Integrand size = 34, antiderivative size = 904 \[ \int \frac {(e+f x)^2 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 (e+f x)^2}{b^3 d}-\frac {(e+f x)^2}{b d}-\frac {a^4 (e+f x)^2}{b^3 \left (a^2+b^2\right ) d}+\frac {(e+f x)^3}{3 b f}-\frac {4 a f (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 d^2}+\frac {4 a^3 f (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac {2 a^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^3 d^2}+\frac {2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac {2 a^4 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {2 i a f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^3}-\frac {2 i a^3 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {2 i a f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^3}+\frac {2 i a^3 f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {2 a^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 a^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac {a^2 f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^3 d^3}+\frac {f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3}+\frac {a^4 f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac {2 a^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 a^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}+\frac {a (e+f x)^2 \text {sech}(c+d x)}{b^2 d}-\frac {a^3 (e+f x)^2 \text {sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x)^2 \tanh (c+d x)}{b^3 d}-\frac {(e+f x)^2 \tanh (c+d x)}{b d}-\frac {a^4 (e+f x)^2 \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d} \]
2*f*(f*x+e)*ln(1+exp(2*d*x+2*c))/b/d^2+a^2*(f*x+e)^2/b^3/d-4*a*f*(f*x+e)*a rctan(exp(d*x+c))/b^2/d^2-2*a^2*f*(f*x+e)*ln(1+exp(2*d*x+2*c))/b^3/d^2+2*a ^3*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d^3- 2*a^3*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d ^3-2*I*a*f^2*polylog(2,I*exp(d*x+c))/b^2/d^3+1/3*(f*x+e)^3/b/f-a^2*f^2*pol ylog(2,-exp(2*d*x+2*c))/b^3/d^3-a^4*(f*x+e)^2/b^3/(a^2+b^2)/d+a*(f*x+e)^2* sech(d*x+c)/b^2/d+a^2*(f*x+e)^2*tanh(d*x+c)/b^3/d+a^4*f^2*polylog(2,-exp(2 *d*x+2*c))/b^3/(a^2+b^2)/d^3-a^3*(f*x+e)^2*sech(d*x+c)/b^2/(a^2+b^2)/d-a^4 *(f*x+e)^2*tanh(d*x+c)/b^3/(a^2+b^2)/d-a^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a- (a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d+a^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+( a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d+2*I*a^3*f^2*polylog(2,I*exp(d*x+c))/b ^2/(a^2+b^2)/d^3+2*I*a*f^2*polylog(2,-I*exp(d*x+c))/b^2/d^3+f^2*polylog(2, -exp(2*d*x+2*c))/b/d^3-(f*x+e)^2*tanh(d*x+c)/b/d+4*a^3*f*(f*x+e)*arctan(ex p(d*x+c))/b^2/(a^2+b^2)/d^2+2*a^4*f*(f*x+e)*ln(1+exp(2*d*x+2*c))/b^3/(a^2+ b^2)/d^2-2*a^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a ^2+b^2)^(3/2)/d^2+2*a^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/ 2)))/b/(a^2+b^2)^(3/2)/d^2-2*I*a^3*f^2*polylog(2,-I*exp(d*x+c))/b^2/(a^2+b ^2)/d^3-(f*x+e)^2/b/d
Time = 2.92 (sec) , antiderivative size = 665, normalized size of antiderivative = 0.74 \[ \int \frac {(e+f x)^2 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {x \left (3 e^2+3 e f x+f^2 x^2\right )}{3 b}-\frac {f \left (4 b d^2 e e^{2 c} x-4 b d^2 e \left (1+e^{2 c}\right ) x+2 b d^2 e^{2 c} f x^2-2 b d^2 \left (1+e^{2 c}\right ) f x^2+4 a d e \left (1+e^{2 c}\right ) \arctan \left (e^{c+d x}\right )+2 b d e \left (1+e^{2 c}\right ) \left (2 d x-\log \left (1+e^{2 (c+d x)}\right )\right )+2 i a \left (1+e^{2 c}\right ) f \left (d x \left (\log \left (1-i e^{c+d x}\right )-\log \left (1+i e^{c+d x}\right )\right )-\operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+\operatorname {PolyLog}\left (2,i e^{c+d x}\right )\right )+b \left (1+e^{2 c}\right ) f \left (2 d x \left (d x-\log \left (1+e^{2 (c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )\right )\right )}{\left (a^2+b^2\right ) d^3 \left (1+e^{2 c}\right )}+\frac {a^3 \left (2 d^2 e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{b \left (a^2+b^2\right )^{3/2} d^3}+\frac {(e+f x)^2 \text {sech}(c+d x) (a-b \text {sech}(c) \sinh (d x))}{\left (a^2+b^2\right ) d} \]
(x*(3*e^2 + 3*e*f*x + f^2*x^2))/(3*b) - (f*(4*b*d^2*e*E^(2*c)*x - 4*b*d^2* e*(1 + E^(2*c))*x + 2*b*d^2*E^(2*c)*f*x^2 - 2*b*d^2*(1 + E^(2*c))*f*x^2 + 4*a*d*e*(1 + E^(2*c))*ArcTan[E^(c + d*x)] + 2*b*d*e*(1 + E^(2*c))*(2*d*x - Log[1 + E^(2*(c + d*x))]) + (2*I)*a*(1 + E^(2*c))*f*(d*x*(Log[1 - I*E^(c + d*x)] - Log[1 + I*E^(c + d*x)]) - PolyLog[2, (-I)*E^(c + d*x)] + PolyLog [2, I*E^(c + d*x)]) + b*(1 + E^(2*c))*f*(2*d*x*(d*x - Log[1 + E^(2*(c + d* x))]) - PolyLog[2, -E^(2*(c + d*x))])))/((a^2 + b^2)*d^3*(1 + E^(2*c))) + (a^3*(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 + 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*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*PolyLo g[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(b*(a^2 + b^2)^(3/2)*d^3) + ((e + f*x)^2*Sech[c + d*x]*(a - b*Sech[c]*Sinh[d*x]))/((a^2 + b^2)*d)
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.
| \(\displaystyle \int \frac {(e+f x)^2 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6115 |
| \(\displaystyle \frac {\int (e+f x)^2 \tanh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int -(e+f x)^2 \tan (i c+i d x)^2dx}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\int (e+f x)^2 \tan (i c+i d x)^2dx}{b}\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {2 i f \int i (e+f x) \tanh (c+d x)dx}{d}-\int (e+f x)^2dx+\frac {(e+f x)^2 \tanh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {2 i f \int i (e+f x) \tanh (c+d x)dx}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {-\frac {2 f \int (e+f x) \tanh (c+d x)dx}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {-\frac {2 f \int -i (e+f x) \tan (i c+i d x)dx}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {2 i f \int (e+f x) \tan (i c+i d x)dx}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {2 i f \left (2 i \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int \log \left (1+e^{2 (c+d x)}\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int e^{-2 (c+d x)} \log \left (1+e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 6101 |
| \(\displaystyle -\frac {a \left (\frac {\int (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 5974 |
| \(\displaystyle -\frac {a \left (\frac {\frac {2 f \int (e+f x) \text {sech}(c+d x)dx}{d}-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (-\frac {i f \int \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {i f \int \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (-\frac {i f \int e^{-c-d x} \log \left (1-i e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {i f \int e^{-c-d x} \log \left (1+i e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d}}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 6117 |
| \(\displaystyle -\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d}}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{b}\right )}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \tanh (c+d x)dx}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \left (-\frac {a \left (\frac {\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {2 f \int (e+f x) \tanh (c+d x)dx}{d}}{b}-\frac {a \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d}}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \tan (i c+i d x)dx}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \int (e+f x) \tan (i c+i d x)dx}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int \log \left (1+e^{2 (c+d x)}\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int e^{-2 (c+d x)} \log \left (1+e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {b^2 \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^2 \text {sech}(c+d x)}{d}+\frac {2 f \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{b}-\frac {a \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {(e+f x)^3}{3 f}}{b}\) |
3.5.12.3.1 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[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*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_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && 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[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Sech[a + b*x]^n/(b*n)) , x] + Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/b Int[(e + f*x)^m*Sech[ c + d*x]*Tanh[c + d*x]^(n - 1), x], x] - Simp[a/b Int[(e + f*x)^m*Sech[c + d*x]*(Tanh[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[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b^2/(a^2 + b^2) Int[(e + f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Simp[1/(a^2 + b^2) Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; F reeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0 ]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/b Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x], x] - S imp[a/b Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*(Tanh[c + d*x]^n/(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[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/b Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1), x], x] - Simp[a/b Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*(Tanh[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] && IGtQ[p, 0]
\[\int \frac {\left (f x +e \right )^{2} \sinh \left (d x +c \right ) \tanh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4196 vs. \(2 (841) = 1682\).
Time = 0.37 (sec) , antiderivative size = 4196, normalized size of antiderivative = 4.64 \[ \int \frac {(e+f x)^2 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
1/3*((a^4 + 2*a^2*b^2 + b^4)*d^3*f^2*x^3 + 3*(a^4 + 2*a^2*b^2 + b^4)*d^3*e *f*x^2 + 3*(a^4 + 2*a^2*b^2 + b^4)*d^3*e^2*x + 6*(a^2*b^2 + b^4)*d^2*e^2 - 12*(a^2*b^2 + b^4)*c*d*e*f + 6*(a^2*b^2 + b^4)*c^2*f^2 + ((a^4 + 2*a^2*b^ 2 + b^4)*d^3*f^2*x^3 - 12*(a^2*b^2 + b^4)*c*d*e*f + 6*(a^2*b^2 + b^4)*c^2* f^2 + 3*((a^4 + 2*a^2*b^2 + b^4)*d^3*e*f - 2*(a^2*b^2 + b^4)*d^2*f^2)*x^2 + 3*((a^4 + 2*a^2*b^2 + b^4)*d^3*e^2 - 4*(a^2*b^2 + b^4)*d^2*e*f)*x)*cosh( d*x + c)^2 + ((a^4 + 2*a^2*b^2 + b^4)*d^3*f^2*x^3 - 12*(a^2*b^2 + b^4)*c*d *e*f + 6*(a^2*b^2 + b^4)*c^2*f^2 + 3*((a^4 + 2*a^2*b^2 + b^4)*d^3*e*f - 2* (a^2*b^2 + b^4)*d^2*f^2)*x^2 + 3*((a^4 + 2*a^2*b^2 + b^4)*d^3*e^2 - 4*(a^2 *b^2 + b^4)*d^2*e*f)*x)*sinh(d*x + c)^2 - 6*(a^3*b*d*f^2*x + a^3*b*d*e*f + (a^3*b*d*f^2*x + a^3*b*d*e*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*f^2*x + a^3*b* d*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d*f^2*x + a^3*b*d*e*f)*sinh(d* x + c)^2)*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) + 6 *(a^3*b*d*f^2*x + a^3*b*d*e*f + (a^3*b*d*f^2*x + a^3*b*d*e*f)*cosh(d*x + c )^2 + 2*(a^3*b*d*f^2*x + a^3*b*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b *d*f^2*x + a^3*b*d*e*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*dilog((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 + 1) + 3*(a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c ^2*f^2 + (a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2)*cosh(d*x + c...
\[ \int \frac {(e+f x)^2 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \sinh {\left (c + d x \right )} \tanh ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
\[ \int \frac {(e+f x)^2 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \sinh \left (d x + c\right ) \tanh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-2*b*e*f*(2*(d*x + c)/((a^2 + b^2)*d^2) - log(e^(2*d*x + 2*c) + 1)/((a^2 + b^2)*d^2)) - 4*a*f^2*integrate(x*e^(d*x + c)/(a^2*d*e^(2*d*x + 2*c) + b^2 *d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 4*b*f^2*integrate(x/(a^2*d*e^(2* d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - (a^3*log((b*e^(- d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(( a^2*b + b^3)*sqrt(a^2 + b^2)*d) - 2*(a*e^(-d*x - c) - b)/((a^2 + b^2 + (a^ 2 + b^2)*e^(-2*d*x - 2*c))*d) - (d*x + c)/(b*d))*e^2 - 4*a*e*f*arctan(e^(d *x + c))/((a^2 + b^2)*d^2) + 1/3*(12*b^2*e*f*x + (a^2*d*f^2 + b^2*d*f^2)*x ^3 + 3*(a^2*d*e*f + (d*e*f + 2*f^2)*b^2)*x^2 + ((a^2*d*f^2*e^(2*c) + b^2*d *f^2*e^(2*c))*x^3 + 3*(a^2*d*e*f*e^(2*c) + b^2*d*e*f*e^(2*c))*x^2)*e^(2*d* x) + 6*(a*b*f^2*x^2*e^c + 2*a*b*e*f*x*e^c)*e^(d*x))/(a^2*b*d + b^3*d + (a^ 2*b*d*e^(2*c) + b^3*d*e^(2*c))*e^(2*d*x)) - integrate(-2*(a^3*f^2*x^2*e^c + 2*a^3*e*f*x*e^c)*e^(d*x)/(a^2*b^2 + b^4 - (a^2*b^2*e^(2*c) + b^4*e^(2*c) )*e^(2*d*x) - 2*(a^3*b*e^c + a*b^3*e^c)*e^(d*x)), x)
Timed out. \[ \int \frac {(e+f x)^2 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x)^2 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {sinh}\left (c+d\,x\right )\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]