Integrand size = 28, antiderivative size = 680 \[ \int \frac {(e+f x)^2 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {(e+f x)^2}{2 a d}-\frac {(e+f x)^3}{3 a f}-\frac {b^2 (e+f x)^3}{3 a^3 f}+\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 a^3 f}+\frac {4 b f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {(e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {b^2 (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {2 b f^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {2 b f^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}-\frac {2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^2}+\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^3 d^2}+\frac {2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {f^2 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3}-\frac {b^2 f^2 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^3 d^3} \] Output:
1/2*(f*x+e)^2/a/d-1/3*(f*x+e)^3/a/f-1/3*b^2*(f*x+e)^3/a^3/f+1/3*(a^2+b^2)* (f*x+e)^3/a^3/f+4*b*f*(f*x+e)*arctanh(exp(d*x+c))/a^2/d^2-f*(f*x+e)*coth(d *x+c)/a/d^2-1/2*(f*x+e)^2*coth(d*x+c)^2/a/d+b*(f*x+e)^2*csch(d*x+c)/a^2/d- (a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/d-(a^2+b^2) *(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/d+(f*x+e)^2*ln(1-exp (2*d*x+2*c))/a/d+b^2*(f*x+e)^2*ln(1-exp(2*d*x+2*c))/a^3/d+f^2*ln(sinh(d*x+ c))/a/d^3+2*b*f^2*polylog(2,-exp(d*x+c))/a^2/d^3-2*b*f^2*polylog(2,exp(d*x +c))/a^2/d^3-2*(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1 /2)))/a^3/d^2-2*(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^( 1/2)))/a^3/d^2+f*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a/d^2+b^2*f*(f*x+e)*pol ylog(2,exp(2*d*x+2*c))/a^3/d^2+2*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/(a- (a^2+b^2)^(1/2)))/a^3/d^3+2*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+ b^2)^(1/2)))/a^3/d^3-1/2*f^2*polylog(3,exp(2*d*x+2*c))/a/d^3-1/2*b^2*f^2*p olylog(3,exp(2*d*x+2*c))/a^3/d^3
Leaf count is larger than twice the leaf count of optimal. \(2403\) vs. \(2(680)=1360\).
Time = 10.75 (sec) , antiderivative size = 2403, normalized size of antiderivative = 3.53 \[ \int \frac {(e+f x)^2 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^2*Coth[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
(b*(e + f*x)^2*Csch[c])/(a^2*d) + ((-e^2 - 2*e*f*x - f^2*x^2)*Csch[c/2 + ( d*x)/2]^2)/(8*a*d) - (12*a^2*d^3*e^2*E^(2*c)*x + 12*b^2*d^3*e^2*E^(2*c)*x + 12*a^2*d*E^(2*c)*f^2*x + 12*a^2*d^3*e*E^(2*c)*f*x^2 + 12*b^2*d^3*e*E^(2* c)*f*x^2 + 4*a^2*d^3*E^(2*c)*f^2*x^3 + 4*b^2*d^3*E^(2*c)*f^2*x^3 + 24*a*b* d*e*f*ArcTanh[E^(c + d*x)] - 24*a*b*d*e*E^(2*c)*f*ArcTanh[E^(c + d*x)] - 1 2*a*b*d*f^2*x*Log[1 - E^(c + d*x)] + 12*a*b*d*E^(2*c)*f^2*x*Log[1 - E^(c + d*x)] + 12*a*b*d*f^2*x*Log[1 + E^(c + d*x)] - 12*a*b*d*E^(2*c)*f^2*x*Log[ 1 + E^(c + d*x)] + 6*a^2*d^2*e^2*Log[1 - E^(2*(c + d*x))] + 6*b^2*d^2*e^2* Log[1 - E^(2*(c + d*x))] - 6*a^2*d^2*e^2*E^(2*c)*Log[1 - E^(2*(c + d*x))] - 6*b^2*d^2*e^2*E^(2*c)*Log[1 - E^(2*(c + d*x))] + 6*a^2*f^2*Log[1 - E^(2* (c + d*x))] - 6*a^2*E^(2*c)*f^2*Log[1 - E^(2*(c + d*x))] + 12*a^2*d^2*e*f* x*Log[1 - E^(2*(c + d*x))] + 12*b^2*d^2*e*f*x*Log[1 - E^(2*(c + d*x))] - 1 2*a^2*d^2*e*E^(2*c)*f*x*Log[1 - E^(2*(c + d*x))] - 12*b^2*d^2*e*E^(2*c)*f* x*Log[1 - E^(2*(c + d*x))] + 6*a^2*d^2*f^2*x^2*Log[1 - E^(2*(c + d*x))] + 6*b^2*d^2*f^2*x^2*Log[1 - E^(2*(c + d*x))] - 6*a^2*d^2*E^(2*c)*f^2*x^2*Log [1 - E^(2*(c + d*x))] - 6*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^(2*(c + d*x))] - 12*a*b*(-1 + E^(2*c))*f^2*PolyLog[2, -E^(c + d*x)] + 12*a*b*(-1 + E^(2* c))*f^2*PolyLog[2, E^(c + d*x)] + 6*a^2*d*e*f*PolyLog[2, E^(2*(c + d*x))] + 6*b^2*d*e*f*PolyLog[2, E^(2*(c + d*x))] - 6*a^2*d*e*E^(2*c)*f*PolyLog[2, E^(2*(c + d*x))] - 6*b^2*d*e*E^(2*c)*f*PolyLog[2, E^(2*(c + d*x))] + 6...
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.
| \(\displaystyle \int \frac {(e+f x)^2 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6103 |
| \(\displaystyle \frac {\int (e+f x)^2 \coth ^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int i (e+f x)^2 \tan \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^3dx}{a}\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {i f \int -\left ((e+f x) \coth ^2(c+d x)\right )dx}{d}-\int i (e+f x)^2 \coth (c+d x)dx+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {i f \int (e+f x) \coth ^2(c+d x)dx}{d}-\int i (e+f x)^2 \coth (c+d x)dx+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {i f \int (e+f x) \coth ^2(c+d x)dx}{d}-i \int (e+f x)^2 \coth (c+d x)dx+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-i \int -i (e+f x)^2 \tan \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {i f \int -\left ((e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )^2\right )dx}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-i \int -i (e+f x)^2 \tan \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {i f \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx+\frac {i f \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)^2}{1+e^{2 c+2 d x-i \pi }}dx+\frac {i f \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int (e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{d}\right )+\frac {i f \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (\frac {i f \int i \coth (c+d x)dx}{d}-\int (e+f x)dx+\frac {(e+f x) \coth (c+d x)}{d}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (\frac {i f \int i \coth (c+d x)dx}{d}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \int \coth (c+d x)dx}{d}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \int -i \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (\frac {i f \int \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^2 \cosh (c+d x)dx+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (\frac {-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}+\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)dx+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5975 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )+\frac {i f \left (-\frac {f \log (-i \sinh (c+d x))}{d^2}+\frac {(e+f x) \coth (c+d x)}{d}-\frac {(e+f x)^2}{2 f}\right )}{d}+\frac {i (e+f x)^2 \coth ^2(c+d x)}{2 d}+\frac {i (e+f x)^3}{3 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {2 f \int (e+f x) \text {csch}(c+d x)dx}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}\right )}{a}\) |
Input:
Int[((e + f*x)^2*Coth[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{2} \coth \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^2*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*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. 7775 vs. \(2 (639) = 1278\).
Time = 0.21 (sec) , antiderivative size = 7775, normalized size of antiderivative = 11.43 \[ \int \frac {(e+f x)^2 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(e+f x)^2 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \coth ^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**2*coth(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)**2*coth(c + d*x)**3/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^2 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \coth \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
-e^2*(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)) + 2*(a*f^2*x + a*e*f + (b*d*f^2*x^2*e^(3*c) + 2*b*d*e*f*x*e^(3*c))*e^(3*d*x) - (a*d*f^ 2*x^2*e^(2*c) + a*e*f*e^(2*c) + (2*d*e*f + f^2)*a*x*e^(2*c))*e^(2*d*x) - ( b*d*f^2*x^2*e^c + 2*b*d*e*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) - (2*b*d*e*f + a*f^2)*x/(a^2*d^2) + (2*b *d*e*f - a*f^2)*x/(a^2*d^2) + (2*b*d*e*f + a*f^2)*log(e^(d*x + c) + 1)/(a^ 2*d^3) - (2*b*d*e*f - a*f^2)*log(e^(d*x + c) - 1)/(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)) )*(a^2*f^2 + b^2*f^2)/(a^3*d^3) + (d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*d ilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*(a^2*f^2 + b^2*f^2)/(a^3*d^ 3) + 2*(a^2*d*e*f + b^2*d*e*f + a*b*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog (-e^(d*x + c)))/(a^3*d^3) + 2*(a^2*d*e*f + b^2*d*e*f - a*b*f^2)*(d*x*log(- e^(d*x + c) + 1) + dilog(e^(d*x + c)))/(a^3*d^3) - 1/3*((a^2*f^2 + b^2*f^2 )*d^3*x^3 + 3*(a^2*d*e*f + b^2*d*e*f + a*b*f^2)*d^2*x^2)/(a^3*d^3) - 1/3*( (a^2*f^2 + b^2*f^2)*d^3*x^3 + 3*(a^2*d*e*f + b^2*d*e*f - a*b*f^2)*d^2*x^2) /(a^3*d^3) + integrate(-2*((a^2*b*f^2 + b^3*f^2)*x^2 + 2*(a^2*b*e*f + b^3* e*f)*x - ((a^3*f^2*e^c + a*b^2*f^2*e^c)*x^2 + 2*(a^3*e*f*e^c + a*b^2*e*...
Timed out. \[ \int \frac {(e+f x)^2 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^2*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^2 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((coth(c + d*x)^3*(e + f*x)^2)/(a + b*sinh(c + d*x)),x)
Output:
int((coth(c + d*x)^3*(e + f*x)^2)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^2 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^2*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
(576*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**3*f**2 + 816*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**3*f**2 + 240*e**(7*c + 4*d*x)*i nt((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**3*b**4*d**3*f **2 + 1152*e**(7*c + 4*d*x)*int((e**(3*d*x)*x)/(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**3*e*f + 1632*e**(7*c + 4*d*x)*int((e**(3*d*x)*x)/(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**3*e*f - 128*e**(7*c + 4*d*x)*in t((e**(3*d*x)*x)/(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**2*f*...