Integrand size = 32, antiderivative size = 462 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {(e+f x)^3}{3 b f}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {2 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {2 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^3} \]
1/3*(f*x+e)^3/b/f-2*(f*x+e)^2*arctanh(exp(d*x+c))/a/d-2*f*(f*x+e)*polylog( 2,-exp(d*x+c))/a/d^2+2*f*(f*x+e)*polylog(2,exp(d*x+c))/a/d^2+2*f^2*polylog (3,-exp(d*x+c))/a/d^3-2*f^2*polylog(3,exp(d*x+c))/a/d^3-(f*x+e)^2*ln(1+b*e xp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d+(f*x+e)^2*ln(1+b*exp( d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d-2*f*(f*x+e)*polylog(2,-b *exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d^2+2*f*(f*x+e)*polyl og(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d^2+2*f^2*poly log(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d^3-2*f^2*pol ylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d^3
Time = 0.71 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.06 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {x \left (3 e^2+3 e f x+f^2 x^2\right )}{3 b}+\frac {(e+f x)^2 \log \left (1-e^{c+d x}\right )-(e+f x)^2 \log \left (1+e^{c+d x}\right )-\frac {2 f \left (d (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )-f \operatorname {PolyLog}\left (3,-e^{c+d x}\right )\right )}{d^2}+\frac {2 f \left (d (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )-f \operatorname {PolyLog}\left (3,e^{c+d x}\right )\right )}{d^2}}{a d}-\frac {\sqrt {a^2+b^2} \left (-2 d^2 e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a b d^3} \]
(x*(3*e^2 + 3*e*f*x + f^2*x^2))/(3*b) + ((e + f*x)^2*Log[1 - E^(c + d*x)] - (e + f*x)^2*Log[1 + E^(c + d*x)] - (2*f*(d*(e + f*x)*PolyLog[2, -E^(c + d*x)] - f*PolyLog[3, -E^(c + d*x)]))/d^2 + (2*f*(d*(e + f*x)*PolyLog[2, E^ (c + d*x)] - f*PolyLog[3, E^(c + d*x)]))/d^2)/(a*d) - (Sqrt[a^2 + b^2]*(-2 *d^2*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + d^2*f^2*x^2*Log[1 + (b*E^(c + d *x))/(a - Sqrt[a^2 + b^2])] - 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqr t[a^2 + b^2])] - d^2*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]) ] + 2*d*f*(e + f*x)*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2 *d*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2*f^ 2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*f^2*PolyLog[3, -( (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(a*b*d^3)
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.
| \(\displaystyle \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle \frac {\int (e+f x)^2 \cosh (c+d x) \coth (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle \frac {\int (e+f x)^2 \sinh (c+d x)dx+\int (e+f x)^2 \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -i (e+f x)^2 \sin (i c+i d x)dx+\int i (e+f x)^2 \csc (i c+i d x)dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \int (e+f x)^2 \sin (i c+i d x)dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (\frac {2 i f \int (e+f x) \log \left (1-e^{c+d x}\right )dx}{d}-\frac {2 i f \int (e+f x) \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2dx}{b^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{b}\right )}{a}+\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}+\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}+\frac {\int -i (e+f x)^2 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \int (e+f x)^2 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{a}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {2 \left (a^2+b^2\right ) \int -\frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{a}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{a}\) |
3.5.26.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 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[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_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( -I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[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[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[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. )*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-a/b^2 Int[(e + f*x)^m*Cos h[c + d*x]^(n - 2), x], x] + (Simp[1/b Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)*Sinh[c + d*x], x], x] + Simp[(a^2 + b^2)/b^2 Int[(e + f*x)^m*(Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 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]
\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right ) \coth \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 992 vs. \(2 (421) = 842\).
Time = 0.28 (sec) , antiderivative size = 992, normalized size of antiderivative = 2.15 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
1/3*(a*d^3*f^2*x^3 + 3*a*d^3*e*f*x^2 + 3*a*d^3*e^2*x + 6*b*f^2*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c ) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 6*b*f^2*sqrt((a^2 + b^2)/ b^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b* sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 6*b*f^2*polylog(3, cosh(d*x + c ) + sinh(d*x + c)) + 6*b*f^2*polylog(3, -cosh(d*x + c) - sinh(d*x + c)) - 6*(b*d*f^2*x + b*d*e*f)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*s inh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 6*(b*d*f^2*x + b*d*e*f)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d *x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 3*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*sqrt((a^ 2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 3*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*sqrt((a^2 + b^2 )/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^ 2) + 2*a) - 3*(b*d^2*f^2*x^2 + 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c^2*f^2)*sq rt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 3*(b*d^2*f^2*x^2 + 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c^2*f^2)*sqrt((a^2 + b^2)/b^2)*log(-(a*co sh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt(( a^2 + b^2)/b^2) - b)/b) + 6*(b*d*f^2*x + b*d*e*f)*dilog(cosh(d*x + c) +...
\[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \cosh {\left (c + d x \right )} \coth {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
\[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right ) \coth \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
e^2*((d*x + c)/(b*d) - log(e^(-d*x - c) + 1)/(a*d) + log(e^(-d*x - c) - 1) /(a*d) - sqrt(a^2 + b^2)*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^( -d*x - c) - a + sqrt(a^2 + b^2)))/(a*b*d)) + 1/3*(f^2*x^3 + 3*e*f*x^2)/b - 2*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*e*f/(a*d^2) + 2*(d*x*l og(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))*e*f/(a*d^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) + (d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*p olylog(3, e^(d*x + c)))*f^2/(a*d^3) - integrate(2*((a^2*f^2*e^c + b^2*f^2* e^c)*x^2 + 2*(a^2*e*f*e^c + b^2*e*f*e^c)*x)*e^(d*x)/(a*b^2*e^(2*d*x + 2*c) + 2*a^2*b*e^(d*x + c) - a*b^2), x)
Timed out. \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {coth}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]