Integrand size = 36, antiderivative size = 1245 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \]
-2*b^2*f*(f*x+e)*polylog(2,-exp(d*x+c))/a^3/d^2+2*b^2*f*(f*x+e)*polylog(2, exp(d*x+c))/a^3/d^2+3*(f*x+e)^2*arctanh(exp(d*x+c))/a/d-3*f^2*polylog(3,-e xp(d*x+c))/a/d^3+3*f^2*polylog(3,exp(d*x+c))/a/d^3-f^2*arctanh(cosh(d*x+c) )/a/d^3-2*b*f*(f*x+e)*ln(1-exp(4*d*x+4*c))/a^2/d^2+2*b^5*f^2*polylog(3,-b* exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(3/2)/d^3+2*I*f^2*polylog(2, I*exp(d*x+c))/a/d^3-2*I*b^2*f^2*polylog(2,I*exp(d*x+c))/a^3/d^3-2*b^5*f^2* polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(3/2)/d^3+2*I*b ^4*f^2*polylog(2,I*exp(d*x+c))/a^3/(a^2+b^2)/d^3+2*I*b^2*f^2*polylog(2,-I* exp(d*x+c))/a^3/d^3-3/2*(f*x+e)^2*sech(d*x+c)/a/d-4*b^2*f*(f*x+e)*arctan(e xp(d*x+c))/a^3/d^2-b^5*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^ 3/(a^2+b^2)^(3/2)/d+b^5*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a ^3/(a^2+b^2)^(3/2)/d+2*b*(f*x+e)^2/a^2/d-2*b^2*(f*x+e)^2*arctanh(exp(d*x+c ))/a^3/d+2*b^2*f^2*polylog(3,-exp(d*x+c))/a^3/d^3-2*b^2*f^2*polylog(3,exp( d*x+c))/a^3/d^3+4*b^4*f*(f*x+e)*arctan(exp(d*x+c))/a^3/(a^2+b^2)/d^2+2*b^3 *f*(f*x+e)*ln(1+exp(2*d*x+2*c))/a^2/(a^2+b^2)/d^2-2*b^5*f*(f*x+e)*polylog( 2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(3/2)/d^2+2*b^5*f*(f*x+ e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(3/2)/d^2-2* I*b^4*f^2*polylog(2,-I*exp(d*x+c))/a^3/(a^2+b^2)/d^3+b^3*f^2*polylog(2,-ex p(2*d*x+2*c))/a^2/(a^2+b^2)/d^3-b^4*(f*x+e)^2*sech(d*x+c)/a^3/(a^2+b^2)/d- b^3*(f*x+e)^2*tanh(d*x+c)/a^2/(a^2+b^2)/d-b^3*(f*x+e)^2/a^2/(a^2+b^2)/d...
Time = 9.79 (sec) , antiderivative size = 2346, normalized size of antiderivative = 1.88 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]
(f*(4*b*d^2*e*E^(2*c)*x - 4*b*d^2*e*(1 + E^(2*c))*x + 2*b*d^2*E^(2*c)*f*x^ 2 - 2*b*d^2*(1 + E^(2*c))*f*x^2 + 4*a*d*e*(1 + E^(2*c))*ArcTan[E^(c + d*x) ] + 2*b*d*e*(1 + E^(2*c))*(2*d*x - Log[1 + E^(2*(c + d*x))]) + (2*I)*a*(1 + E^(2*c))*f*(d*x*(Log[1 - I*E^(c + d*x)] - Log[1 + I*E^(c + d*x)]) - Poly Log[2, (-I)*E^(c + d*x)] + PolyLog[2, I*E^(c + d*x)]) + b*(1 + E^(2*c))*f* (2*d*x*(d*x - Log[1 + E^(2*(c + d*x))]) - PolyLog[2, -E^(2*(c + d*x))])))/ ((a^2 + b^2)*d^3*(1 + E^(2*c))) + (8*a*b*d^2*e*E^(2*c)*f*x + 4*a*b*d^2*E^( 2*c)*f^2*x^2 - 6*a^2*d^2*e^2*ArcTanh[E^(c + d*x)] + 4*b^2*d^2*e^2*ArcTanh[ E^(c + d*x)] + 6*a^2*d^2*e^2*E^(2*c)*ArcTanh[E^(c + d*x)] - 4*b^2*d^2*e^2* E^(2*c)*ArcTanh[E^(c + d*x)] + 4*a^2*f^2*ArcTanh[E^(c + d*x)] - 4*a^2*E^(2 *c)*f^2*ArcTanh[E^(c + d*x)] + 6*a^2*d^2*e*f*x*Log[1 - E^(c + d*x)] - 4*b^ 2*d^2*e*f*x*Log[1 - E^(c + d*x)] - 6*a^2*d^2*e*E^(2*c)*f*x*Log[1 - E^(c + d*x)] + 4*b^2*d^2*e*E^(2*c)*f*x*Log[1 - E^(c + d*x)] + 3*a^2*d^2*f^2*x^2*L og[1 - E^(c + d*x)] - 2*b^2*d^2*f^2*x^2*Log[1 - E^(c + d*x)] - 3*a^2*d^2*E ^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] + 2*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^ (c + d*x)] - 6*a^2*d^2*e*f*x*Log[1 + E^(c + d*x)] + 4*b^2*d^2*e*f*x*Log[1 + E^(c + d*x)] + 6*a^2*d^2*e*E^(2*c)*f*x*Log[1 + E^(c + d*x)] - 4*b^2*d^2* e*E^(2*c)*f*x*Log[1 + E^(c + d*x)] - 3*a^2*d^2*f^2*x^2*Log[1 + E^(c + d*x) ] + 2*b^2*d^2*f^2*x^2*Log[1 + E^(c + d*x)] + 3*a^2*d^2*E^(2*c)*f^2*x^2*Log [1 + E^(c + d*x)] - 2*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(c + d*x)] + 4*...
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.
| \(\displaystyle \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6123 |
| \(\displaystyle \frac {\int (e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5985 |
| \(\displaystyle \frac {-2 f \int \frac {1}{2} (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 6123 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (\frac {4 \int (e+f x)^2 \text {csch}^2(2 c+2 d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {4 \int -(e+f x)^2 \csc (2 i c+2 i d x)^2dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \int (e+f x)^2 \csc (2 i c+2 i d x)^2dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}-\frac {i f \int -i (e+f x) \coth (2 c+2 d x)dx}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}-\frac {f \int (e+f x) \coth (2 c+2 d x)dx}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}-\frac {f \int -i (e+f x) \tan \left (2 i c+2 i d x+\frac {\pi }{2}\right )dx}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \int (e+f x) \tan \left (\frac {1}{2} (4 i c+\pi )+2 i d x\right )dx}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \int \frac {e^{4 c+4 d x-i \pi } (e+f x)}{1+e^{4 c+4 d x-i \pi }}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}-\frac {f \int \log \left (1+e^{4 c+4 d x-i \pi }\right )dx}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}-\frac {f \int e^{-4 c-4 d x+i \pi } \log \left (1+e^{4 c+4 d x-i \pi }\right )de^{4 c+4 d x-i \pi }}{16 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6123 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {\int (e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5985 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {-2 f \int -\left ((e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {b^2 \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {2 b^2 \int -\frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \int \frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-f \int (e+f x) \left (-\frac {\text {sech}(c+d x) \text {csch}^2(c+d x)}{d}+\frac {3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 \text {sech}(c+d x)}{d}\right )dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {-f \int \frac {(e+f x) \left (-\text {sech}(c+d x) \text {csch}^2(c+d x)+3 \text {arctanh}(\cosh (c+d x))-3 \text {sech}(c+d x)\right )}{d}dx+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {2 f \int \frac {(e+f x) (\text {arctanh}(\cosh (c+d x))-\text {sech}(c+d x))}{d}dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {f \int (e+f x) \left (-\text {sech}(c+d x) \text {csch}^2(c+d x)+3 \text {arctanh}(\cosh (c+d x))-3 \text {sech}(c+d x)\right )dx}{d}+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {\frac {2 f \int (e+f x) (\text {arctanh}(\cosh (c+d x))-\text {sech}(c+d x))dx}{d}-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
3.5.96.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( -I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csch[2*a + 2*b*x ]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> With[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Simp[(c + d*x)^m u, x] - Simp[d*m Int[(c + d*x)^(m - 1)*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n , p]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b^2/(a^2 + b^2) Int[(e + f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Simp[1/(a^2 + b^2) Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; F reeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0 ]
Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/a Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Simp[b/ a Int[(e + f*x)^m*Sech[c + d*x]^p*(Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {\left (f x +e \right )^{2} \operatorname {csch}\left (d x +c \right )^{3} \operatorname {sech}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 29722 vs. \(2 (1152) = 2304\).
Time = 0.78 (sec) , antiderivative size = 29722, normalized size of antiderivative = 23.87 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right )^{3} \operatorname {sech}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
2*b*e*f*(2*(d*x + c)/((a^2 + b^2)*d^2) - log(e^(2*d*x + 2*c) + 1)/((a^2 + b^2)*d^2)) + 4*a*f^2*integrate(x*e^(d*x + c)/(a^2*d*e^(2*d*x + 2*c) + b^2* d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) + 4*b*f^2*integrate(x/(a^2*d*e^(2*d *x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 1/2*(2*b^5*log((b *e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2) ))/((a^5 + a^3*b^2)*sqrt(a^2 + b^2)*d) + 2*(4*a^2*b*e^(-2*d*x - 2*c) + 2*b ^3*e^(-4*d*x - 4*c) - 4*a^2*b - 2*b^3 + (3*a^3 + a*b^2)*e^(-d*x - c) - 2*( a^3 - a*b^2)*e^(-3*d*x - 3*c) + (3*a^3 + a*b^2)*e^(-5*d*x - 5*c))/((a^4 + a^2*b^2 - (a^4 + a^2*b^2)*e^(-2*d*x - 2*c) - (a^4 + a^2*b^2)*e^(-4*d*x - 4 *c) + (a^4 + a^2*b^2)*e^(-6*d*x - 6*c))*d) - (3*a^2 - 2*b^2)*log(e^(-d*x - c) + 1)/(a^3*d) + (3*a^2 - 2*b^2)*log(e^(-d*x - c) - 1)/(a^3*d))*e^2 + 4* a*e*f*arctan(e^(d*x + c))/((a^2 + b^2)*d^2) - (2*(2*a^2*b*d*f^2 + b^3*d*f^ 2)*x^2 + 4*(2*a^2*b*d*e*f + b^3*d*e*f)*x + (2*a^3*e*f*e^(5*c) + 2*a*b^2*e* f*e^(5*c) + (3*a^3*d*f^2*e^(5*c) + a*b^2*d*f^2*e^(5*c))*x^2 + 2*((3*d*e*f + f^2)*a^3*e^(5*c) + (d*e*f + f^2)*a*b^2*e^(5*c))*x)*e^(5*d*x) - 2*(b^3*d* f^2*x^2*e^(4*c) + 2*b^3*d*e*f*x*e^(4*c))*e^(4*d*x) - 2*((a^3*d*f^2*e^(3*c) - a*b^2*d*f^2*e^(3*c))*x^2 + 2*(a^3*d*e*f*e^(3*c) - a*b^2*d*e*f*e^(3*c))* x)*e^(3*d*x) - 4*(a^2*b*d*f^2*x^2*e^(2*c) + 2*a^2*b*d*e*f*x*e^(2*c))*e^(2* d*x) - (2*a^3*e*f*e^c + 2*a*b^2*e*f*e^c - (3*a^3*d*f^2*e^c + a*b^2*d*f^2*e ^c)*x^2 - 2*((3*d*e*f - f^2)*a^3*e^c + (d*e*f - f^2)*a*b^2*e^c)*x)*e^(d...
Exception generated. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Not invertible Error: Bad Argument Value
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]