Integrand size = 28, antiderivative size = 972 \[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:
3/2*f*(f*x+e)^2*polylog(2,exp(2*d*x+2*c))/a/d^2-3/2*f^2*(f*x+e)*polylog(3, exp(2*d*x+2*c))/a/d^3+3/4*f^3*polylog(4,exp(2*d*x+2*c))/a/d^4-6*(a^2+b^2)* f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/d^4-(a^2+b^2)*(f*x+e) ^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/d-(a^2+b^2)*(f*x+e)^3*ln(1+b *exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/d-6*(a^2+b^2)*f^3*polylog(4,-b*exp(d* x+c)/(a+(a^2+b^2)^(1/2)))/a^3/d^4+6*b*f*(f*x+e)^2*arctanh(exp(d*x+c))/a^2/ d^2+6*b*f^2*(f*x+e)*polylog(2,-exp(d*x+c))/a^2/d^3-6*b*f^2*(f*x+e)*polylog (2,exp(d*x+c))/a^2/d^3+3/2*b^2*f*(f*x+e)^2*polylog(2,exp(2*d*x+2*c))/a^3/d ^2-3/2*b^2*f^2*(f*x+e)*polylog(3,exp(2*d*x+2*c))/a^3/d^3-3/2*f*(f*x+e)^2*c oth(d*x+c)/a/d^2+3*f^2*(f*x+e)*ln(1-exp(2*d*x+2*c))/a/d^3-6*b*f^3*polylog( 3,-exp(d*x+c))/a^2/d^4+6*b*f^3*polylog(3,exp(d*x+c))/a^2/d^4+3/4*b^2*f^3*p olylog(4,exp(2*d*x+2*c))/a^3/d^4-3/2*f*(f*x+e)^2/a/d^2+(f*x+e)^3*ln(1-exp( 2*d*x+2*c))/a/d+b*(f*x+e)^3*csch(d*x+c)/a^2/d+b^2*(f*x+e)^3*ln(1-exp(2*d*x +2*c))/a^3/d+3/2*f^3*polylog(2,exp(2*d*x+2*c))/a/d^4-1/4*b^2*(f*x+e)^4/a^3 /f+1/4*(a^2+b^2)*(f*x+e)^4/a^3/f-1/2*(f*x+e)^3*coth(d*x+c)^2/a/d+1/2*(f*x+ e)^3/a/d-1/4*(f*x+e)^4/a/f-3*(a^2+b^2)*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c) /(a+(a^2+b^2)^(1/2)))/a^3/d^2-3*(a^2+b^2)*f*(f*x+e)^2*polylog(2,-b*exp(d*x +c)/(a-(a^2+b^2)^(1/2)))/a^3/d^2+6*(a^2+b^2)*f^2*(f*x+e)*polylog(3,-b*exp( d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/d^3+6*(a^2+b^2)*f^2*(f*x+e)*polylog(3,-b*e xp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/d^3
Leaf count is larger than twice the leaf count of optimal. \(3868\) vs. \(2(972)=1944\).
Time = 12.17 (sec) , antiderivative size = 3868, normalized size of antiderivative = 3.98 \[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^3*Coth[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
(b*(e + f*x)^3*Csch[c])/(a^2*d) + ((-e^3 - 3*e^2*f*x - 3*e*f^2*x^2 - f^3*x ^3)*Csch[c/2 + (d*x)/2]^2)/(8*a*d) - (8*a^2*d^4*e^3*E^(2*c)*x + 8*b^2*d^4* e^3*E^(2*c)*x + 24*a^2*d^2*e*E^(2*c)*f^2*x + 12*a^2*d^4*e^2*E^(2*c)*f*x^2 + 12*b^2*d^4*e^2*E^(2*c)*f*x^2 + 12*a^2*d^2*E^(2*c)*f^3*x^2 + 8*a^2*d^4*e* E^(2*c)*f^2*x^3 + 8*b^2*d^4*e*E^(2*c)*f^2*x^3 + 2*a^2*d^4*E^(2*c)*f^3*x^4 + 2*b^2*d^4*E^(2*c)*f^3*x^4 + 24*a*b*d^2*e^2*f*ArcTanh[E^(c + d*x)] - 24*a *b*d^2*e^2*E^(2*c)*f*ArcTanh[E^(c + d*x)] - 24*a*b*d^2*e*f^2*x*Log[1 - E^( c + d*x)] + 24*a*b*d^2*e*E^(2*c)*f^2*x*Log[1 - E^(c + d*x)] - 12*a*b*d^2*f ^3*x^2*Log[1 - E^(c + d*x)] + 12*a*b*d^2*E^(2*c)*f^3*x^2*Log[1 - E^(c + d* x)] + 24*a*b*d^2*e*f^2*x*Log[1 + E^(c + d*x)] - 24*a*b*d^2*e*E^(2*c)*f^2*x *Log[1 + E^(c + d*x)] + 12*a*b*d^2*f^3*x^2*Log[1 + E^(c + d*x)] - 12*a*b*d ^2*E^(2*c)*f^3*x^2*Log[1 + E^(c + d*x)] + 4*a^2*d^3*e^3*Log[1 - E^(2*(c + d*x))] + 4*b^2*d^3*e^3*Log[1 - E^(2*(c + d*x))] - 4*a^2*d^3*e^3*E^(2*c)*Lo g[1 - E^(2*(c + d*x))] - 4*b^2*d^3*e^3*E^(2*c)*Log[1 - E^(2*(c + d*x))] + 12*a^2*d*e*f^2*Log[1 - E^(2*(c + d*x))] - 12*a^2*d*e*E^(2*c)*f^2*Log[1 - E ^(2*(c + d*x))] + 12*a^2*d^3*e^2*f*x*Log[1 - E^(2*(c + d*x))] + 12*b^2*d^3 *e^2*f*x*Log[1 - E^(2*(c + d*x))] - 12*a^2*d^3*e^2*E^(2*c)*f*x*Log[1 - E^( 2*(c + d*x))] - 12*b^2*d^3*e^2*E^(2*c)*f*x*Log[1 - E^(2*(c + d*x))] + 12*a ^2*d*f^3*x*Log[1 - E^(2*(c + d*x))] - 12*a^2*d*E^(2*c)*f^3*x*Log[1 - E^(2* (c + d*x))] + 12*a^2*d^3*e*f^2*x^2*Log[1 - E^(2*(c + d*x))] + 12*b^2*d^...
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6103 |
| \(\displaystyle \frac {\int (e+f x)^3 \coth ^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int i (e+f x)^3 \tan \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^3dx}{a}\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {3 i f \int -(e+f x)^2 \coth ^2(c+d x)dx}{2 d}-\int i (e+f x)^3 \coth (c+d x)dx+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {3 i f \int (e+f x)^2 \coth ^2(c+d x)dx}{2 d}-\int i (e+f x)^3 \coth (c+d x)dx+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {3 i f \int (e+f x)^2 \coth ^2(c+d x)dx}{2 d}-i \int (e+f x)^3 \coth (c+d x)dx+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-i \int -i (e+f x)^3 \tan \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {3 i f \int -(e+f x)^2 \tan \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-i \int -i (e+f x)^3 \tan \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {3 i f \int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\int (e+f x)^3 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx+\frac {3 i f \int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)^3}{1+e^{2 c+2 d x-i \pi }}dx+\frac {3 i f \int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \int (e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )+\frac {3 i f \int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \int i (e+f x) \coth (c+d x)dx}{d}-\int (e+f x)^2dx+\frac {(e+f x)^2 \coth (c+d x)}{d}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \int i (e+f x) \coth (c+d x)dx}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (-\frac {2 f \int (e+f x) \coth (c+d x)dx}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (-\frac {2 f \int -i (e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {3 i f \left (\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int e^{-2 c-2 d x+i \pi } \log \left (1+e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^3 \cosh (c+d x)dx+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {3 i f \int -i (e+f x)^2 \sinh (c+d x)dx}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (\frac {-\frac {3 f \int (e+f x)^2 \sinh (c+d x)dx}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {3 f \int -i (e+f x)^2 \sin (i c+i d x)dx}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 i f \int (e+f x)^2 \sin (i c+i d x)dx}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )+\frac {3 i f \left (\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}\right )}{2 d}+\frac {i (e+f x)^3 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^4}{4 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}+\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{a}\right )}{a}\) |
Input:
Int[((e + f*x)^3*Coth[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{3} \coth \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^3*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 13683 vs. \(2 (908) = 1816\).
Time = 0.36 (sec) , antiderivative size = 13683, normalized size of antiderivative = 14.08 \[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \coth ^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**3*coth(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)**3*coth(c + d*x)**3/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \coth \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
-e^3*(2*(b*e^(-d*x - c) - a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((2*a^2 *e^(-2*d*x - 2*c) - a^2*e^(-4*d*x - 4*c) - a^2)*d) + (a^2 + b^2)*log(-2*a* e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^3*d) - (a^2 + b^2)*log(e^(-d*x - c) + 1)/(a^3*d) - (a^2 + b^2)*log(e^(-d*x - c) - 1)/(a^3*d)) + (3*a*f^3*x ^2 + 6*a*e*f^2*x + 3*a*e^2*f + 2*(b*d*f^3*x^3*e^(3*c) + 3*b*d*e*f^2*x^2*e^ (3*c) + 3*b*d*e^2*f*x*e^(3*c))*e^(3*d*x) - (2*a*d*f^3*x^3*e^(2*c) + 3*a*e^ 2*f*e^(2*c) + 3*(2*d*e*f^2 + f^3)*a*x^2*e^(2*c) + 6*(d*e^2*f + e*f^2)*a*x* e^(2*c))*e^(2*d*x) - 2*(b*d*f^3*x^3*e^c + 3*b*d*e*f^2*x^2*e^c + 3*b*d*e^2* f*x*e^c)*e^(d*x))/(a^2*d^2*e^(4*d*x + 4*c) - 2*a^2*d^2*e^(2*d*x + 2*c) + a ^2*d^2) - 3*(b*d*e^2*f + a*e*f^2)*x/(a^2*d^2) + 3*(b*d*e^2*f - a*e*f^2)*x/ (a^2*d^2) + 3*(b*d*e^2*f + a*e*f^2)*log(e^(d*x + c) + 1)/(a^2*d^3) - 3*(b* d*e^2*f - a*e*f^2)*log(e^(d*x + c) - 1)/(a^2*d^3) + (d^3*x^3*log(e^(d*x + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -e^(d*x + c)) + 6*polylog(4, -e^(d*x + c)))*(a^2*f^3 + b^2*f^3)/(a^3*d^4) + (d^3*x^3*log( -e^(d*x + c) + 1) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x*polylog(3, e^(d*x + c)) + 6*polylog(4, e^(d*x + c)))*(a^2*f^3 + b^2*f^3)/(a^3*d^4) + 3*(a^2 *d*e*f^2 + b^2*d*e*f^2 + a*b*f^3)*(d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*di log(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))/(a^3*d^4) + 3*(a^2*d*e*f^2 + b^2*d*e*f^2 - a*b*f^3)*(d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^( d*x + c)) - 2*polylog(3, e^(d*x + c)))/(a^3*d^4) + 3*(b^2*d^2*e^2*f + 2...
Timed out. \[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^3*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((coth(c + d*x)^3*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)
Output:
int((coth(c + d*x)^3*(e + f*x)^3)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^3*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
(2304*e**(7*c + 4*d*x)*int((e**(3*d*x)*x**3)/(e**(8*c + 8*d*x)*b + 2*e**(7 *c + 7*d*x)*a - 4*e**(6*c + 6*d*x)*b - 6*e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x)*b + 6*e**(3*c + 3*d*x)*a - 4*e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**7*d**4*f**3 + 3264*e**(7*c + 4*d*x)*int((e**(3*d*x)*x**3)/(e**( 8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a - 4*e**(6*c + 6*d*x)*b - 6*e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x)*b + 6*e**(3*c + 3*d*x)*a - 4*e**(2*c + 2*d* x)*b - 2*e**(c + d*x)*a + b),x)*a**5*b**2*d**4*f**3 + 960*e**(7*c + 4*d*x) *int((e**(3*d*x)*x**3)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a - 4*e**( 6*c + 6*d*x)*b - 6*e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x)*b + 6*e**(3*c + 3*d*x)*a - 4*e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**3*b**4*d**4 *f**3 + 6912*e**(7*c + 4*d*x)*int((e**(3*d*x)*x**2)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a - 4*e**(6*c + 6*d*x)*b - 6*e**(5*c + 5*d*x)*a + 6*e** (4*c + 4*d*x)*b + 6*e**(3*c + 3*d*x)*a - 4*e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**7*d**4*e*f**2 + 9792*e**(7*c + 4*d*x)*int((e**(3*d*x)*x* *2)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a - 4*e**(6*c + 6*d*x)*b - 6* e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x)*b + 6*e**(3*c + 3*d*x)*a - 4*e**(2 *c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**5*b**2*d**4*e*f**2 - 768*e**(7 *c + 4*d*x)*int((e**(3*d*x)*x**2)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x) *a - 4*e**(6*c + 6*d*x)*b - 6*e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x)*b + 6*e**(3*c + 3*d*x)*a - 4*e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*...