Integrand size = 28, antiderivative size = 1053 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:
-6*b^2*f^3*polylog(4,-exp(d*x+c))/a^3/d^4+6*b^2*f^3*polylog(4,exp(d*x+c))/ a^3/d^4+3/2*b*f^3*polylog(3,exp(2*d*x+2*c))/a^2/d^4-2*b^2*(f*x+e)^3*arctan h(exp(d*x+c))/a^3/d+(f*x+e)^3*arctanh(exp(d*x+c))/a/d+3*f^3*polylog(4,-exp (d*x+c))/a/d^4-3*f^3*polylog(4,exp(d*x+c))/a/d^4-6*b^3*f^2*(f*x+e)*polylog (3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(1/2)/d^3+6*b^3*f^2*(f *x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(1/2)/d^3 +3*b^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b ^2)^(1/2)/d^2-3*b^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2) ))/a^3/(a^2+b^2)^(1/2)/d^2+6*b^2*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/a^3/d^ 3-3*b^2*f*(f*x+e)^2*polylog(2,-exp(d*x+c))/a^3/d^2-3*b*f*(f*x+e)^2*ln(1-ex p(2*d*x+2*c))/a^2/d^2-6*b^2*f^2*(f*x+e)*polylog(3,exp(d*x+c))/a^3/d^3-3*b* f^2*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a^2/d^3+3*b^2*f*(f*x+e)^2*polylog(2, exp(d*x+c))/a^3/d^2+b*(f*x+e)^3/a^2/d-6*f^2*(f*x+e)*arctanh(exp(d*x+c))/a/ d^3-3/2*f*(f*x+e)^2*csch(d*x+c)/a/d^2-1/2*(f*x+e)^3*coth(d*x+c)*csch(d*x+c )/a/d+3/2*f*(f*x+e)^2*polylog(2,-exp(d*x+c))/a/d^2-3/2*f*(f*x+e)^2*polylog (2,exp(d*x+c))/a/d^2-3*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/a/d^3+3*f^2*(f*x +e)*polylog(3,exp(d*x+c))/a/d^3+b*(f*x+e)^3*coth(d*x+c)/a^2/d-3*f^3*polylo g(2,-exp(d*x+c))/a/d^4+3*f^3*polylog(2,exp(d*x+c))/a/d^4+6*b^3*f^3*polylog (4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(1/2)/d^4-6*b^3*f^3*po lylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(1/2)/d^4+b^3*...
Leaf count is larger than twice the leaf count of optimal. \(2801\) vs. \(2(1053)=2106\).
Time = 9.00 (sec) , antiderivative size = 2801, normalized size of antiderivative = 2.66 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^3*Csch[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
(12*a*b*d^3*e^2*E^(2*c)*f*x + 12*a*b*d^3*e*E^(2*c)*f^2*x^2 + 4*a*b*d^3*E^( 2*c)*f^3*x^3 - 2*a^2*d^3*e^3*ArcTanh[E^(c + d*x)] + 4*b^2*d^3*e^3*ArcTanh[ E^(c + d*x)] + 2*a^2*d^3*e^3*E^(2*c)*ArcTanh[E^(c + d*x)] - 4*b^2*d^3*e^3* E^(2*c)*ArcTanh[E^(c + d*x)] + 12*a^2*d*e*f^2*ArcTanh[E^(c + d*x)] - 12*a^ 2*d*e*E^(2*c)*f^2*ArcTanh[E^(c + d*x)] + 3*a^2*d^3*e^2*f*x*Log[1 - E^(c + d*x)] - 6*b^2*d^3*e^2*f*x*Log[1 - E^(c + d*x)] - 3*a^2*d^3*e^2*E^(2*c)*f*x *Log[1 - E^(c + d*x)] + 6*b^2*d^3*e^2*E^(2*c)*f*x*Log[1 - E^(c + d*x)] - 6 *a^2*d*f^3*x*Log[1 - E^(c + d*x)] + 6*a^2*d*E^(2*c)*f^3*x*Log[1 - E^(c + d *x)] + 3*a^2*d^3*e*f^2*x^2*Log[1 - E^(c + d*x)] - 6*b^2*d^3*e*f^2*x^2*Log[ 1 - E^(c + d*x)] - 3*a^2*d^3*e*E^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] + 6*b^ 2*d^3*e*E^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] + a^2*d^3*f^3*x^3*Log[1 - E^( c + d*x)] - 2*b^2*d^3*f^3*x^3*Log[1 - E^(c + d*x)] - a^2*d^3*E^(2*c)*f^3*x ^3*Log[1 - E^(c + d*x)] + 2*b^2*d^3*E^(2*c)*f^3*x^3*Log[1 - E^(c + d*x)] - 3*a^2*d^3*e^2*f*x*Log[1 + E^(c + d*x)] + 6*b^2*d^3*e^2*f*x*Log[1 + E^(c + d*x)] + 3*a^2*d^3*e^2*E^(2*c)*f*x*Log[1 + E^(c + d*x)] - 6*b^2*d^3*e^2*E^ (2*c)*f*x*Log[1 + E^(c + d*x)] + 6*a^2*d*f^3*x*Log[1 + E^(c + d*x)] - 6*a^ 2*d*E^(2*c)*f^3*x*Log[1 + E^(c + d*x)] - 3*a^2*d^3*e*f^2*x^2*Log[1 + E^(c + d*x)] + 6*b^2*d^3*e*f^2*x^2*Log[1 + E^(c + d*x)] + 3*a^2*d^3*e*E^(2*c)*f ^2*x^2*Log[1 + E^(c + d*x)] - 6*b^2*d^3*e*E^(2*c)*f^2*x^2*Log[1 + E^(c + d *x)] - a^2*d^3*f^3*x^3*Log[1 + E^(c + d*x)] + 2*b^2*d^3*f^3*x^3*Log[1 +...
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.
| \(\displaystyle \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle \frac {\int (e+f x)^3 \text {csch}^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -i (e+f x)^3 \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \int (e+f x)^3 \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (-\frac {3 f^2 \int -i (e+f x) \text {csch}(c+d x)dx}{d^2}+\frac {1}{2} \int -i (e+f x)^3 \text {csch}(c+d x)dx-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {3 i f^2 \int (e+f x) \text {csch}(c+d x)dx}{d^2}-\frac {1}{2} i \int (e+f x)^3 \text {csch}(c+d x)dx-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {3 i f^2 \int i (e+f x) \csc (i c+i d x)dx}{d^2}-\frac {1}{2} i \int i (e+f x)^3 \csc (i c+i d x)dx-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (-\frac {3 f^2 \int (e+f x) \csc (i c+i d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \csc (i c+i d x)dx-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (-\frac {3 f^2 \left (\frac {i f \int \log \left (1-e^{c+d x}\right )dx}{d}-\frac {i f \int \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (-\frac {3 f^2 \left (\frac {i f \int e^{-c-d x} \log \left (1-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i f \int e^{-c-d x} \log \left (1+e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {1}{2} \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^3 \text {csch}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -(e+f x)^3 \csc (i c+i d x)^2dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\int (e+f x)^3 \csc (i c+i d x)^2dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {3 i f \int -i (e+f x)^2 \coth (c+d x)dx}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {3 f \int (e+f x)^2 \coth (c+d x)dx}{d}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {3 f \int -i (e+f x)^2 \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)^2}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int (e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (\frac {\int (e+f x)^3 \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (\frac {\int i (e+f x)^3 \csc (i c+i d x)dx}{a}-\frac {b \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{a}\right )}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}-\frac {b \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{a}\right )}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (-\frac {2 b \int -\frac {e^{c+d x} (e+f x)^3}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (\frac {2 b \int \frac {e^{c+d x} (e+f x)^3}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \int -\frac {e^{c+d x} (e+f x)^3}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)^3}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
Input:
Int[((e + f*x)^3*Csch[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{3} \operatorname {csch}\left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^3*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 18159 vs. \(2 (977) = 1954\).
Time = 0.44 (sec) , antiderivative size = 18159, normalized size of antiderivative = 17.25 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**3*csch(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \operatorname {csch}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
-1/2*e^3*(2*b^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3*d) - 2*(a*e^(-d*x - c) + 2*b *e^(-2*d*x - 2*c) + a*e^(-3*d*x - 3*c) - 2*b)/((2*a^2*e^(-2*d*x - 2*c) - a ^2*e^(-4*d*x - 4*c) - a^2)*d) - (a^2 - 2*b^2)*log(e^(-d*x - c) + 1)/(a^3*d ) + (a^2 - 2*b^2)*log(e^(-d*x - c) - 1)/(a^3*d)) - (2*b*d*f^3*x^3 + 6*b*d* e*f^2*x^2 + 6*b*d*e^2*f*x + (a*d*f^3*x^3*e^(3*c) + 3*a*e^2*f*e^(3*c) + 3*( d*e*f^2 + f^3)*a*x^2*e^(3*c) + 3*(d*e^2*f + 2*e*f^2)*a*x*e^(3*c))*e^(3*d*x ) - 2*(b*d*f^3*x^3*e^(2*c) + 3*b*d*e*f^2*x^2*e^(2*c) + 3*b*d*e^2*f*x*e^(2* c))*e^(2*d*x) + (a*d*f^3*x^3*e^c - 3*a*e^2*f*e^c + 3*(d*e*f^2 - f^3)*a*x^2 *e^c + 3*(d*e^2*f - 2*e*f^2)*a*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) + 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) + 1/2*(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)))*(a^2*f^3 - 2 *b^2*f^3)/(a^3*d^4) - 1/2*(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))) *(a^2*f^3 - 2*b^2*f^3)/(a^3*d^4) + 3/2*(a^2*d*e*f^2 - 2*b^2*d*e*f^2 - 2*a* b*f^3)*(d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polyl og(3, -e^(d*x + c)))/(a^3*d^4) - 3/2*(a^2*d*e*f^2 - 2*b^2*d*e*f^2 + 2*a...
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^3*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \] Input:
int((e + f*x)^3/(sinh(c + d*x)^3*(a + b*sinh(c + d*x))),x)
Output:
int((e + f*x)^3/(sinh(c + d*x)^3*(a + b*sinh(c + d*x))), x)
\[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^3*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
( - 16*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sq rt(a**2 + b**2))*b**4*d**3*e**3*i + 32*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**4*d**3*e**3*i - 16*sqr t(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*b**4*d**3* e**3*i - 256*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**6*d**4*f**3 - 256*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**4*b**2*d**4*f**3 - 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**6*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**4*b**2*d**4*e*f**2 - 768*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...