Integrand size = 36, antiderivative size = 1212 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:
-1/2*(f*x+e)^2*coth(d*x+c)*csch(d*x+c)/a/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*I*f^2*polylog(2,I*ex p(d*x+c))/a/d^3-3*f^2*polylog(3,-exp(d*x+c))/a/d^3+3*f^2*polylog(3,exp(d*x +c))/a/d^3+3*(f*x+e)^2*arctanh(exp(d*x+c))/a/d+3*f*(f*x+e)*polylog(2,-exp( d*x+c))/a/d^2-3*f*(f*x+e)*polylog(2,exp(d*x+c))/a/d^2-2*b^2*f^2*polylog(3, exp(d*x+c))/a^3/d^3-2*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-b^3*(f*x+e)^2/a^2/(a^2+b^2)/d+b^3*f ^2*polylog(2,-exp(2*d*x+2*c))/a^2/(a^2+b^2)/d^3+b^2*(f*x+e)^2*sech(d*x+c)/ a^3/d-b^3*(f*x+e)^2*tanh(d*x+c)/a^2/(a^2+b^2)/d-b^4*(f*x+e)^2*sech(d*x+c)/ a^3/(a^2+b^2)/d+4*f*(f*x+e)*arctan(exp(d*x+c))/a/d^2+2*b*(f*x+e)^2/a^2/d-( f*x+e)^2*sech(d*x+c)/a/d+2*I*b^4*f^2*polylog(2,I*exp(d*x+c))/a^3/(a^2+b^2) /d^3-f*(f*x+e)*csch(d*x+c)/a/d^2-2*b*f*(f*x+e)*ln(1-exp(4*d*x+4*c))/a^2/d^ 2-4*b^2*f*(f*x+e)*arctan(exp(d*x+c))/a^3/d^2-2*I*b^2*f^2*polylog(2,I*exp(d *x+c))/a^3/d^3-1/2*b*f^2*polylog(2,exp(4*d*x+4*c))/a^2/d^3+2*b*(f*x+e)^2*c oth(2*d*x+2*c)/a^2/d-2*I*f^2*polylog(2,-I*exp(d*x+c))/a/d^3-f^2*arctanh(co sh(d*x+c))/a/d^3+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^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*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*b^3*f*(f*x+e)*ln(1+exp(2*d*x+2*c))/a^2/(...
Time = 9.43 (sec) , antiderivative size = 2346, normalized size of antiderivative = 1.94 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^2*Csch[c + d*x]^3*Sech[c + d*x]^2)/(a + b*Sinh[c + d* x]),x]
Output:
(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 |
| \(\displaystyle \frac {-\frac {f \int (e+f x) \left (-\text {sech}(c+d x) \text {csch}^2(c+d x)+3 \text {arctanh}(\cosh (c+d x))-3 \text {sech}(c+d x)\right )dx}{d}+\frac {3 (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \left (\frac {\frac {2 f \int (e+f x) (\text {arctanh}(\cosh (c+d x))-\text {sech}(c+d x))dx}{d}-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\right )}{a}\) |
Input:
Int[((e + f*x)^2*Csch[c + d*x]^3*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{2} \operatorname {csch}\left (d x +c \right )^{3} \operatorname {sech}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 29722 vs. \(2 (1121) = 2242\).
Time = 0.65 (sec) , antiderivative size = 29722, normalized size of antiderivative = 24.52 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algor ithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**2*csch(d*x+c)**3*sech(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right )^{3} \operatorname {sech}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algor ithm="maxima")
Output:
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. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algor ithm="giac")
Output:
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. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \] Input:
int((e + f*x)^2/(cosh(c + d*x)^2*sinh(c + d*x)^3*(a + b*sinh(c + d*x))),x)
Output:
int((e + f*x)^2/(cosh(c + d*x)^2*sinh(c + d*x)^3*(a + b*sinh(c + d*x))), x )
\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
( - 36*e**(6*c + 6*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sq rt(a**2 + b**2))*b**6*d**2*e**2*i + 36*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)* atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*b**6*d**2*e**2*i + 36*e** (2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*b**6*d**2*e**2*i - 36*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a* i)/sqrt(a**2 + b**2))*b**6*d**2*e**2*i - 2304*e**(11*c + 6*d*x)*int((e**(5 *d*x)*x**2)/(e**(12*c + 12*d*x)*b + 2*e**(11*c + 11*d*x)*a - 2*e**(10*c + 10*d*x)*b - 2*e**(9*c + 9*d*x)*a - e**(8*c + 8*d*x)*b - 4*e**(7*c + 7*d*x) *a + 4*e**(6*c + 6*d*x)*b + 4*e**(5*c + 5*d*x)*a - e**(4*c + 4*d*x)*b + 2* e**(3*c + 3*d*x)*a - 2*e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**8* d**3*f**2 - 4608*e**(11*c + 6*d*x)*int((e**(5*d*x)*x**2)/(e**(12*c + 12*d* x)*b + 2*e**(11*c + 11*d*x)*a - 2*e**(10*c + 10*d*x)*b - 2*e**(9*c + 9*d*x )*a - e**(8*c + 8*d*x)*b - 4*e**(7*c + 7*d*x)*a + 4*e**(6*c + 6*d*x)*b + 4 *e**(5*c + 5*d*x)*a - e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a - 2*e**(2* c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**6*b**2*d**3*f**2 - 2304*e**(11* c + 6*d*x)*int((e**(5*d*x)*x**2)/(e**(12*c + 12*d*x)*b + 2*e**(11*c + 11*d *x)*a - 2*e**(10*c + 10*d*x)*b - 2*e**(9*c + 9*d*x)*a - e**(8*c + 8*d*x)*b - 4*e**(7*c + 7*d*x)*a + 4*e**(6*c + 6*d*x)*b + 4*e**(5*c + 5*d*x)*a - e* *(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a - 2*e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**4*b**4*d**3*f**2 - 4608*e**(11*c + 6*d*x)*int((e**(5...