Integrand size = 29, antiderivative size = 214 \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {i (e+f x) \coth (c+d x)}{a d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {2 i f \log \left (\cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}-\frac {i f \log (\sinh (c+d x))}{a d^2}+\frac {3 f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {i (e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]
3*(f*x+e)*arctanh(exp(d*x+c))/a/d+I*(f*x+e)*coth(d*x+c)/a/d-1/2*f*csch(d*x +c)/a/d^2-1/2*(f*x+e)*coth(d*x+c)*csch(d*x+c)/a/d-2*I*f*ln(cosh(1/2*c+1/4* I*Pi+1/2*d*x))/a/d^2-I*f*ln(sinh(d*x+c))/a/d^2+3/2*f*polylog(2,-exp(d*x+c) )/a/d^2-3/2*f*polylog(2,exp(d*x+c))/a/d^2+I*(f*x+e)*tanh(1/2*c+1/4*I*Pi+1/ 2*d*x)/a/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(461\) vs. \(2(214)=428\).
Time = 3.54 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.15 \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left (2 i (i f+2 d (e+f x)) \cosh \left (\frac {1}{2} (c+d x)\right ) \left (i+\coth \left (\frac {1}{2} (c+d x)\right )\right )-d (e+f x) \left (i+\coth \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right )-8 f (c+d x) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )+16 f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )+4 \left (-2 i f (c+d x)-(2 i f+3 d (e+f x)) \log \left (1-e^{-c-d x}\right )+(-2 i f+3 d (e+f x)) \log \left (1+e^{-c-d x}\right )-3 f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+3 f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )+16 i d (e+f x) \sinh \left (\frac {1}{2} (c+d x)\right )+8 f \log (\cosh (c+d x)) \left (-i \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )+2 (f+2 i d (e+f x)) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \tanh \left (\frac {1}{2} (c+d x)\right )-i d (e+f x) \text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (-i+\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{8 d^2 (a+i a \sinh (c+d x))} \]
((Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])*((2*I)*(I*f + 2*d*(e + f*x))*Co sh[(c + d*x)/2]*(I + Coth[(c + d*x)/2]) - d*(e + f*x)*(I + Coth[(c + d*x)/ 2])*Csch[(c + d*x)/2] - 8*f*(c + d*x)*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x )/2]) + 16*f*ArcTan[Tanh[(c + d*x)/2]]*(Cosh[(c + d*x)/2] + I*Sinh[(c + d* x)/2]) + 4*((-2*I)*f*(c + d*x) - ((2*I)*f + 3*d*(e + f*x))*Log[1 - E^(-c - d*x)] + ((-2*I)*f + 3*d*(e + f*x))*Log[1 + E^(-c - d*x)] - 3*f*PolyLog[2, -E^(-c - d*x)] + 3*f*PolyLog[2, E^(-c - d*x)])*(Cosh[(c + d*x)/2] + I*Sin h[(c + d*x)/2]) + (16*I)*d*(e + f*x)*Sinh[(c + d*x)/2] + 8*f*Log[Cosh[c + d*x]]*((-I)*Cosh[(c + d*x)/2] + Sinh[(c + d*x)/2]) + 2*(f + (2*I)*d*(e + f *x))*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])*Tanh[(c + d*x)/2] - I*d*(e + f*x)*Sech[(c + d*x)/2]*(-I + Tanh[(c + d*x)/2])))/(8*d^2*(a + I*a*Sinh[c + d*x]))
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.
| \(\displaystyle \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle \frac {\int (e+f x) \text {csch}^3(c+d x)dx}{a}-i \int \frac {(e+f x) \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -i (e+f x) \csc (i c+i d x)^3dx}{a}-i \int \frac {(e+f x) \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \int (e+f x) \csc (i c+i d x)^3dx}{a}-i \int \frac {(e+f x) \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4673 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \int -i (e+f x) \text {csch}(c+d x)dx-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x) \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (-\frac {1}{2} i \int (e+f x) \text {csch}(c+d x)dx-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x) \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (-\frac {1}{2} i \int i (e+f x) \csc (i c+i d x)dx-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x) \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \int (e+f x) \csc (i c+i d x)dx-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x) \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (\frac {i f \int \log \left (1-e^{c+d x}\right )dx}{d}-\frac {i f \int \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x) \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (\frac {i f \int e^{-c-d x} \log \left (1-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i f \int e^{-c-d x} \log \left (1+e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x) \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -i \int \frac {(e+f x) \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -i \left (\frac {\int (e+f x) \text {csch}^2(c+d x)dx}{a}-i \int \frac {(e+f x) \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {\int -\left ((e+f x) \csc (i c+i d x)^2\right )dx}{a}-i \int \frac {(e+f x) \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (-\frac {\int (e+f x) \csc (i c+i d x)^2dx}{a}-i \int \frac {(e+f x) \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -i \left (-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {i f \int -i \coth (c+d x)dx}{d}}{a}-i \int \frac {(e+f x) \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \int \coth (c+d x)dx}{d}}{a}-i \int \frac {(e+f x) \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \int -i \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{a}-i \int \frac {(e+f x) \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (-\frac {\frac {(e+f x) \coth (c+d x)}{d}+\frac {i f \int \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}}{a}-i \int \frac {(e+f x) \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -i \left (-i \int \frac {(e+f x) \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -i \left (-i \left (\frac {\int (e+f x) \text {csch}(c+d x)dx}{a}-i \int \frac {e+f x}{i \sinh (c+d x) a+a}dx\right )-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (-i \left (\frac {\int i (e+f x) \csc (i c+i d x)dx}{a}-i \int \frac {e+f x}{\sin (i c+i d x) a+a}dx\right )-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (-i \left (\frac {i \int (e+f x) \csc (i c+i d x)dx}{a}-i \int \frac {e+f x}{\sin (i c+i d x) a+a}dx\right )-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle -i \left (-i \left (\frac {i \int (e+f x) \csc (i c+i d x)dx}{a}-\frac {i \int -\left ((e+f x) \text {csch}^2\left (\frac {c}{2}+\frac {d x}{2}-\frac {i \pi }{4}\right )\right )dx}{2 a}\right )-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (-i \left (\frac {i \int (e+f x) \csc (i c+i d x)dx}{a}+\frac {i \int -\left ((e+f x) \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )dx}{2 a}\right )-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (-i \left (\frac {i \int (e+f x) \csc (i c+i d x)dx}{a}-\frac {i \int (e+f x) \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}\right )-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (-i \left (\frac {i \int (e+f x) \csc (i c+i d x)dx}{a}-\frac {i \int (e+f x) \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -i \left (-i \left (\frac {i \left (\frac {i f \int \log \left (1-e^{c+d x}\right )dx}{d}-\frac {i f \int \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \int (e+f x) \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -i \left (-i \left (\frac {i \left (\frac {i f \int e^{-c-d x} \log \left (1-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i f \int e^{-c-d x} \log \left (1+e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \int (e+f x) \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -i \left (-i \left (\frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{a}-\frac {i \int (e+f x) \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -i \left (-i \left (\frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{a}-\frac {i \left (\frac {2 (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 i f \int -i \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (-i \left (\frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{a}-\frac {i \left (\frac {2 (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 f \int \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x) \coth (c+d x)}{d}-\frac {f \log (-i \sinh (c+d x))}{d^2}}{a}\right )-\frac {i \left (\frac {1}{2} \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-\frac {i f \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
3.3.19.3.1 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[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[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[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Simp[(2*a)^n Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-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[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S imp[b^2*((n - 2)/(n - 1)) Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a Int[(e + f*x)^m*Csch[ c + d*x]^n, x], x] - Simp[b/a Int[(e + f*x)^m*(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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (185 ) = 370\).
Time = 2.38 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.06
| method | result | size |
| risch | \(-\frac {-3 i d e \,{\mathrm e}^{3 d x +3 c}-5 \,{\mathrm e}^{2 d x +2 c} d f x +3 d f x \,{\mathrm e}^{4 d x +4 c}+i d e \,{\mathrm e}^{d x +c}-5 \,{\mathrm e}^{2 d x +2 c} d e +3 d e \,{\mathrm e}^{4 d x +4 c}+i d f x \,{\mathrm e}^{d x +c}+4 d f x +f \,{\mathrm e}^{4 d x +4 c}-i {\mathrm e}^{3 d x +3 c} f +i {\mathrm e}^{d x +c} f +4 d e -f \,{\mathrm e}^{2 d x +2 c}-3 i d f x \,{\mathrm e}^{3 d x +3 c}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} d^{2} \left ({\mathrm e}^{d x +c}-i\right ) a}-\frac {3 f \ln \left (1-{\mathrm e}^{d x +c}\right ) x}{2 a d}+\frac {3 f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{2 a d}-\frac {3 e \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a d}+\frac {4 i f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}-\frac {i f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}-\frac {i f \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{2}}-\frac {i f \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{2}}+\frac {3 e \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a d}-\frac {3 f \ln \left (1-{\mathrm e}^{d x +c}\right ) c}{2 a \,d^{2}}+\frac {3 c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a \,d^{2}}+\frac {2 f \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {3 f \operatorname {polylog}\left (2, -{\mathrm e}^{d x +c}\right )}{2 a \,d^{2}}-\frac {3 f \operatorname {polylog}\left (2, {\mathrm e}^{d x +c}\right )}{2 a \,d^{2}}\) | \(441\) |
-(-3*I*d*e*exp(3*d*x+3*c)-5*exp(2*d*x+2*c)*d*f*x+3*d*f*x*exp(4*d*x+4*c)+I* d*e*exp(d*x+c)-5*exp(2*d*x+2*c)*d*e+3*d*e*exp(4*d*x+4*c)+I*d*f*x*exp(d*x+c )+4*d*f*x+f*exp(4*d*x+4*c)-I*exp(3*d*x+3*c)*f+I*exp(d*x+c)*f+4*d*e-f*exp(2 *d*x+2*c)-3*I*d*f*x*exp(3*d*x+3*c))/(exp(2*d*x+2*c)-1)^2/d^2/(exp(d*x+c)-I )/a-3/2/a/d*f*ln(1-exp(d*x+c))*x+3/2/a/d*f*ln(exp(d*x+c)+1)*x-3/2/a/d*e*ln (exp(d*x+c)-1)+4*I/a/d^2*f*ln(exp(d*x+c))-I/a/d^2*f*ln(exp(d*x+c)-1)-I/a/d ^2*f*ln(exp(d*x+c)+1)-I/a/d^2*f*ln(1+exp(2*d*x+2*c))+3/2/a/d*e*ln(exp(d*x+ c)+1)-3/2/a/d^2*f*ln(1-exp(d*x+c))*c+3/2/a/d^2*c*f*ln(exp(d*x+c)-1)+2/a/d^ 2*f*arctan(exp(d*x+c))+3/2*f*polylog(2,-exp(d*x+c))/a/d^2-3/2*f*polylog(2, exp(d*x+c))/a/d^2
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 817 vs. \(2 (181) = 362\).
Time = 0.28 (sec) , antiderivative size = 817, normalized size of antiderivative = 3.82 \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]
-1/2*(8*d*e - 4*c*f - 3*(f*e^(5*d*x + 5*c) - I*f*e^(4*d*x + 4*c) - 2*f*e^( 3*d*x + 3*c) + 2*I*f*e^(2*d*x + 2*c) + f*e^(d*x + c) - I*f)*dilog(-e^(d*x + c)) + 3*(f*e^(5*d*x + 5*c) - I*f*e^(4*d*x + 4*c) - 2*f*e^(3*d*x + 3*c) + 2*I*f*e^(2*d*x + 2*c) + f*e^(d*x + c) - I*f)*dilog(e^(d*x + c)) + 4*(-2*I *d*f*x - I*c*f)*e^(5*d*x + 5*c) - 2*(d*f*x - 3*d*e + (2*c - 1)*f)*e^(4*d*x + 4*c) + 2*(5*I*d*f*x - 3*I*d*e + (4*I*c - I)*f)*e^(3*d*x + 3*c) + 2*(3*d *f*x - 5*d*e + (4*c - 1)*f)*e^(2*d*x + 2*c) + 2*(-3*I*d*f*x + I*d*e + (-2* I*c + I)*f)*e^(d*x + c) - (-3*I*d*f*x - 3*I*d*e + (3*d*f*x + 3*d*e - 2*I*f )*e^(5*d*x + 5*c) + (-3*I*d*f*x - 3*I*d*e - 2*f)*e^(4*d*x + 4*c) - 2*(3*d* f*x + 3*d*e - 2*I*f)*e^(3*d*x + 3*c) - 2*(-3*I*d*f*x - 3*I*d*e - 2*f)*e^(2 *d*x + 2*c) + (3*d*f*x + 3*d*e - 2*I*f)*e^(d*x + c) - 2*f)*log(e^(d*x + c) + 1) + 4*(I*f*e^(5*d*x + 5*c) + f*e^(4*d*x + 4*c) - 2*I*f*e^(3*d*x + 3*c) - 2*f*e^(2*d*x + 2*c) + I*f*e^(d*x + c) + f)*log(e^(d*x + c) - I) - (3*I* d*e + (-3*I*c - 2)*f - (3*d*e - (3*c - 2*I)*f)*e^(5*d*x + 5*c) + (3*I*d*e + (-3*I*c - 2)*f)*e^(4*d*x + 4*c) + 2*(3*d*e - (3*c - 2*I)*f)*e^(3*d*x + 3 *c) - 2*(3*I*d*e + (-3*I*c - 2)*f)*e^(2*d*x + 2*c) - (3*d*e - (3*c - 2*I)* f)*e^(d*x + c))*log(e^(d*x + c) - 1) + 3*(-I*d*f*x - I*c*f + (d*f*x + c*f) *e^(5*d*x + 5*c) + (-I*d*f*x - I*c*f)*e^(4*d*x + 4*c) - 2*(d*f*x + c*f)*e^ (3*d*x + 3*c) + 2*(I*d*f*x + I*c*f)*e^(2*d*x + 2*c) + (d*f*x + c*f)*e^(d*x + c))*log(-e^(d*x + c) + 1))/(a*d^2*e^(5*d*x + 5*c) - I*a*d^2*e^(4*d*x...
\[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \left (\int \frac {e \operatorname {csch}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f x \operatorname {csch}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \]
-I*(Integral(e*csch(c + d*x)**3/(sinh(c + d*x) - I), x) + Integral(f*x*csc h(c + d*x)**3/(sinh(c + d*x) - I), x))/a
\[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {csch}\left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]
-(24*d*integrate(1/16*x/(a*d*e^(d*x + c) + a*d), x) + 24*d*integrate(1/16* x/(a*d*e^(d*x + c) - a*d), x) + 8*(2*d*x*e^(5*d*x + 5*c) + 2*I*d*x + (I*d* x*e^(4*c) + I*e^(4*c))*e^(4*d*x) - (d*x*e^(3*c) - e^(3*c))*e^(3*d*x) + (-I *d*x*e^(2*c) - I*e^(2*c))*e^(2*d*x) + (d*x*e^c - e^c)*e^(d*x))/(8*I*a*d^2* e^(5*d*x + 5*c) + 8*a*d^2*e^(4*d*x + 4*c) - 16*I*a*d^2*e^(3*d*x + 3*c) - 1 6*a*d^2*e^(2*d*x + 2*c) + 8*I*a*d^2*e^(d*x + c) + 8*a*d^2) - 2*I*(d*x + c) /(a*d^2) + 2*I*log((e^(d*x + c) - I)*e^(-c))/(a*d^2) + I*log(e^(d*x + c) + 1)/(a*d^2) + I*log(e^(d*x + c) - 1)/(a*d^2))*f - 1/2*e*(2*(-I*e^(-d*x - c ) - 5*e^(-2*d*x - 2*c) + 3*I*e^(-3*d*x - 3*c) + 3*e^(-4*d*x - 4*c) + 4)/(( a*e^(-d*x - c) - 2*I*a*e^(-2*d*x - 2*c) - 2*a*e^(-3*d*x - 3*c) + I*a*e^(-4 *d*x - 4*c) + a*e^(-5*d*x - 5*c) + I*a)*d) - 3*log(e^(-d*x - c) + 1)/(a*d) + 3*log(e^(-d*x - c) - 1)/(a*d))
\[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {csch}\left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x) \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {e+f\,x}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]