Integrand size = 34, antiderivative size = 1038 \[ \int \frac {(e+f x)^3 \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 b \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {6 b \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {3 b f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^3 d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}+\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^3 d^4}-\frac {6 b \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {6 b \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^4} \]
-3*b*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1 /2)/a^3/d^2+3*b*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*( a^2+b^2)^(1/2)/a^3/d^2+6*b*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2 )^(1/2)))*(a^2+b^2)^(1/2)/a^3/d^3-3*b^2*f*(f*x+e)^2*polylog(2,-exp(d*x+c)) /a^3/d^2+3*b^2*f*(f*x+e)^2*polylog(2,exp(d*x+c))/a^3/d^2-3*b*f^2*(f*x+e)*p olylog(2,exp(2*d*x+2*c))/a^2/d^3+6*b^2*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/ a^3/d^3-6*b^2*f^2*(f*x+e)*polylog(3,exp(d*x+c))/a^3/d^3-3*b*f*(f*x+e)^2*ln (1-exp(2*d*x+2*c))/a^2/d^2-3*f^3*polylog(4,-exp(d*x+c))/a/d^4+3*f^3*polylo g(4,exp(d*x+c))/a/d^4-6*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,ex p(d*x+c))/a/d^3-6*b*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+ b^2)^(1/2)/a^3/d^4+6*b*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a ^2+b^2)^(1/2)/a^3/d^4-b*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*( a^2+b^2)^(1/2)/a^3/d+b*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a ^2+b^2)^(1/2)/a^3/d+b*(f*x+e)^3*coth(d*x+c)/a^2/d+b*(f*x+e)^3/a^2/d-3*f^3* polylog(2,-exp(d*x+c))/a/d^4+3*f^3*polylog(2,exp(d*x+c))/a/d^4-2*b^2*(f*x+ e)^3*arctanh(exp(d*x+c))/a^3/d+3/2*b*f^3*polylog(3,exp(2*d*x+2*c))/a^2/d^4 -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...
Leaf count is larger than twice the leaf count of optimal. \(2799\) vs. \(2(1038)=2076\).
Time = 9.49 (sec) , antiderivative size = 2799, normalized size of antiderivative = 2.70 \[ \int \frac {(e+f x)^3 \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]
(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.
| \(\displaystyle \int \frac {(e+f x)^3 \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6121 |
| \(\displaystyle \frac {\int (e+f x)^3 \coth ^2(c+d x) \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5980 |
| \(\displaystyle \frac {\int (e+f x)^3 \text {csch}^3(c+d x)dx+\int (e+f x)^3 \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int i (e+f x)^3 \csc (i c+i d x)dx+\int -i (e+f x)^3 \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \int (e+f x)^3 \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \int (e+f x)^3 \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \int (e+f x)^3 \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (-\frac {3 f^2 \int -i (e+f x) \text {csch}(c+d x)dx}{d^2}+\frac {1}{2} \int -i (e+f x)^3 \text {csch}(c+d x)dx-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {3 i f^2 \int (e+f x) \text {csch}(c+d x)dx}{d^2}-\frac {1}{2} i \int (e+f x)^3 \text {csch}(c+d x)dx-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {3 i f^2 \int i (e+f x) \csc (i c+i d x)dx}{d^2}-\frac {1}{2} i \int i (e+f x)^3 \csc (i c+i d x)dx-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (-\frac {3 f^2 \int (e+f x) \csc (i c+i d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \csc (i c+i d x)dx-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (-\frac {3 f^2 \left (\frac {i f \int \log \left (1-e^{c+d x}\right )dx}{d}-\frac {i f \int \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (-\frac {3 f^2 \left (\frac {i f \int e^{-c-d x} \log \left (1-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i f \int e^{-c-d x} \log \left (1+e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 6103 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^3 \coth ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -(e+f x)^3 \tan \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\int (e+f x)^3 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \int i (e+f x)^2 \coth (c+d x)dx}{d}-\int (e+f x)^3dx+\frac {(e+f x)^3 \coth (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \int i (e+f x)^2 \coth (c+d x)dx}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {-\frac {3 f \int (e+f x)^2 \coth (c+d x)dx}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\right )}{a}+\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {-\frac {3 f \int -i (e+f x)^2 \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)^2}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int (e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (\frac {\int (e+f x)^3 \cosh (c+d x) \coth (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (\frac {\int (e+f x)^3 \sinh (c+d x)dx+\int (e+f x)^3 \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -i (e+f x)^3 \sin (i c+i d x)dx+\int i (e+f x)^3 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^3 \csc (i c+i d x)dx-i \int (e+f x)^3 \sin (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{a}\right )}{a}\) |
3.5.81.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
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_.)*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[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
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_)]*(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[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b* x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Int[Coth[(a_.) + (b_.)*(x_)]^(p_)*Csch[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*( x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Csch[a + b*x]*Coth[a + b*x]^(p - 2 ), x] + Int[(c + d*x)^m*Csch[a + b*x]^3*Coth[a + b*x]^(p - 2), x] /; FreeQ[ {a, b, c, d, m}, x] && IGtQ[p/2, 0]
Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a Int[(e + f*x)^m*Coth[ c + d*x]^n, x], x] - Simp[b/a Int[(e + f*x)^m*Cosh[c + d*x]*(Coth[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]
Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/a Int[(e + f*x)^m*Cosh[c + d*x]^p*Coth[c + d*x]^n, x], x] - Simp[b/ a Int[(e + f*x)^m*Cosh[c + d*x]^(p + 1)*(Coth[c + d*x]^(n - 1)/(a + b*Sin h[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[ n, 0] && IGtQ[p, 0]
Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*Csch[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/a Int[(e + f*x)^m*Csch[c + d*x]^p*Coth[c + d*x]^n, x], x] - Simp[b/ a Int[(e + f*x)^m*Csch[c + d*x]^(p - 1)*(Coth[c + d*x]^n/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
\[\int \frac {\left (f x +e \right )^{3} \coth \left (d x +c \right )^{2} \operatorname {csch}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 13504 vs. \(2 (962) = 1924\).
Time = 0.52 (sec) , antiderivative size = 13504, normalized size of antiderivative = 13.01 \[ \int \frac {(e+f x)^3 \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(e+f x)^3 \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(e+f x)^3 \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \coth \left (d x + c\right )^{2} \operatorname {csch}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
1/2*e^3*(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*(a^2*b + 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*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*polylog(3, -e^(d*x + c)))/(a^3*d^4) + 3/2*(a^2*d*e*f^2 + 2*b^2*d*e*...
Timed out. \[ \int \frac {(e+f x)^3 \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x)^3 \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]