Integrand size = 34, antiderivative size = 752 \[ \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {3 f (e+f x)^2}{2 a d^2}+\frac {6 b f (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 a d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {b^2 (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2}-\frac {6 b f^3 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^4}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {3 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^3 d^3}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^4}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {3 b^2 f^3 \operatorname {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a^3 d^4} \] Output:
-3/2*f*(f*x+e)^2/a/d^2+6*b*f*(f*x+e)^2*arctanh(exp(d*x+c))/a^2/d^2-3/2*f*( f*x+e)^2*coth(d*x+c)/a/d^2+b*(f*x+e)^3*csch(d*x+c)/a^2/d-1/2*(f*x+e)^3*csc h(d*x+c)^2/a/d-b^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/d- b^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/d+3*f^2*(f*x+e)*l n(1-exp(2*d*x+2*c))/a/d^3+b^2*(f*x+e)^3*ln(1-exp(2*d*x+2*c))/a^3/d+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*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*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+3/2*f^3*polylog(2,exp(2*d*x+2*c))/a/d^4+3/2*b^2*f*(f*x+e)^2*polylog( 2,exp(2*d*x+2*c))/a^3/d^2-6*b*f^3*polylog(3,-exp(d*x+c))/a^2/d^4+6*b*f^3*p olylog(3,exp(d*x+c))/a^2/d^4+6*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*b^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^ 2+b^2)^(1/2)))/a^3/d^3-3/2*b^2*f^2*(f*x+e)*polylog(3,exp(2*d*x+2*c))/a^3/d ^3-6*b^2*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/d^4-6*b^2*f^ 3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/d^4+3/4*b^2*f^3*polylog (4,exp(2*d*x+2*c))/a^3/d^4
Leaf count is larger than twice the leaf count of optimal. \(3254\) vs. \(2(752)=1504\).
Time = 11.18 (sec) , antiderivative size = 3254, normalized size of antiderivative = 4.33 \[ \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^3*Coth[c + d*x]*Csch[c + d*x]^2)/(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*b^2*d^4*e^3*E^(2*c)*x + 24*a^2*d^2 *e*E^(2*c)*f^2*x + 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*b^2*d^4*e*E^(2*c)*f^2*x^3 + 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*Lo g[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*b^2*d^3*e^3*Log[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*b^2*d^3*e^2*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*b^2*d^3*e*f^2*x^2*Log[1 - E^(2*(c + d*x))] - 12*b^2*d^3*e*E^(2*c)* f^2*x^2*Log[1 - E^(2*(c + d*x))] + 4*b^2*d^3*f^3*x^3*Log[1 - E^(2*(c + d*x ))] - 4*b^2*d^3*E^(2*c)*f^3*x^3*Log[1 - E^(2*(c + d*x))] - 24*a*b*d*(-1 + E^(2*c))*f^2*(e + f*x)*PolyLog[2, -E^(c + d*x)] + 24*a*b*d*(-1 + E^(2*c))* f^2*(e + f*x)*PolyLog[2, E^(c + d*x)] + 6*b^2*d^2*e^2*f*PolyLog[2, E^(2...
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.
| \(\displaystyle \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6121 |
| \(\displaystyle \frac {\int (e+f x)^3 \coth (c+d x) \text {csch}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5975 |
| \(\displaystyle \frac {\frac {3 f \int (e+f x)^2 \text {csch}^2(c+d x)dx}{2 d}-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}+\frac {3 f \int -(e+f x)^2 \csc (i c+i d x)^2dx}{2 d}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \int (e+f x)^2 \csc (i c+i d x)^2dx}{2 d}}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \coth (c+d x)dx}{d}\right )}{2 d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {2 f \int (e+f x) \coth (c+d x)dx}{d}\right )}{2 d}-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {2 f \int -i (e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{2 d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}\right )}{2 d}}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int e^{-2 c-2 d x+i \pi } \log \left (1+e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}\) |
| \(\Big \downarrow \) 6121 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}\) |
| \(\Big \downarrow \) 5975 |
| \(\displaystyle -\frac {b \left (\frac {\frac {3 f \int (e+f x)^2 \text {csch}(c+d x)dx}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 f \int i (e+f x)^2 \csc (i c+i d x)dx}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \int (e+f x)^2 \csc (i c+i d x)dx}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \left (\frac {2 i f \int (e+f x) \log \left (1-e^{c+d x}\right )dx}{d}-\frac {2 i f \int (e+f x) \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6103 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {\int (e+f x)^3 \coth (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -i (e+f x)^3 \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \int (e+f x)^3 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)^3}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^4}{4 f}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \int (e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}}{a}-\frac {b \left (-\frac {b \left (\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^4}{4 b f}\right )}{a}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{a}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\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}\right )}{2 d}}{a}-\frac {b \left (\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \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 )}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\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 {i (e+f x)^4}{4 f}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \text {csch}^2(c+d x)}{2 d}-\frac {3 f \left (\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (\frac {-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}+\frac {3 i f \left (\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}\right )}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{a}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{a}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {-\frac {\text {csch}^2(c+d x) (e+f x)^3}{2 d}-\frac {3 f \left (\frac {\coth (c+d x) (e+f x)^2}{d}+\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 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}}{a}-\frac {b \left (\frac {\frac {3 i f \left (\frac {2 i \text {arctanh}\left (e^{c+d x}\right ) (e+f x)^2}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{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 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\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 \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int \operatorname {PolyLog}\left (3,-e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )}{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)^4}{4 f}\right )}{a}\right )}{a}\right )}{a}\) |
Input:
Int[((e + f*x)^3*Coth[c + d*x]*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{3} \coth \left (d x +c \right ) \operatorname {csch}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^3*coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 11595 vs. \(2 (703) = 1406\).
Time = 0.30 (sec) , antiderivative size = 11595, normalized size of antiderivative = 15.42 \[ \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorit hm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**3*coth(d*x+c)*csch(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \coth \left (d x + c\right ) \operatorname {csch}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorit hm="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) + b^2*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^3*d) - b^2*log(e^(-d*x - c) + 1)/(a^3*d) - 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) + (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)))*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)))*b^2*f^3/(a^3*d^4) - 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) + 3*(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*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*a*b* d*e*f^2 + a^2*f^3)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^...
Timed out. \[ \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^3*coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorit hm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {coth}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \] Input:
int((coth(c + d*x)*(e + f*x)^3)/(sinh(c + d*x)^2*(a + b*sinh(c + d*x))),x)
Output:
int((coth(c + d*x)*(e + f*x)^3)/(sinh(c + d*x)^2*(a + b*sinh(c + d*x))), x )
\[ \int \frac {(e+f x)^3 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^3*coth(d*x+c)*csch(d*x+c)^2/(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**5*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**2*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**5*d**4*e*f* *2 + 2880*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**3*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)*a**3*b**2*d**3*f**3 + 6912*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**...