Integrand size = 31, antiderivative size = 368 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 i (e+f x)^2}{a d}+\frac {3 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {i (e+f x)^2 \coth (c+d x)}{a d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {4 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {3 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {3 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]
2*I*(f*x+e)^2/a/d+3*(f*x+e)^2*arctanh(exp(d*x+c))/a/d-f^2*arctanh(cosh(d*x +c))/a/d^3+I*(f*x+e)^2*coth(d*x+c)/a/d-f*(f*x+e)*csch(d*x+c)/a/d^2-1/2*(f* x+e)^2*coth(d*x+c)*csch(d*x+c)/a/d-4*I*f*(f*x+e)*ln(1+I*exp(d*x+c))/a/d^2- 2*I*f*(f*x+e)*ln(1-exp(2*d*x+2*c))/a/d^2+3*f*(f*x+e)*polylog(2,-exp(d*x+c) )/a/d^2-4*I*f^2*polylog(2,-I*exp(d*x+c))/a/d^3-3*f*(f*x+e)*polylog(2,exp(d *x+c))/a/d^2-I*f^2*polylog(2,exp(2*d*x+2*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+I*(f*x+e)^2*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. \(1496\) vs. \(2(368)=736\).
Time = 8.80 (sec) , antiderivative size = 1496, normalized size of antiderivative = 4.07 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx =\text {Too large to display} \]
(-4*E^c*f*((e + f*x)^2/(2*E^c*f) + ((I + E^(-c))*(e + f*x)*Log[1 - I*E^(-c - d*x)])/d - ((1 + I*E^c)*f*PolyLog[2, I*E^(-c - d*x)])/(d^2*E^c)))/(a*d* (-I + E^c)) + (-(d*(-1 + E^(2*c))*(3*d^2*e^2 - (4*I)*d*e*f - 2*f^2)*x) + d *(-1 + E^(2*c))*(3*d^2*e^2 + (4*I)*d*e*f - 2*f^2)*x + (4*I)*d^2*(e + f*x)^ 2 - 2*d*(-1 + E^(2*c))*(3*d*e + (2*I)*f)*f*x*Log[1 - E^(-c - d*x)] - 3*d^2 *(-1 + E^(2*c))*f^2*x^2*Log[1 - E^(-c - d*x)] + 2*d*(-1 + E^(2*c))*(3*d*e - (2*I)*f)*f*x*Log[1 + E^(-c - d*x)] + 3*d^2*(-1 + E^(2*c))*f^2*x^2*Log[1 + E^(-c - d*x)] - (-1 + E^(2*c))*(3*d^2*e^2 + (4*I)*d*e*f - 2*f^2)*Log[1 - E^(c + d*x)] + (-1 + E^(2*c))*(3*d^2*e^2 - (4*I)*d*e*f - 2*f^2)*Log[1 + E ^(c + d*x)] - 2*(-1 + E^(2*c))*(3*d*e - (2*I)*f)*f*PolyLog[2, -E^(-c - d*x )] - 6*d*(-1 + E^(2*c))*f^2*x*PolyLog[2, -E^(-c - d*x)] + 2*(-1 + E^(2*c)) *(3*d*e + (2*I)*f)*f*PolyLog[2, E^(-c - d*x)] + 6*d*(-1 + E^(2*c))*f^2*x*P olyLog[2, E^(-c - d*x)] - 6*(-1 + E^(2*c))*f^2*PolyLog[3, -E^(-c - d*x)] + 6*(-1 + E^(2*c))*f^2*PolyLog[3, E^(-c - d*x)])/(2*a*d^3*(-1 + E^(2*c))) + (Csch[c]*Csch[c + d*x]^2*(2*e*f*Cosh[(d*x)/2] + 2*f^2*x*Cosh[(d*x)/2] + 2 *e*f*Cosh[(3*d*x)/2] + 2*f^2*x*Cosh[(3*d*x)/2] + (5*I)*d*e^2*Cosh[c - (d*x )/2] + (10*I)*d*e*f*x*Cosh[c - (d*x)/2] + (5*I)*d*f^2*x^2*Cosh[c - (d*x)/2 ] - I*d*e^2*Cosh[c + (d*x)/2] - (2*I)*d*e*f*x*Cosh[c + (d*x)/2] - I*d*f^2* x^2*Cosh[c + (d*x)/2] - 2*e*f*Cosh[2*c + (d*x)/2] - 2*f^2*x*Cosh[2*c + (d* x)/2] + I*d*e^2*Cosh[c + (3*d*x)/2] + (2*I)*d*e*f*x*Cosh[c + (3*d*x)/2]...
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle \frac {\int (e+f x)^2 \text {csch}^3(c+d x)dx}{a}-i \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -i (e+f x)^2 \csc (i c+i d x)^3dx}{a}-i \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \int (e+f x)^2 \csc (i c+i d x)^3dx}{a}-i \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle -\frac {i \left (-\frac {f^2 \int -i \text {csch}(c+d x)dx}{d^2}+\frac {1}{2} \int -i (e+f x)^2 \text {csch}(c+d x)dx-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {i f^2 \int \text {csch}(c+d x)dx}{d^2}-\frac {1}{2} i \int (e+f x)^2 \text {csch}(c+d x)dx-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {i f^2 \int i \csc (i c+i d x)dx}{d^2}-\frac {1}{2} i \int i (e+f x)^2 \csc (i c+i d x)dx-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (-\frac {f^2 \int \csc (i c+i d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^2 \csc (i c+i d x)dx-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \int (e+f x)^2 \csc (i c+i d x)dx-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (\frac {2 i f \int (e+f x) \log \left (1-e^{c+d x}\right )dx}{d}-\frac {2 i f \int (e+f x) \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (\frac {\int (e+f x)^2 \text {csch}^2(c+d x)dx}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (\frac {\int -(e+f x)^2 \csc (i c+i d x)^2dx}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\int (e+f x)^2 \csc (i c+i d x)^2dx}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \coth (c+d x)dx}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {2 f \int (e+f x) \coth (c+d x)dx}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {2 f \int -i (e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int e^{-2 c-2 d x+i \pi } \log \left (1+e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-i \left (\frac {\int (e+f x)^2 \text {csch}(c+d x)dx}{a}-i \int \frac {(e+f x)^2}{i \sinh (c+d x) a+a}dx\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-i \left (\frac {\int i (e+f x)^2 \csc (i c+i d x)dx}{a}-i \int \frac {(e+f x)^2}{\sin (i c+i d x) a+a}dx\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-i \left (\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}-i \int \frac {(e+f x)^2}{\sin (i c+i d x) a+a}dx\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-i \left (\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}-\frac {i \int -(e+f x)^2 \text {csch}^2\left (\frac {c}{2}+\frac {d x}{2}-\frac {i \pi }{4}\right )dx}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-i \left (\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}+\frac {i \int -(e+f x)^2 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-i \left (\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}-\frac {i \int (e+f x)^2 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-i \left (\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}-\frac {i \int (e+f x)^2 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -i \left (-i \left (\frac {i \left (\frac {2 i f \int (e+f x) \log \left (1-e^{c+d x}\right )dx}{d}-\frac {2 i f \int (e+f x) \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \int (e+f x)^2 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )-\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
3.3.18.3.1 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[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*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_))^(m_), x_Symbo l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ (n - 1)) Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c , d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
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. 1146 vs. \(2 (343 ) = 686\).
Time = 2.78 (sec) , antiderivative size = 1147, normalized size of antiderivative = 3.12
-4*I*f^2*polylog(2,-I*exp(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^3-3/2/a/d*e^2*ln(exp(d*x+c)-1)+3/2/a/d*e^ 2*ln(exp(d*x+c)+1)-3/a/d^2*e*f*ln(1-exp(d*x+c))*c-3/a/d*e*f*ln(1-exp(d*x+c ))*x+3/a/d*e*f*ln(exp(d*x+c)+1)*x+3/a/d^2*e*c*f*ln(exp(d*x+c)-1)+2*I/a/d^3 *f^2*c*ln(1+exp(2*d*x+2*c))+8*I/a/d^2*e*f*ln(exp(d*x+c))-2*I/a/d^2*e*f*ln( exp(d*x+c)-1)-2*I/a/d^2*e*f*ln(exp(d*x+c)+1)-2*I/a/d^2*e*f*ln(1+exp(2*d*x+ 2*c))-2*I/a/d^2*f^2*ln(1-exp(d*x+c))*x-2*I/a/d^2*f^2*ln(exp(d*x+c)+1)*x-4* I/a/d^2*f^2*ln(1+I*exp(d*x+c))*x+8*I/a/d^2*f^2*c*x-8*I/a/d^3*c*f^2*ln(exp( d*x+c))-4*I/a/d^3*f^2*ln(1+I*exp(d*x+c))*c-2*I/a/d^3*f^2*ln(1-exp(d*x+c))* c+2*I/a/d^3*f^2*c*ln(exp(d*x+c)-1)-3/a/d^2*e*f*polylog(2,exp(d*x+c))+3/a/d ^2*e*f*polylog(2,-exp(d*x+c))+3/2/a/d^3*f^2*ln(1-exp(d*x+c))*c^2-3/2/a/d*f ^2*ln(1-exp(d*x+c))*x^2-3/a/d^2*f^2*polylog(2,exp(d*x+c))*x+3/2/a/d*f^2*ln (exp(d*x+c)+1)*x^2+3/a/d^2*f^2*polylog(2,-exp(d*x+c))*x-4/a/d^3*c*f^2*arct an(exp(d*x+c))+4/a/d^2*e*f*arctan(exp(d*x+c))-2*I/a/d^3*f^2*polylog(2,-exp (d*x+c))+4*I/a/d^3*c^2*f^2+4*I/a/d*f^2*x^2-2*I/a/d^3*f^2*polylog(2,exp(d*x +c))-3/2/a/d^3*c^2*f^2*ln(exp(d*x+c)-1)-(4*d*e^2+3*d*e^2*exp(4*d*x+4*c)+2* f^2*x*exp(4*d*x+4*c)+2*e*f*exp(4*d*x+4*c)-2*f^2*x*exp(2*d*x+2*c)-2*e*f*exp (2*d*x+2*c)+4*d*f^2*x^2+3*d*f^2*x^2*exp(4*d*x+4*c)+2*I*f^2*x*exp(d*x+c)-5* f^2*x^2*d*exp(2*d*x+2*c)+6*d*e*f*x*exp(4*d*x+4*c)-3*I*d*f^2*x^2*exp(3*d*x+ 3*c)-2*I*e*f*exp(3*d*x+3*c)+2*I*d*e*f*x*exp(d*x+c)-3*I*d*e^2*exp(3*d*x+...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2215 vs. \(2 (329) = 658\).
Time = 0.28 (sec) , antiderivative size = 2215, normalized size of antiderivative = 6.02 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]
-1/2*(8*d^2*e^2 - 16*c*d*e*f + 8*c^2*f^2 + 8*(I*f^2*e^(5*d*x + 5*c) + f^2* e^(4*d*x + 4*c) - 2*I*f^2*e^(3*d*x + 3*c) - 2*f^2*e^(2*d*x + 2*c) + I*f^2* e^(d*x + c) + f^2)*dilog(-I*e^(d*x + c)) + 2*(3*I*d*f^2*x + 3*I*d*e*f + 2* f^2 - (3*d*f^2*x + 3*d*e*f - 2*I*f^2)*e^(5*d*x + 5*c) + (3*I*d*f^2*x + 3*I *d*e*f + 2*f^2)*e^(4*d*x + 4*c) + 2*(3*d*f^2*x + 3*d*e*f - 2*I*f^2)*e^(3*d *x + 3*c) + 2*(-3*I*d*f^2*x - 3*I*d*e*f - 2*f^2)*e^(2*d*x + 2*c) - (3*d*f^ 2*x + 3*d*e*f - 2*I*f^2)*e^(d*x + c))*dilog(-e^(d*x + c)) + 2*(-3*I*d*f^2* x - 3*I*d*e*f + 2*f^2 + (3*d*f^2*x + 3*d*e*f + 2*I*f^2)*e^(5*d*x + 5*c) + (-3*I*d*f^2*x - 3*I*d*e*f + 2*f^2)*e^(4*d*x + 4*c) - 2*(3*d*f^2*x + 3*d*e* f + 2*I*f^2)*e^(3*d*x + 3*c) + 2*(3*I*d*f^2*x + 3*I*d*e*f - 2*f^2)*e^(2*d* x + 2*c) + (3*d*f^2*x + 3*d*e*f + 2*I*f^2)*e^(d*x + c))*dilog(e^(d*x + c)) + 8*(-I*d^2*f^2*x^2 - 2*I*d^2*e*f*x - 2*I*c*d*e*f + I*c^2*f^2)*e^(5*d*x + 5*c) - 2*(d^2*f^2*x^2 - 3*d^2*e^2 + 2*(4*c - 1)*d*e*f - 4*c^2*f^2 + 2*(d^ 2*e*f - d*f^2)*x)*e^(4*d*x + 4*c) + 2*(5*I*d^2*f^2*x^2 - 3*I*d^2*e^2 + 2*( 8*I*c - I)*d*e*f - 8*I*c^2*f^2 + 2*(5*I*d^2*e*f - I*d*f^2)*x)*e^(3*d*x + 3 *c) + 2*(3*d^2*f^2*x^2 - 5*d^2*e^2 + 2*(8*c - 1)*d*e*f - 8*c^2*f^2 + 2*(3* d^2*e*f - d*f^2)*x)*e^(2*d*x + 2*c) + 2*(-3*I*d^2*f^2*x^2 + I*d^2*e^2 + 2* (-4*I*c + I)*d*e*f + 4*I*c^2*f^2 + 2*(-3*I*d^2*e*f + I*d*f^2)*x)*e^(d*x + c) - (-3*I*d^2*f^2*x^2 - 3*I*d^2*e^2 - 4*d*e*f + 2*I*f^2 - 2*(3*I*d^2*e*f + 2*d*f^2)*x + (3*d^2*f^2*x^2 + 3*d^2*e^2 - 4*I*d*e*f - 2*f^2 + 2*(3*d^...
\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \left (\int \frac {e^{2} \operatorname {csch}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f^{2} x^{2} \operatorname {csch}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {2 e f x \operatorname {csch}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \]
-I*(Integral(e**2*csch(c + d*x)**3/(sinh(c + d*x) - I), x) + Integral(f**2 *x**2*csch(c + d*x)**3/(sinh(c + d*x) - I), x) + Integral(2*e*f*x*csch(c + d*x)**3/(sinh(c + d*x) - I), x))/a
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 863 vs. \(2 (329) = 658\).
Time = 0.46 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.35 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]
-1/2*e^2*(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)) + 2*I*f^2*x^2 /(a*d) + 4*I*e*f*x/(a*d) - (4*d*f^2*x^2 + 8*d*e*f*x + (3*d*f^2*x^2*e^(4*c) + 2*e*f*e^(4*c) + 2*(3*d*e*f + f^2)*x*e^(4*c))*e^(4*d*x) + (-3*I*d*f^2*x^ 2*e^(3*c) - 2*I*e*f*e^(3*c) - 2*(3*I*d*e*f + I*f^2)*x*e^(3*c))*e^(3*d*x) - (5*d*f^2*x^2*e^(2*c) + 2*e*f*e^(2*c) + 2*(5*d*e*f + f^2)*x*e^(2*c))*e^(2* d*x) + (I*d*f^2*x^2*e^c + 2*I*e*f*e^c - 2*(-I*d*e*f - I*f^2)*x*e^c)*e^(d*x ))/(a*d^2*e^(5*d*x + 5*c) - I*a*d^2*e^(4*d*x + 4*c) - 2*a*d^2*e^(3*d*x + 3 *c) + 2*I*a*d^2*e^(2*d*x + 2*c) + a*d^2*e^(d*x + c) - I*a*d^2) - 4*I*e*f*l og(I*e^(d*x + c) + 1)/(a*d^2) + 3/2*(d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x* dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*f^2/(a*d^3) - 3/2*(d^2*x ^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*f^2/(a*d^3) - 4*I*(d*x*log(I*e^(d*x + c) + 1) + dilog(-I*e^(d*x + c )))*f^2/(a*d^3) + (2*I*d*e*f + f^2)*x/(a*d^2) + (2*I*d*e*f - f^2)*x/(a*d^2 ) + (3*d*e*f - 2*I*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/( a*d^3) - (3*d*e*f + 2*I*f^2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c )))/(a*d^3) - (2*I*d*e*f + f^2)*log(e^(d*x + c) + 1)/(a*d^3) - (2*I*d*e*f - f^2)*log(e^(d*x + c) - 1)/(a*d^3) + 1/2*(d^3*f^2*x^3 + (3*d*e*f + 2*I...
\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]