 + 307475*a^6*b^8 - 37740* 
a^4*b^10 + 2320*a^2*b^12 - 64*b^14)*b^4/((a^2 - b^2)^10*a^10*d^6))^(1/3) + 729 
*(I*sqrt(3) + 1)*(-1/19683*(35*a^6*b^2 + 195*a^4*b^4 + 18*a^2*b^6 - 5*b^8)^3/( 
a^12*d^2 - 5*a^10*b^2*d^2 + 10*a^8*b^4*d^2 - 10*a^6*b^6*d^2 + 5*a^4*b^8*d^2...
 

Integral number [403] $$\int \frac{{\sec }^4(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 140.116 (sec), size = 133123 ,normalized size = 5787.96 $$\text{Too large to display}$$

[In]

integrate(sec(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

output

1/108*(36*(2*a^5*b - 30*a^3*b^3 - 17*a*b^5)*cos(d*x + c)^6 - 36*a^5*b + 72*a^3 
*b^3 - 36*a*b^5 - 108*(a^5*b - 21*a^3*b^3 - 10*a*b^5)*cos(d*x + c)^4 + sqrt(2/ 
3)*sqrt(1/6)*((a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*cos(d*x + c)^3 - ((a^7 
*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*cos(d*x + c)^5 - (a^7*b - 3*a^5*b^3 + 3* 
a^3*b^5 - a*b^7)*d*cos(d*x + c)^3)*sin(d*x + c))*sqrt(-(573480*a^8*b^4 + 42933 
24*a^6*b^6 + 3847662*a^4*b^8 + 159894*a^2*b^10 - 17010*b^12 - (a^16 - 7*a^14*b 
^2 + 21*a^12*b^4 - 35*a^10*b^6 + 35*a^8*b^8 - 21*a^6*b^10 + 7*a^4*b^12 - a^2*b 
^14)*((-I*sqrt(3) + 1)*((1180*a^8*b^4 + 8834*a^6*b^6 + 7917*a^4*b^8 + 329*a^2* 
b^10 - 35*b^12)^2/(a^16*d^2 - 7*a^14*b^2*d^2 + 21*a^12*b^4*d^2 - 35*a^10*b^6*d 
^2 + 35*a^8*b^8*d^2 - 21*a^6*b^10*d^2 + 7*a^4*b^12*d^2 - a^2*b^14*d^2)^2 + 15* 
(1029*a^4*b^6 - 3173*a^2*b^8 + 119*b^10)/(a^18*d^4 - 7*a^16*b^2*d^4 + 21*a^14* 
b^4*d^4 - 35*a^12*b^6*d^4 + 35*a^10*b^8*d^4 - 21*a^8*b^10*d^4 + 7*a^6*b^12*d^4 
 - a^4*b^14*d^4))/(-1/531441*(1180*a^8*b^4 + 8834*a^6*b^6 + 7917*a^4*b^8 + 329 
*a^2*b^10 - 35*b^12)^3/(a^16*d^2 - 7*a^14*b^2*d^2 + 21*a^12*b^4*d^2 - 35*a^10* 
b^6*d^2 + 35*a^8*b^8*d^2 - 21*a^6*b^10*d^2 + 7*a^4*b^12*d^2 - a^2*b^14*d^2)^3 
- 1/1062882*(117649*a^4*b^8 - 5488*a^2*b^10 + 64*b^12)/(a^24*d^6 - 7*a^22*b^2* 
d^6 + 21*a^20*b^4*d^6 - 35*a^18*b^6*d^6 + 35*a^16*b^8*d^6 - 21*a^14*b^10*d^6 + 
 7*a^12*b^12*d^6 - a^10*b^14*d^6) - 5/118098*(1180*a^8*b^4 + 8834*a^6*b^6 + 79 
17*a^4*b^8 + 329*a^2*b^10 - 35*b^12)*(1029*a^4*b^6 - 3173*a^2*b^8 + 119*b^10)/ 
((a^18*d^4 - 7*a^16*b^2*d^4 + 21*a^14*b^4*d^4 - 35*a^12*b^6*d^4 + 35*a^10*b...
 

4.5.3 Mupad

Integral number [399] $$\int \frac{{\cos }^4(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[B]   time = 15.0511 (sec), size = -1 ,normalized size = -0.04 $$\text{Too large to display}$$

[In]

int(cos(c + d*x)^4/(a + b*sin(c + d*x)^3)^2,x)

output

2/(3*d*(a*b + 8*b^2*tan(c/2 + (d*x)/2)^3 + 3*a*b*tan(c/2 + (d*x)/2)^2 + 3*a*b* 
tan(c/2 + (d*x)/2)^4 + a*b*tan(c/2 + (d*x)/2)^6)) + symsum(log((638976*a^2*b^4 
 - 655360*b^6 - 8192*a^6 + 24576*a^4*b^2 - 2949120*root(531441*a^10*b^8*d^6 + 
59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, 
 d, k)*a^3*b^5 + 2138112*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a 
^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)*a^5*b^3 - 9437184*r 
oot(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 
15*a^4*b^2 + a^6 - 64*b^6, d, k)*b^8*tan(c/2 + (d*x)/2) - 786432*a*b^5*tan(c/2 
 + (d*x)/2) + 98304*a^5*b*tan(c/2 + (d*x)/2) - 21233664*root(531441*a^10*b^8*d 
^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64 
*b^6, d, k)^2*a^2*b^8 + 18579456*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 
+ 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^2*a^4*b^6 + 
 2654208*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48* 
a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^2*a^6*b^4 - 167215104*root(531441*a 
^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + 
 a^6 - 64*b^6, d, k)^3*a^5*b^7 + 113467392*root(531441*a^10*b^8*d^6 + 59049*a^ 
8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^3 
*a^7*b^5 - 107495424*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b 
^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^4*a^6*b^8 + 107495424*r 
oot(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4...
 

Integral number [400] $$\int \frac{{\cos }^2(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[B]   time = 15.2435 (sec), size = -1 ,normalized size = -0.04 $$\text{Too large to display}$$

[In]

int(cos(c + d*x)^2/(a + b*sin(c + d*x)^3)^2,x)

output

symsum(log(-((131072*b^2)/243 - (16384*a^2)/243 + (8192*root(531441*a^12*b^4*d 
^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + 
a^4 + 64*b^4, d, k)*a^4*tan(c/2 + (d*x)/2))/27 + (1048576*root(531441*a^12*b^4 
*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 
+ a^4 + 64*b^4, d, k)*b^4*tan(c/2 + (d*x)/2))/27 + (262144*root(531441*a^12*b^ 
4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 
 + a^4 + 64*b^4, d, k)^2*a^2*b^4)/3 - (131072*root(531441*a^12*b^4*d^6 - 53144 
1*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b 
^4, d, k)^2*a^4*b^2)/3 - 98304*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 
+ 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^3*a^5 
*b^3 + 442368*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d 
^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^4*a^6*b^4 + 221184*roo 
t(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2* 
d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^4*a^8*b^2 + 7962624*root(531441*a^12*b^ 
4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 
 + a^4 + 64*b^4, d, k)^5*a^7*b^5 - 5971968*root(531441*a^12*b^4*d^6 - 531441*a 
^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, 
 d, k)^5*a^9*b^3 + (131072*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19 
683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)*a*b^3)/27 
 - (65536*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^...
 

Integral number [401] $$\int \frac{1}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[B]   time = 17.0233 (sec), size = -1 ,normalized size = -0.07 $$\text{Too large to display}$$

[In]

int(1/(a + b*sin(c + d*x)^3)^2,x)

output

symsum(log(- (8192*(80*b^6 - 270*a^2*b^4))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) - 
 root(1594323*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 5314 
41*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019* 
a^6*b^2*d^2 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*((8192*(14 
4*a*b^7 + 648*a^3*b^5 - 2187*a^5*b^3))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) - roo 
t(1594323*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 531441*a 
^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6* 
b^2*d^2 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*(root(1594323* 
a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 531441*a^16*d^6 - 
59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*b^2*d^2 - 
19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*((8192*(26973*a^7*b^5 - 
20412*a^5*b^7 + 39366*a^9*b^3))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) - root(15943 
23*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 531441*a^16*d^6 
 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*b^2*d^2 
 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*(root(1594323*a^14*b^ 
2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 531441*a^16*d^6 - 59049*a 
^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*b^2*d^2 - 19683*a 
^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*((8192*(236196*a^7*b^9 - 649539 
*a^9*b^7 + 590490*a^11*b^5 - 177147*a^13*b^3))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2 
)) + (8192*tan(c/2 + (d*x)/2)*(6561*a^8*b^8 - 13122*a^10*b^6 + 6561*a^12*b^...
 

Integral number [402] $$\int \frac{{\sec }^2(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[B]   time = 19.147 (sec), size = -1 ,normalized size = -0.04 $$\text{Too large to display}$$

[In]

int(1/(cos(c + d*x)^2*(a + b*sin(c + d*x)^3)^2),x)

output

symsum(log(5479612416*a^8*b^36 - 180486144*a^6*b^38 - root(5314410*a^16*b^4*d^ 
6 - 5314410*a^14*b^6*d^6 - 2657205*a^18*b^2*d^6 + 2657205*a^12*b^8*d^6 - 53144 
1*a^10*b^10*d^6 + 531441*a^20*d^6 + 11514555*a^12*b^4*d^4 + 2066715*a^14*b^2*d 
^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 984150*a^8*b^4*d^2 - 98415*a^ 
6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 64*b^8, d, k)*(tan(c/2 + (d*x)/2)*( 
764411904*a^6*b^40 - 27805483008*a^8*b^38 + 437297356800*a^10*b^36 - 367246172 
1600*a^12*b^34 + 19250011791360*a^14*b^32 - 69150635753472*a^16*b^30 + 1801658 
72001024*a^18*b^28 - 352655758540800*a^20*b^26 + 529923028377600*a^22*b^24 - 6 
18699706859520*a^24*b^22 + 563713761042432*a^26*b^20 - 399760062234624*a^28*b^ 
18 + 218398602240000*a^30*b^16 - 90108039168000*a^32*b^14 + 27130620764160*a^3 
4*b^12 - 5617221156864*a^36*b^10 + 713536708608*a^38*b^8 - 41803776000*a^40*b^ 
6) - root(5314410*a^16*b^4*d^6 - 5314410*a^14*b^6*d^6 - 2657205*a^18*b^2*d^6 + 
 2657205*a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*a^20*d^6 + 11514555*a^12 
*b^4*d^4 + 2066715*a^14*b^2*d^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 
984150*a^8*b^4*d^2 - 98415*a^6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 64*b^8 
, d, k)*(root(5314410*a^16*b^4*d^6 - 5314410*a^14*b^6*d^6 - 2657205*a^18*b^2*d 
^6 + 2657205*a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*a^20*d^6 + 11514555* 
a^12*b^4*d^4 + 2066715*a^14*b^2*d^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^ 
4 + 984150*a^8*b^4*d^2 - 98415*a^6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 64 
*b^8, d, k)*(tan(c/2 + (d*x)/2)*(157695787008*a^12*b^38 - 4039140556800*a^1...
 

Integral number [403] $$\int \frac{{\sec }^4(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[B]   time = 23.8272 (sec), size = -1 ,normalized size = -0.04 $$\text{Too large to display}$$

[In]

int(1/(cos(c + d*x)^4*(a + b*sin(c + d*x)^3)^2),x)

output

symsum(log(26838024192*a^8*b^54 - tan(c/2 + (d*x)/2)*(7962624000*a^7*b^55 - 50 
8612608000*a^9*b^53 + 8841498624000*a^11*b^51 - 82283765760000*a^13*b^49 + 501 
714984960000*a^15*b^47 - 2205295497216000*a^17*b^45 + 7379181637632000*a^19*b^ 
43 - 19451488075776000*a^21*b^41 + 41318016122880000*a^23*b^39 - 7181143216128 
0000*a^25*b^37 + 103155513237504000*a^27*b^35 - 123224906907648000*a^29*b^33 + 
 122756816093184000*a^31*b^31 - 101967282708480000*a^33*b^29 + 703968720076800 
00*a^35*b^27 - 40129785593856000*a^37*b^25 + 18687625592832000*a^39*b^23 - 699 
4754113536000*a^41*b^21 + 2053854351360000*a^43*b^19 - 455730831360000*a^45*b^ 
17 + 71860690944000*a^47*b^15 - 7177310208000*a^49*b^13 + 341397504000*a^51*b^ 
11) - 392822784*a^6*b^56 - root(18600435*a^18*b^6*d^6 - 18600435*a^16*b^8*d^6 
- 11160261*a^20*b^4*d^6 + 11160261*a^14*b^10*d^6 + 3720087*a^22*b^2*d^6 - 3720 
087*a^12*b^12*d^6 + 531441*a^10*b^14*d^6 - 531441*a^24*d^6 - 173879622*a^14*b^ 
6*d^4 - 155830311*a^12*b^8*d^4 - 23225940*a^16*b^4*d^4 - 6475707*a^10*b^10*d^4 
 + 688905*a^8*b^12*d^4 - 11565585*a^8*b^8*d^2 + 3750705*a^10*b^6*d^2 + 433755* 
a^6*b^10*d^2 - 117649*a^4*b^8 + 5488*a^2*b^10 - 64*b^12, d, k)*(tan(c/2 + (d*x 
)/2)*(764411904*a^6*b^58 - 61439606784*a^8*b^56 + 2110475575296*a^10*b^54 - 33 
643637121024*a^12*b^52 + 319697763065856*a^14*b^50 - 2067381036048384*a^16*b^4 
8 + 9810082122817536*a^18*b^46 - 35797302942326784*a^20*b^44 + 103613766013034 
496*a^22*b^42 - 243004699498881024*a^24*b^40 + 468678655511248896*a^26*b^38 - 
750973819695611904*a^28*b^36 + 1006348379003928576*a^30*b^34 - 113202827820...
 

4.6 Test file Number [151]

4.6.1 Mathematica

Integral number [65] $$\int \frac{\arctan (a+bx)}{\sqrt[3]{1+a^2+2abx+b^2{x}^2}}dx$$

[B]   time = 0.31135 (sec), size = 163 ,normalized size = 5.82 $$\frac{6\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)(15+10(a+bx)\arctan (a+bx)+\frac{4(a+bx)\arctan (a+bx)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{4}{3},\frac{11}{6},\frac{1}{1+(a+bx{)}^2})}{1+(a+bx{)}^2})+\frac{5\sqrt[3]{2}\sqrt{\pi }\mathrm{Gamma}\left(\frac{5}{3}\right){}_3{F}_2(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{1+(a+bx{)}^2})}{1+(a+bx{)}^2}}{20b\sqrt[3]{1+a^2+2abx+b^2{x}^2}\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)}$$

[In]

Integrate[ArcTan[a + b*x]/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]

output

(6*Gamma[11/6]*Gamma[7/3]*(15 + 10*(a + b*x)*ArcTan[a + b*x] + (4*(a + b*x)*Ar 
cTan[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + (a + b*x)^2)^(-1)])/(1 + (a 
 + b*x)^2)) + (5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, 
{11/6, 7/3}, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2))/(20*b*(1 + a^2 + 2*a* 
b*x + b^2*x^2)^(1/3)*Gamma[11/6]*Gamma[7/3])
 

Integral number [66] $$\int \frac{\arctan (a+bx)}{\sqrt[3]{(1+a^2)c+2abcx+b^{2}c{x}^2}}dx$$

[B]   time = 0.151768 (sec), size = 165 ,normalized size = 5. $$\frac{6\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)(15+10(a+bx)\arctan (a+bx)+\frac{4(a+bx)\arctan (a+bx)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{4}{3},\frac{11}{6},\frac{1}{1+(a+bx{)}^2})}{1+(a+bx{)}^2})+\frac{5\sqrt[3]{2}\sqrt{\pi }\mathrm{Gamma}\left(\frac{5}{3}\right){}_3{F}_2(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{1+(a+bx{)}^2})}{1+(a+bx{)}^2}}{20b\sqrt[3]{c(1+a^2+2abx+b^2{x}^2)}\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)}$$

[In]

Integrate[ArcTan[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]

output

(6*Gamma[11/6]*Gamma[7/3]*(15 + 10*(a + b*x)*ArcTan[a + b*x] + (4*(a + b*x)*Ar 
cTan[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + (a + b*x)^2)^(-1)])/(1 + (a 
 + b*x)^2)) + (5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, 
{11/6, 7/3}, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2))/(20*b*(c*(1 + a^2 + 2 
*a*b*x + b^2*x^2))^(1/3)*Gamma[11/6]*Gamma[7/3])
 

Integral number [69] $$\int \frac{(a+bx{)}^2\arctan (a+bx)}{\sqrt[3]{1+a^2+2abx+b^2{x}^2}}dx$$

[B]   time = 4.75302 (sec), size = 181 ,normalized size = 5.17 $$-\frac{3{(1+(a+bx{)}^2)}^{2/3}(\frac{5\sqrt[3]{2}\sqrt{\pi }\mathrm{Gamma}\left(\frac{5}{3}\right){}_3{F}_2(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{1+(a+bx{)}^2})}{{(1+(a+bx{)}^2)}^2}+\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)(15+\frac{90}{1+(a+bx{)}^2}+\frac{24(a+bx)\arctan (a+bx)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{4}{3},\frac{11}{6},\frac{1}{1+(a+bx{)}^2})}{{(1+(a+bx{)}^2)}^2}+5\arctan (a+bx)(-4(a+bx)+6\sin (2\arctan (a+bx)))))}{140b\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)}$$

[In]

Integrate[((a + b*x)^2*ArcTan[a + b*x])/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]

output

(-3*(1 + (a + b*x)^2)^(2/3)*((5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[ 
{1, 4/3, 4/3}, {11/6, 7/3}, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2)^2 + Gam 
ma[11/6]*Gamma[7/3]*(15 + 90/(1 + (a + b*x)^2) + (24*(a + b*x)*ArcTan[a + b*x] 
*Hypergeometric2F1[1, 4/3, 11/6, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2)^2 
+ 5*ArcTan[a + b*x]*(-4*(a + b*x) + 6*Sin[2*ArcTan[a + b*x]]))))/(140*b*Gamma[ 
11/6]*Gamma[7/3])
 

Integral number [70] $$\int \frac{(a+bx{)}^2\arctan (a+bx)}{\sqrt[3]{(1+a^2)c+2abcx+b^{2}c{x}^2}}dx$$

[B]   time = 0.727785 (sec), size = 225 ,normalized size = 5.62 $$-\frac{3\sqrt[3]{1+a^2+2abx+b^2{x}^2}{(1+(a+bx{)}^2)}^{2/3}(\frac{5\sqrt[3]{2}\sqrt{\pi }\mathrm{Gamma}\left(\frac{5}{3}\right){}_3{F}_2(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{1+(a+bx{)}^2})}{{(1+(a+bx{)}^2)}^2}+\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)(15+\frac{90}{1+(a+bx{)}^2}+\frac{24(a+bx)\arctan (a+bx)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{4}{3},\frac{11}{6},\frac{1}{1+(a+bx{)}^2})}{{(1+(a+bx{)}^2)}^2}+5\arctan (a+bx)(-4(a+bx)+6\sin (2\arctan (a+bx)))))}{140b\sqrt[3]{c(1+a^2+2abx+b^2{x}^2)}\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)}$$

[In]

Integrate[((a + b*x)^2*ArcTan[a + b*x])/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]

output

(-3*(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3)*(1 + (a + b*x)^2)^(2/3)*((5*2^(1/3)*Sq 
rt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + (a + b*x) 
^2)^(-1)])/(1 + (a + b*x)^2)^2 + Gamma[11/6]*Gamma[7/3]*(15 + 90/(1 + (a + b*x 
)^2) + (24*(a + b*x)*ArcTan[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + (a + 
 b*x)^2)^(-1)])/(1 + (a + b*x)^2)^2 + 5*ArcTan[a + b*x]*(-4*(a + b*x) + 6*Sin[ 
2*ArcTan[a + b*x]]))))/(140*b*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))^(1/3)*Gamma[11 
/6]*Gamma[7/3])
 

4.7 Test file Number [154]

4.7.1 Mathematica

Integral number [116] $$\int \frac{{\cot }^{-1}(a+bx)}{\sqrt[3]{1+a^2+2abx+b^2{x}^2}}dx$$

[B]   time = 0.30209 (sec), size = 177 ,normalized size = 6.32 $$\frac{6\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)(5(1+a^2+2abx+b^2{x}^2)(-3+2(a+bx){\cot }^{-1}(a+bx))+4(a+bx){\cot }^{-1}(a+bx)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{4}{3},\frac{11}{6},\frac{1}{1+a^2+2abx+b^2{x}^2}))-5\sqrt[3]{2}\sqrt{\pi }\mathrm{Gamma}\left(\frac{5}{3}\right){}_3{F}_2(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{1+a^2+2abx+b^2{x}^2})}{20b{(1+a^2+2abx+b^2{x}^2)}^{4/3}\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)}$$

[In]

Integrate[ArcCot[a + b*x]/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]

output

(6*Gamma[11/6]*Gamma[7/3]*(5*(1 + a^2 + 2*a*b*x + b^2*x^2)*(-3 + 2*(a + b*x)*A 
rcCot[a + b*x]) + 4*(a + b*x)*ArcCot[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, 
(1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) - 5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*Hypergeom 
etricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)])/(20* 
b*(1 + a^2 + 2*a*b*x + b^2*x^2)^(4/3)*Gamma[11/6]*Gamma[7/3])
 

Integral number [117] $$\int \frac{{\cot }^{-1}(a+bx)}{\sqrt[3]{(1+a^2)c+2abcx+b^{2}c{x}^2}}dx$$

[B]   time = 0.141477 (sec), size = 180 ,normalized size = 5.45 $$\frac{c(6\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)(5(1+a^2+2abx+b^2{x}^2)(-3+2(a+bx){\cot }^{-1}(a+bx))+4(a+bx){\cot }^{-1}(a+bx)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{4}{3},\frac{11}{6},\frac{1}{1+a^2+2abx+b^2{x}^2}))-5\sqrt[3]{2}\sqrt{\pi }\mathrm{Gamma}\left(\frac{5}{3}\right){}_3{F}_2(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{1+a^2+2abx+b^2{x}^2}))}{20b{\left(c(1+a^2+2abx+b^2{x}^2)\right)}^{4/3}\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)}$$

[In]

Integrate[ArcCot[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]

output

(c*(6*Gamma[11/6]*Gamma[7/3]*(5*(1 + a^2 + 2*a*b*x + b^2*x^2)*(-3 + 2*(a + b*x 
)*ArcCot[a + b*x]) + 4*(a + b*x)*ArcCot[a + b*x]*Hypergeometric2F1[1, 4/3, 11/ 
6, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) - 5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*Hyperg 
eometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]))/ 
(20*b*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))^(4/3)*Gamma[11/6]*Gamma[7/3])
 

Integral number [120] $$\int \frac{(a+bx{)}^2{\cot }^{-1}(a+bx)}{\sqrt[3]{1+a^2+2abx+b^2{x}^2}}dx$$

[B]   time = 0.740671 (sec), size = 198 ,normalized size = 5.66 $$\frac{3(\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)(5(1+(a+bx{)}^2)(3(7+(a+bx{)}^2)+4(a+bx)(-2+(a+bx{)}^2){\cot }^{-1}(a+bx))-24(a+bx){\cot }^{-1}(a+bx)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{4}{3},\frac{11}{6},\frac{1}{1+a^2+2abx+b^2{x}^2}))+5\sqrt[3]{2}\sqrt{\pi }\mathrm{Gamma}\left(\frac{5}{3}\right){}_3{F}_2(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{1+a^2+2abx+b^2{x}^2}))}{140b\sqrt[3]{1+a^2+2abx+b^2{x}^2}(1+(a+bx{)}^2)\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)}$$

[In]

Integrate[((a + b*x)^2*ArcCot[a + b*x])/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3),x]

output

(3*(Gamma[11/6]*Gamma[7/3]*(5*(1 + (a + b*x)^2)*(3*(7 + (a + b*x)^2) + 4*(a + 
b*x)*(-2 + (a + b*x)^2)*ArcCot[a + b*x]) - 24*(a + b*x)*ArcCot[a + b*x]*Hyperg 
eometric2F1[1, 4/3, 11/6, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) + 5*2^(1/3)*Sqr 
t[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a* 
b*x + b^2*x^2)^(-1)]))/(140*b*(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3)*(1 + (a + b* 
x)^2)*Gamma[11/6]*Gamma[7/3])
 

Integral number [121] $$\int \frac{(a+bx{)}^2{\cot }^{-1}(a+bx)}{\sqrt[3]{(1+a^2)c+2abcx+b^{2}c{x}^2}}dx$$

[B]   time = 0.321534 (sec), size = 200 ,normalized size = 5. $$\frac{3(\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)(5(1+(a+bx{)}^2)(3(7+(a+bx{)}^2)+4(a+bx)(-2+(a+bx{)}^2){\cot }^{-1}(a+bx))-24(a+bx){\cot }^{-1}(a+bx)\mathrm{Hypergeometric}2\mathrm{F}1(1,\frac{4}{3},\frac{11}{6},\frac{1}{1+a^2+2abx+b^2{x}^2}))+5\sqrt[3]{2}\sqrt{\pi }\mathrm{Gamma}\left(\frac{5}{3}\right){}_3{F}_2(1,\frac{4}{3},\frac{4}{3};\frac{11}{6},\frac{7}{3};\frac{1}{1+a^2+2abx+b^2{x}^2}))}{140b\sqrt[3]{c(1+a^2+2abx+b^2{x}^2)}(1+(a+bx{)}^2)\mathrm{Gamma}\left(\frac{11}{6}\right)\mathrm{Gamma}\left(\frac{7}{3}\right)}$$

[In]

Integrate[((a + b*x)^2*ArcCot[a + b*x])/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]

output

(3*(Gamma[11/6]*Gamma[7/3]*(5*(1 + (a + b*x)^2)*(3*(7 + (a + b*x)^2) + 4*(a + 
b*x)*(-2 + (a + b*x)^2)*ArcCot[a + b*x]) - 24*(a + b*x)*ArcCot[a + b*x]*Hyperg 
eometric2F1[1, 4/3, 11/6, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) + 5*2^(1/3)*Sqr 
t[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a* 
b*x + b^2*x^2)^(-1)]))/(140*b*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))^(1/3)*(1 + (a 
+ b*x)^2)*Gamma[11/6]*Gamma[7/3])
 

4.8 Test file Number [173]

4.8.1 Mathematica

Integral number [74] $$\int \frac{{\sinh }^3(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 1.06392 (sec), size = 826 ,normalized size = 35.91 $$\frac{-9a(a^2+3b^2)\cosh (c+dx)+a^3\cosh (3(c+dx))-a{b}^2\cosh (3(c+dx))-2ab\text{RootSum}[a-b+3a{\mathrm{\#}\text{1}}^2+3b{\mathrm{\#}\text{1}}^2+3a{\mathrm{\#}\text{1}}^4-3b{\mathrm{\#}\text{1}}^4+a{\mathrm{\#}\text{1}}^6+b{\mathrm{\#}\text{1}}^6\mathrm{\&},\frac{3a^{2}c+3abc+3b^{2}c+3a^{2}dx+3abdx+3b^{2}dx+6a^2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})+6ab\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})+6b^2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})+2a^{2}c{\mathrm{\#}\text{1}}^2-2b^{2}c{\mathrm{\#}\text{1}}^2+2a^{2}dx{\mathrm{\#}\text{1}}^2-2b^{2}dx{\mathrm{\#}\text{1}}^2+4a^2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^2-4b^2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^2+3a^{2}c{\mathrm{\#}\text{1}}^4-3abc{\mathrm{\#}\text{1}}^4+3b^{2}c{\mathrm{\#}\text{1}}^4+3a^{2}dx{\mathrm{\#}\text{1}}^4-3abdx{\mathrm{\#}\text{1}}^4+3b^{2}dx{\mathrm{\#}\text{1}}^4+6a^2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4-6ab\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4+6b^2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4}{a\mathrm{\#}\text{1}+b\mathrm{\#}\text{1}+2a{\mathrm{\#}\text{1}}^3-2b{\mathrm{\#}\text{1}}^3+a{\mathrm{\#}\text{1}}^5+b{\mathrm{\#}\text{1}}^5}\mathrm{\&}]+27a^{2}b\sinh (c+dx)+9b^3\sinh (c+dx)-a^{2}b\sinh (3(c+dx))+b^3\sinh (3(c+dx))}{12(a-b{)}^2(a+b{)}^{2}d}$$

[In]

Integrate[Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^3),x]

output

(-9*a*(a^2 + 3*b^2)*Cosh[c + d*x] + a^3*Cosh[3*(c + d*x)] - a*b^2*Cosh[3*(c + 
d*x)] - 2*a*b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1 
^6 + b*#1^6 & , (3*a^2*c + 3*a*b*c + 3*b^2*c + 3*a^2*d*x + 3*a*b*d*x + 3*b^2*d 
*x + 6*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - 
 Sinh[(c + d*x)/2]*#1] + 6*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Co 
sh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 6*b^2*Log[-Cosh[(c + d*x)/2] - Si 
nh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2*a^2*c*#1^2 
- 2*b^2*c*#1^2 + 2*a^2*d*x*#1^2 - 2*b^2*d*x*#1^2 + 4*a^2*Log[-Cosh[(c + d*x)/2 
] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 4* 
b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[( 
c + d*x)/2]*#1]*#1^2 + 3*a^2*c*#1^4 - 3*a*b*c*#1^4 + 3*b^2*c*#1^4 + 3*a^2*d*x* 
#1^4 - 3*a*b*d*x*#1^4 + 3*b^2*d*x*#1^4 + 6*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[( 
c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 6*a*b*Log[-C 
osh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2 
]*#1]*#1^4 + 6*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x) 
/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1 
^5 + b*#1^5) & ] + 27*a^2*b*Sinh[c + d*x] + 9*b^3*Sinh[c + d*x] - a^2*b*Sinh[3 
*(c + d*x)] + b^3*Sinh[3*(c + d*x)])/(12*(a - b)^2*(a + b)^2*d)
 

Integral number [76] $$\int \frac{\sinh (c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 0.718557 (sec), size = 409 ,normalized size = 19.48 $$\frac{6a\cosh (c+dx)+b\text{RootSum}[a-b+3a{\mathrm{\#}\text{1}}^2+3b{\mathrm{\#}\text{1}}^2+3a{\mathrm{\#}\text{1}}^4-3b{\mathrm{\#}\text{1}}^4+a{\mathrm{\#}\text{1}}^6+b{\mathrm{\#}\text{1}}^6\mathrm{\&},\frac{2ac+bc+2adx+bdx+4a\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})+2b\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})+2ac{\mathrm{\#}\text{1}}^4-bc{\mathrm{\#}\text{1}}^4+2adx{\mathrm{\#}\text{1}}^4-bdx{\mathrm{\#}\text{1}}^4+4a\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4-2b\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4}{a\mathrm{\#}\text{1}+b\mathrm{\#}\text{1}+2a{\mathrm{\#}\text{1}}^3-2b{\mathrm{\#}\text{1}}^3+a{\mathrm{\#}\text{1}}^5+b{\mathrm{\#}\text{1}}^5}\mathrm{\&}]-6b\sinh (c+dx)}{6(a-b)(a+b)d}$$

[In]

Integrate[Sinh[c + d*x]/(a + b*Tanh[c + d*x]^3),x]

output

(6*a*Cosh[c + d*x] + b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1 
^4 + a*#1^6 + b*#1^6 & , (2*a*c + b*c + 2*a*d*x + b*d*x + 4*a*Log[-Cosh[(c + d 
*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2* 
b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c 
+ d*x)/2]*#1] + 2*a*c*#1^4 - b*c*#1^4 + 2*a*d*x*#1^4 - b*d*x*#1^4 + 4*a*Log[-C 
osh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2 
]*#1]*#1^4 - 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2 
]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1^5 
 + b*#1^5) & ] - 6*b*Sinh[c + d*x])/(6*(a - b)*(a + b)*d)
 

Integral number [77] $$\int \frac{\text{csch}(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 0.62824 (sec), size = 331 ,normalized size = 15.76 $$-\frac{6\log (\cosh (\frac{1}{2}(c+dx)))-6\log (\sinh (\frac{1}{2}(c+dx)))+b\text{RootSum}[a-b+3a{\mathrm{\#}\text{1}}^2+3b{\mathrm{\#}\text{1}}^2+3a{\mathrm{\#}\text{1}}^4-3b{\mathrm{\#}\text{1}}^4+a{\mathrm{\#}\text{1}}^6+b{\mathrm{\#}\text{1}}^6\mathrm{\&},\frac{c+dx+2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})-2c{\mathrm{\#}\text{1}}^2-2dx{\mathrm{\#}\text{1}}^2-4\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^2+c{\mathrm{\#}\text{1}}^4+dx{\mathrm{\#}\text{1}}^4+2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4}{a\mathrm{\#}\text{1}+b\mathrm{\#}\text{1}+2a{\mathrm{\#}\text{1}}^3-2b{\mathrm{\#}\text{1}}^3+a{\mathrm{\#}\text{1}}^5+b{\mathrm{\#}\text{1}}^5}\mathrm{\&}]}{6ad}$$

[In]

Integrate[Csch[c + d*x]/(a + b*Tanh[c + d*x]^3),x]

output

-1/6*(6*Log[Cosh[(c + d*x)/2]] - 6*Log[Sinh[(c + d*x)/2]] + b*RootSum[a - b + 
3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + b*#1^6 & , (c + d*x + 2*L 
og[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d 
*x)/2]*#1] - 2*c*#1^2 - 2*d*x*#1^2 - 4*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x) 
/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + c*#1^4 + d*x*#1^4 + 
2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c 
+ d*x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1^5 + b*#1^5) & ]) 
/(a*d)
 

Integral number [79] $$\int \frac{{\text{csch}}^3(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 0.918625 (sec), size = 214 ,normalized size = 9.3 $$-\frac{16b\text{RootSum}[a-b+3a{\mathrm{\#}\text{1}}^2+3b{\mathrm{\#}\text{1}}^2+3a{\mathrm{\#}\text{1}}^4-3b{\mathrm{\#}\text{1}}^4+a{\mathrm{\#}\text{1}}^6+b{\mathrm{\#}\text{1}}^6\mathrm{\&},\frac{c\mathrm{\#}\text{1}+dx\mathrm{\#}\text{1}+2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})\mathrm{\#}\text{1}}{a+b+2a{\mathrm{\#}\text{1}}^2-2b{\mathrm{\#}\text{1}}^2+a{\mathrm{\#}\text{1}}^4+b{\mathrm{\#}\text{1}}^4}\mathrm{\&}]+3({\text{csch}}^2(\frac{1}{2}(c+dx))-4\log (\cosh (\frac{1}{2}(c+dx)))+4\log (\sinh (\frac{1}{2}(c+dx)))+{\text{sech}}^2(\frac{1}{2}(c+dx)))}{24ad}$$

[In]

Integrate[Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^3),x]

output

-1/24*(16*b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 
 + b*#1^6 & , (c*#1 + d*x*#1 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + 
Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1)/(a + b + 2*a*#1^2 - 2*b*#1^2 
+ a*#1^4 + b*#1^4) & ] + 3*(Csch[(c + d*x)/2]^2 - 4*Log[Cosh[(c + d*x)/2]] + 4 
*Log[Sinh[(c + d*x)/2]] + Sech[(c + d*x)/2]^2))/(a*d)
 

4.8.2 Fricas

Integral number [74] $$\int \frac{{\sinh }^3(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[C]   time = 4.16929 (sec), size = 62017 ,normalized size = 2696.39 $$\text{Too large to display}$$

[In]

integrate(sinh(d*x+c)^3/(a+b*tanh(d*x+c)^3),x, algorithm=""fricas"")

output

1/24*((a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^6 + 6*(a^3 - a^2*b - a*b^2 + b 
^3)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^3 - a^2*b - a*b^2 + b^3)*sinh(d*x + c)^ 
6 - 9*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^4 - 3*(3*a^3 - 9*a^2*b + 9 
*a*b^2 - 3*b^3 - 5*(a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^ 
4 + 4*(5*(a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^3 - 9*(a^3 - 3*a^2*b + 3*a* 
b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*sqrt(2/3)*sqrt(1/6)*((a^4 - 2*a^ 
2*b^2 + b^4)*d*cosh(d*x + c)^3 + 3*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2*s 
inh(d*x + c) + 3*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c)^2 + (a^ 
4 - 2*a^2*b^2 + b^4)*d*sinh(d*x + c)^3)*sqrt(-(810*a^6*b^2 + 2754*a^4*b^4 + 81 
0*a^2*b^6 - (a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10)*(( 
5*a^2*b^2/(a^8*d^4 - 4*a^6*b^2*d^4 + 6*a^4*b^4*d^4 - 4*a^2*b^6*d^4 + b^8*d^4) 
+ 9*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)^2/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6* 
b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)^2)*(-I*sqrt(3) + 1)/(-1/1 
458*a^2*b^2/(a^10*d^6 - 5*a^8*b^2*d^6 + 10*a^6*b^4*d^6 - 10*a^4*b^6*d^6 + 5*a^ 
2*b^8*d^6 - b^10*d^6) - 5/162*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)*a^2*b^2/((a 
^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^ 
10*d^2)*(a^8*d^4 - 4*a^6*b^2*d^4 + 6*a^4*b^4*d^4 - 4*a^2*b^6*d^4 + b^8*d^4)) - 
 1/27*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)^3/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^ 
6*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)^3 + 1/1458*(a^10 - 30*a 
^8*b^2 - 700*a^6*b^4 - 700*a^4*b^6 - 30*a^2*b^8 + b^10)*a^2*b^2/((a^2 - b^2...
 

Integral number [76] $$\int \frac{\sinh (c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[C]   time = 1.73806 (sec), size = 40923 ,normalized size = 1948.71 $$\text{Too large to display}$$

[In]

integrate(sinh(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm=""fricas"")

output

-1/6*(sqrt(2/3)*sqrt(1/2)*((a^2 - b^2)*d*cosh(d*x + c) + (a^2 - b^2)*d*sinh(d* 
x + c))*sqrt(-(108*a^2*b^2 + 54*b^4 - (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*((b^ 
2/(a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4) + 3*(2*a^2*b^2 + b^4)^2/ 
(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^2)*(-I*sqrt(3) + 1)/(-1/14 
58*b^2/(a^8*d^6 - 3*a^6*b^2*d^6 + 3*a^4*b^4*d^6 - a^2*b^6*d^6) - 1/54*(2*a^2*b 
^2 + b^4)*b^2/((a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4)*(a^6*d^2 - 
3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^6*d^2 
- 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^3 - 1/1458*(a^6 - 3*a^4*b^2 - 24*a^ 
2*b^4 - b^6)*b^2/((a^2 - b^2)^6*a^2*d^6))^(1/3) + 27*(-1/1458*b^2/(a^8*d^6 - 3 
*a^6*b^2*d^6 + 3*a^4*b^4*d^6 - a^2*b^6*d^6) - 1/54*(2*a^2*b^2 + b^4)*b^2/((a^6 
*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4)*(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a 
^2*b^4*d^2 - b^6*d^2)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^6*d^2 - 3*a^4*b^2*d^2 + 3 
*a^2*b^4*d^2 - b^6*d^2)^3 - 1/1458*(a^6 - 3*a^4*b^2 - 24*a^2*b^4 - b^6)*b^2/(( 
a^2 - b^2)^6*a^2*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*(2*a^2*b^2 + b^4)/(a^6*d^2 - 
 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2))*d^2 + 3*sqrt(1/3)*(a^6 - 3*a^4*b^2 
+ 3*a^2*b^4 - b^6)*d^2*sqrt((432*a^6*b^2 + 2592*a^4*b^4 + 5184*a^2*b^6 + 540*b 
^8 - (a^12 - 6*a^10*b^2 + 15*a^8*b^4 - 20*a^6*b^6 + 15*a^4*b^8 - 6*a^2*b^10 + 
b^12)*((b^2/(a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4) + 3*(2*a^2*b^2 
 + b^4)^2/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^2)*(-I*sqrt(3) + 
 1)/(-1/1458*b^2/(a^8*d^6 - 3*a^6*b^2*d^6 + 3*a^4*b^4*d^6 - a^2*b^6*d^6) - ...
 

Integral number [77] $$\int \frac{\text{csch}(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[C]   time = 1.72563 (sec), size = 20085 ,normalized size = 956.43 $$\text{Too large to display}$$

[In]

integrate(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm=""fricas"")

output

-1/6*(sqrt(2/3)*sqrt(1/6)*a*d*sqrt(((a^4 - a^2*b^2)*((b^4/(a^4*d^2 - a^2*b^2*d 
^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3) + 1)/(-1/729*b^6/(a^4*d^2 - a 
^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 
1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3) 
 + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^ 
4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/ 
((a^2 - b^2)^2*a^4*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^2/(a^4*d^2 - a^2*b^2*d^2 
))*d^2 + 3*sqrt(1/3)*(a^4 - a^2*b^2)*d^2*sqrt(-((a^8 - 2*a^6*b^2 + a^4*b^4)*(( 
b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3) + 1)/ 
(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^ 
4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 
- b^2)^2*a^4*d^6))^(1/3) + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^ 
4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^ 
6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^ 
2/(a^4*d^2 - a^2*b^2*d^2))^2*d^4 - 1296*a^2*b^2 + 324*b^4 - 36*(a^4*b^2 - a^2* 
b^4)*((b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3 
) + 1)/(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d 
^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2 
/((a^2 - b^2)^2*a^4*d^6))^(1/3) + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1 
/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^...
 

Integral number [79] $$\int \frac{{\text{csch}}^3(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[C]   time = 2.9369 (sec), size = 6846 ,normalized size = 297.65 $$\text{Too large to display}$$

[In]

integrate(csch(d*x+c)^3/(a+b*tanh(d*x+c)^3),x, algorithm=""fricas"")

output

-1/6*(6*cosh(d*x + c)^3 + 18*cosh(d*x + c)*sinh(d*x + c)^2 + 6*sinh(d*x + c)^3 
 + (a*d*cosh(d*x + c)^4 + 4*a*d*cosh(d*x + c)*sinh(d*x + c)^3 + a*d*sinh(d*x + 
 c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh(d*x + c)^2 - a*d)*sinh(d*x + c)^ 
2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*cosh(d*x + c))*sinh(d*x + c))*sqrt((1/2 
)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^ 
6))^(1/3) + 3*sqrt(1/3)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/( 
a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) 
+ 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)) 
)^2*a^6*d^4 + 16*b^2)/(a^6*d^4)) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4 
*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))*log(4*(a^3* 
b + a^2*b^2 + a*b^3 + b^4)*cosh(d*x + c) + 4*(a^3*b + a^2*b^2 + a*b^3 + b^4)*s 
inh(d*x + c) + 1/2*(((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - 
 (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d 
^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))^2*a^9*d^5 
 + (a^7 - a^5*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - 
(a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^ 
4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))*d^3 + 4*(a 
^4*b + 2*a^3*b^2 + a^2*b^3)*d - 3*sqrt(1/3)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^ 
2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^ 
2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(...
 

4.8.3 Mupad

Integral number [76] $$\int \frac{\sinh (c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 83.4069 (sec), size = -1 ,normalized size = -0.05 $$\text{Too large to display}$$

[In]

int(sinh(c + d*x)/(a + b*tanh(c + d*x)^3),x)

output

exp(- c - d*x)/(2*(a*d - b*d)) + symsum(log((81920*a^2*b^5*exp(d*x)*exp(root(2 
187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6 
*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, 
z, k)) + 221184*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6 
*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a 
^2*b^2*d^2*z^2 - b^2, z, k)^3*a^2*b^8*d^3 - 3538944*root(2187*a^6*b^2*d^6*z^6 
- 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2* 
d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^3*b^7*d^3 
+ 1990656*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z 
^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2 
*d^2*z^2 - b^2, z, k)^3*a^4*b^6*d^3 + 3538944*root(2187*a^6*b^2*d^6*z^6 - 2187 
*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^ 
4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^5*b^5*d^3 - 2211 
840*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 7 
29*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z 
^2 - b^2, z, k)^3*a^6*b^4*d^3 + 7962624*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b 
^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 72 
9*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^3*b^9*d^5 + 15925248*r 
oot(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^ 
8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2...
 

Integral number [77] $$\int \frac{\text{csch}(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 15.2722 (sec), size = -1 ,normalized size = -0.05 $$\text{Too large to display}$$

[In]

int(1/(sinh(c + d*x)*(a + b*tanh(c + d*x)^3)),x)

output

symsum(log(-(1409286144*b^6*exp(d*x)*exp(root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^ 
6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)) + 134217728*roo 
t(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2 
*z^2 - b^2, z, k)*b^7*d + 1879048192*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^ 
6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)*a*b^6*d - 2818572288 
*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2 
*d^2*z^2 - b^2, z, k)^3*a^2*b^7*d^3 - 40869298176*root(729*a^6*b^2*d^6*z^6 - 7 
29*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^3*b 
^6*d^3 + 28185722880*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2* 
d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^4*b^5*d^3 + 15502147584*root(729 
*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 
- b^2, z, k)^3*a^5*b^4*d^3 + 18119393280*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^ 
6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^4*b^7*d^5 + 
235552112640*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 
+ 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^5*b^6*d^5 + 14495514624*root(729*a^6*b^2 
*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z 
, k)^5*a^6*b^5*d^5 - 219244658688*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 
 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^7*b^4*d^5 - 4892236 
1856*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2 
*b^2*d^2*z^2 - b^2, z, k)^5*a^8*b^3*d^5 - 32614907904*root(729*a^6*b^2*d^6*...
 

Integral number [79] $$\int \frac{{\text{csch}}^3(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 27.7824 (sec), size = -1 ,normalized size = -0.04 $$\text{Too large to display}$$

[In]

int(1/(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^3)),x)

output

exp(c + d*x)/(a*d - a*d*exp(2*c + 2*d*x)) - (2*exp(c + d*x))/(a*d - 2*a*d*exp( 
2*c + 2*d*x) + a*d*exp(4*c + 4*d*x)) + symsum(log((570425344*a^4*b^6*exp(d*x)* 
exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 33554 
432*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)*a*b^10*d 
 - 553648128*a^2*b^8*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + 
 a^2*b^2 - b^4, z, k)) - 167772160*a^3*b^7*exp(d*x)*exp(root(729*a^10*d^6*z^6 
+ 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 16777216*b^10*exp(d*x)*exp(root 
(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 192937984*a^5 
*b^5*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, 
z, k)) + 2617245696*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4 
, z, k)^3*a^5*b^8*d^3 - 150994944*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + 
 a^2*b^2 - b^4, z, k)^3*a^6*b^7*d^3 - 1384120320*root(729*a^10*d^6*z^6 + 27*a^ 
4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^3*a^7*b^6*d^3 + 2415919104*root(729*a^10* 
d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^3*a^8*b^5*d^3 - 3498049536 
*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^3*a^9*b^4*d 
^3 + 5435817984*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, 
 k)^5*a^8*b^7*d^5 + 679477248*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2 
*b^2 - b^4, z, k)^5*a^9*b^6*d^5 - 70665633792*root(729*a^10*d^6*z^6 + 27*a^4*b 
^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^5*a^10*b^5*d^5 + 52319748096*root(729*a^10*d 
^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^5*a^11*b^4*d^5 + 1223059...
 

4.9 Test file Number [206]

4.9.1 Mathematica

Integral number [115] $$\int \frac{\cos (bx)\mathrm{CosIntegral}(bx)}{x^2}dx$$

[A]   time = 0.0255593 (sec), size = 13 ,normalized size = 1.08 $$-\frac{\cos (bx)\mathrm{CosIntegral}(bx)}{x}$$

[In]

Integrate[(Cos[b*x]*CosIntegral[b*x])/x^2,x]

output

-((Cos[b*x]*CosIntegral[b*x])/x)
 

4.9.2 Fricas

Integral number [16] $$\int \frac{\text{Si}(bx{)}^2}{x^3}dx$$

[C]   time = 0.255927 (sec), size = 74 ,normalized size = 7.4 $$\frac{4b^2{x}^2\mathrm{Ci}\left(2bx\right)-2bx\cos \left(bx\right)\mathrm{Si}\left(bx\right)-(b^2{x}^2+2)\mathrm{Si}{\left(bx\right)}^2+\cos {\left(bx\right)}^2-2(2bx\cos \left(bx\right)+\mathrm{Si}\left(bx\right))\sin \left(bx\right)-1}{4x^2}$$

[In]

integrate(sin_integral(b*x)^2/x^3,x, algorithm=""fricas"")

output

1/4*(4*b^2*x^2*cos_integral(2*b*x) - 2*b*x*cos(b*x)*sin_integral(b*x) - (b^2*x 
^2 + 2)*sin_integral(b*x)^2 + cos(b*x)^2 - 2*(2*b*x*cos(b*x) + sin_integral(b* 
x))*sin(b*x) - 1)/x^2