Integrand size = 28, antiderivative size = 517 \[ \int \frac {(e+f x)^2 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^2}{a d}+\frac {2 b (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^2 \coth (c+d x)}{a d}+\frac {\sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {2 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {2 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {2 b f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {2 b f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3} \] Output:
-(f*x+e)^2/a/d+2*b*(f*x+e)^2*arctanh(exp(d*x+c))/a^2/d-(f*x+e)^2*coth(d*x+ c)/a/d+(a^2+b^2)^(1/2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^ 2/d-(a^2+b^2)^(1/2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/d +2*f*(f*x+e)*ln(1-exp(2*d*x+2*c))/a/d^2+2*b*f*(f*x+e)*polylog(2,-exp(d*x+c ))/a^2/d^2-2*b*f*(f*x+e)*polylog(2,exp(d*x+c))/a^2/d^2+2*(a^2+b^2)^(1/2)*f *(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d^2-2*(a^2+b^2)^ (1/2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/d^2+f^2*p olylog(2,exp(2*d*x+2*c))/a/d^3-2*b*f^2*polylog(3,-exp(d*x+c))/a^2/d^3+2*b* f^2*polylog(3,exp(d*x+c))/a^2/d^3-2*(a^2+b^2)^(1/2)*f^2*polylog(3,-b*exp(d *x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d^3+2*(a^2+b^2)^(1/2)*f^2*polylog(3,-b*exp( d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/d^3
Time = 6.82 (sec) , antiderivative size = 917, normalized size of antiderivative = 1.77 \[ \int \frac {(e+f x)^2 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^2*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
-((-(d^2*e*(-1 + E^(2*c))*(b*d*e - 2*a*f)*x) + d^2*e*(-1 + E^(2*c))*(b*d*e + 2*a*f)*x + 2*a*d^2*(e + f*x)^2 + 2*d*(-1 + E^(2*c))*f*(b*d*e - a*f)*x*L og[1 - E^(-c - d*x)] + b*d^2*(-1 + E^(2*c))*f^2*x^2*Log[1 - E^(-c - d*x)] - 2*d*(-1 + E^(2*c))*f*(b*d*e + a*f)*x*Log[1 + E^(-c - d*x)] - b*d^2*(-1 + E^(2*c))*f^2*x^2*Log[1 + E^(-c - d*x)] + d*e*(-1 + E^(2*c))*(b*d*e - 2*a* f)*Log[1 - E^(c + d*x)] - d*e*(-1 + E^(2*c))*(b*d*e + 2*a*f)*Log[1 + E^(c + d*x)] + 2*(-1 + E^(2*c))*f*(b*d*e + a*f)*PolyLog[2, -E^(-c - d*x)] + 2*b *d*(-1 + E^(2*c))*f^2*x*PolyLog[2, -E^(-c - d*x)] + 2*(-1 + E^(2*c))*f*(-( b*d*e) + a*f)*PolyLog[2, E^(-c - d*x)] - 2*b*d*(-1 + E^(2*c))*f^2*x*PolyLo g[2, E^(-c - d*x)] + 2*b*(-1 + E^(2*c))*f^2*PolyLog[3, -E^(-c - d*x)] - 2* b*(-1 + E^(2*c))*f^2*PolyLog[3, E^(-c - d*x)])/(a^2*d^3*(-1 + E^(2*c)))) + (Sqrt[a^2 + b^2]*(-2*d^2*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + d^2*f^2*x^ 2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 2*d^2*e*f*x*Log[1 + (b* E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - d^2*f^2*x^2*Log[1 + (b*E^(c + d*x))/ (a + Sqrt[a^2 + b^2])] + 2*d*f*(e + f*x)*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*d*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt [a^2 + b^2]))] - 2*f^2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(a^2*d^3) + (Sech[c/2]*Sech[c/2 + (d*x)/2]*(-(e^2*Sinh[(d*x)/2]) - 2*e*f*x*Sinh[(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.
| \(\displaystyle \int \frac {(e+f x)^2 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6103 |
| \(\displaystyle \frac {\int (e+f x)^2 \coth ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -(e+f x)^2 \tan \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{a}\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {2 i f \int i (e+f x) \coth (c+d x)dx}{d}-\int (e+f x)^2dx+\frac {(e+f x)^2 \coth (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {2 i f \int i (e+f x) \coth (c+d x)dx}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {-\frac {2 f \int (e+f x) \coth (c+d x)dx}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {-\frac {2 f \int -i (e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {2 i f \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {2 i f \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int e^{-2 c-2 d x+i \pi } \log \left (1+e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^2 \cosh (c+d x) \coth (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^2 \sinh (c+d x)dx+\int (e+f x)^2 \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -i (e+f x)^2 \sin (i c+i d x)dx+\int i (e+f x)^2 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \int (e+f x)^2 \sin (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {2 i f \int (e+f x) \log \left (1-e^{c+d x}\right )dx}{d}-\frac {2 i f \int (e+f x) \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2dx}{b^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{b}\right )}{a}+\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}+\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}+\frac {\int -i (e+f x)^2 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \int (e+f x)^2 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\right )}{a}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{a}\) |
Input:
Int[((e + f*x)^2*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{2} \coth \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^2*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 2729 vs. \(2 (477) = 954\).
Time = 0.15 (sec) , antiderivative size = 2729, normalized size of antiderivative = 5.28 \[ \int \frac {(e+f x)^2 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
-(2*a*d^2*e^2 - 4*a*c*d*e*f + 2*a*c^2*f^2 + 2*(a*d^2*f^2*x^2 + 2*a*d^2*e*f *x + 2*a*c*d*e*f - a*c^2*f^2)*cosh(d*x + c)^2 + 4*(a*d^2*f^2*x^2 + 2*a*d^2 *e*f*x + 2*a*c*d*e*f - a*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + 2*(a*d^2*f ^2*x^2 + 2*a*d^2*e*f*x + 2*a*c*d*e*f - a*c^2*f^2)*sinh(d*x + c)^2 + 2*(b*d *f^2*x + b*d*e*f - (b*d*f^2*x + b*d*e*f)*cosh(d*x + c)^2 - 2*(b*d*f^2*x + b*d*e*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f^2*x + b*d*e*f)*sinh(d*x + c) ^2)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*co sh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 2*(b*d* f^2*x + b*d*e*f - (b*d*f^2*x + b*d*e*f)*cosh(d*x + c)^2 - 2*(b*d*f^2*x + b *d*e*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f^2*x + b*d*e*f)*sinh(d*x + c)^ 2)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cos h(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - (b*d^2*e ^2 - 2*b*c*d*e*f + b*c^2*f^2 - (b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*cosh( d*x + c)^2 - 2*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*cosh(d*x + c)*sinh(d* x + c) - (b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^ 2)/b^2) + 2*a) + (b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2 - (b*d^2*e^2 - 2*b*c *d*e*f + b*c^2*f^2)*cosh(d*x + c)^2 - 2*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f ^2)*cosh(d*x + c)*sinh(d*x + c) - (b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*si nh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d...
\[ \int \frac {(e+f x)^2 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \coth ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**2*coth(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)**2*coth(c + d*x)**2/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^2 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \coth \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
e^2*(b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d) + s qrt(a^2 + b^2)*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(a^2*d) + 2/((a*e^(-2*d*x - 2*c) - a)*d)) - 4*e*f* x/(a*d) - 2*(f^2*x^2 + 2*e*f*x)/(a*d*e^(2*d*x + 2*c) - a*d) + 2*e*f*log(e^ (d*x + c) + 1)/(a*d^2) + 2*e*f*log(e^(d*x + c) - 1)/(a*d^2) + (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)) )*b*f^2/(a^2*d^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)))*b*f^2/(a^2*d^3) + 2*(b*d*e*f + a*f^2)*(d* x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^2*d^3) - 2*(b*d*e*f - a*f ^2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))/(a^2*d^3) - 1/3*(b*d^ 3*f^2*x^3 + 3*(b*d*e*f + a*f^2)*d^2*x^2)/(a^2*d^3) + 1/3*(b*d^3*f^2*x^3 + 3*(b*d*e*f - a*f^2)*d^2*x^2)/(a^2*d^3) + integrate(2*((a^2*f^2*e^c + b^2*f ^2*e^c)*x^2 + 2*(a^2*e*f*e^c + b^2*e*f*e^c)*x)*e^(d*x)/(a^2*b*e^(2*d*x + 2 *c) + 2*a^3*e^(d*x + c) - a^2*b), x)
Timed out. \[ \int \frac {(e+f x)^2 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^2*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^2 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((coth(c + d*x)^2*(e + f*x)^2)/(a + b*sinh(c + d*x)),x)
Output:
int((coth(c + d*x)^2*(e + f*x)^2)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^2 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^2*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
(2*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a **2 + b**2))*b**2*d**2*e**2*i - 2*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*b**2*d**2*e**2*i + 8*e**(5*c + 2*d*x)*int((e**( 3*d*x)*x**2)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - 3*e**(4*c + 4*d* x)*b - 4*e**(3*c + 3*d*x)*a + 3*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b) ,x)*a**4*d**3*f**2 + 4*e**(5*c + 2*d*x)*int((e**(3*d*x)*x**2)/(e**(6*c + 6 *d*x)*b + 2*e**(5*c + 5*d*x)*a - 3*e**(4*c + 4*d*x)*b - 4*e**(3*c + 3*d*x) *a + 3*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**2*b**2*d**3*f**2 + 16*e**(5*c + 2*d*x)*int((e**(3*d*x)*x)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - 3*e**(4*c + 4*d*x)*b - 4*e**(3*c + 3*d*x)*a + 3*e**(2*c + 2*d*x )*b + 2*e**(c + d*x)*a - b),x)*a**4*d**3*e*f + 8*e**(5*c + 2*d*x)*int((e** (3*d*x)*x)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - 3*e**(4*c + 4*d*x) *b - 4*e**(3*c + 3*d*x)*a + 3*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x )*a**2*b**2*d**3*e*f - 8*e**(5*c + 2*d*x)*int((e**(3*d*x)*x)/(e**(6*c + 6* d*x)*b + 2*e**(5*c + 5*d*x)*a - 3*e**(4*c + 4*d*x)*b - 4*e**(3*c + 3*d*x)* a + 3*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**2*b**2*d**2*f**2 - 8*e**(4*c + 2*d*x)*int((e**(2*d*x)*x**2)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - 3*e**(4*c + 4*d*x)*b - 4*e**(3*c + 3*d*x)*a + 3*e**(2*c + 2*d* x)*b + 2*e**(c + d*x)*a - b),x)*a**3*b*d**3*f**2 - 16*e**(4*c + 2*d*x)*int ((e**(2*d*x)*x)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - 3*e**(4*c ...