Integrand size = 32, antiderivative size = 631 \[ \int \frac {(e+f x) \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a (e+f x)^2}{2 b^2 f}+\frac {2 a^2 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {f \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {a^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {i a^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^3 d^2}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac {i a^4 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {i a^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^3 d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}-\frac {i a^4 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac {a^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {(e+f x) \sinh (c+d x)}{b d} \]
1/2*a*(f*x+e)^2/b^2/f+2*a^2*(f*x+e)*arctan(exp(d*x+c))/b^3/d-2*(f*x+e)*arc tan(exp(d*x+c))/b/d-2*a^4*(f*x+e)*arctan(exp(d*x+c))/b^3/(a^2+b^2)/d-f*cos h(d*x+c)/b/d^2-a*(f*x+e)*ln(1+exp(2*d*x+2*c))/b^2/d+a^3*(f*x+e)*ln(1+exp(2 *d*x+2*c))/b^2/(a^2+b^2)/d-a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2 )))/b^2/(a^2+b^2)/d-a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2 /(a^2+b^2)/d+I*a^2*f*polylog(2,I*exp(d*x+c))/b^3/d^2+I*f*polylog(2,-I*exp( d*x+c))/b/d^2-I*a^4*f*polylog(2,I*exp(d*x+c))/b^3/(a^2+b^2)/d^2+I*a^4*f*po lylog(2,-I*exp(d*x+c))/b^3/(a^2+b^2)/d^2-I*f*polylog(2,I*exp(d*x+c))/b/d^2 -I*a^2*f*polylog(2,-I*exp(d*x+c))/b^3/d^2-1/2*a*f*polylog(2,-exp(2*d*x+2*c ))/b^2/d^2+1/2*a^3*f*polylog(2,-exp(2*d*x+2*c))/b^2/(a^2+b^2)/d^2-a^3*f*po lylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^2-a^3*f*polylog (2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^2+(f*x+e)*sinh(d*x+c )/b/d
Time = 8.64 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.01 \[ \int \frac {(e+f x) \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {f \cosh (c+d x)}{b d^2}-\frac {a^3 \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 b^2 \left (a^2+b^2\right ) d^2}-\frac {-a d e (c+d x)+a c f (c+d x)-\frac {1}{2} a f (c+d x)^2+2 b d e \arctan \left (e^{c+d x}\right )-2 b c f \arctan \left (e^{c+d x}\right )+i b f (c+d x) \log \left (1-i e^{c+d x}\right )-i b f (c+d x) \log \left (1+i e^{c+d x}\right )+a d e \log \left (1+e^{2 (c+d x)}\right )-a c f \log \left (1+e^{2 (c+d x)}\right )+a f (c+d x) \log \left (1+e^{2 (c+d x)}\right )-i b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+i b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+\frac {1}{2} a f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {(d e-c f+f (c+d x)) \sinh (c+d x)}{b d^2} \]
-((f*Cosh[c + d*x])/(b*d^2)) - (a^3*(-2*d*e*(c + d*x) + 2*c*f*(c + d*x) - f*(c + d*x)^2 + (4*a*Sqrt[a^2 + b^2]*d*e*ArcTan[(a + b*E^(c + d*x))/Sqrt[- a^2 - b^2]])/Sqrt[-(a^2 + b^2)^2] - (4*a*Sqrt[-(a^2 + b^2)^2]*d*e*ArcTanh[ (a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/(-a^2 - b^2)^(3/2) + 2*f*(c + d*x)*L og[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*f*(c + d*x)*Log[1 + (b*E ^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*c*f*Log[b - 2*a*E^(c + d*x) - b*E^( 2*(c + d*x))] + 2*d*e*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] + 2* f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*f*PolyLog[2, -((b *E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(2*b^2*(a^2 + b^2)*d^2) - (-(a*d*e *(c + d*x)) + a*c*f*(c + d*x) - (a*f*(c + d*x)^2)/2 + 2*b*d*e*ArcTan[E^(c + d*x)] - 2*b*c*f*ArcTan[E^(c + d*x)] + I*b*f*(c + d*x)*Log[1 - I*E^(c + d *x)] - I*b*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + a*d*e*Log[1 + E^(2*(c + d* x))] - a*c*f*Log[1 + E^(2*(c + d*x))] + a*f*(c + d*x)*Log[1 + E^(2*(c + d* x))] - I*b*f*PolyLog[2, (-I)*E^(c + d*x)] + I*b*f*PolyLog[2, I*E^(c + d*x) ] + (a*f*PolyLog[2, -E^(2*(c + d*x))])/2)/((a^2 + b^2)*d^2) + ((d*e - c*f + f*(c + d*x))*Sinh[c + d*x])/(b*d^2)
Time = 3.60 (sec) , antiderivative size = 570, normalized size of antiderivative = 0.90, number of steps used = 31, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.938, Rules used = {6115, 5972, 3042, 3777, 26, 3042, 26, 3118, 4668, 2715, 2838, 6115, 3042, 26, 4201, 2620, 2715, 2838, 6101, 3042, 4668, 2715, 2838, 6107, 6095, 2620, 2715, 2838, 7293, 2009}
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.
| \(\displaystyle \int \frac {(e+f x) \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6115 |
| \(\displaystyle \frac {\int (e+f x) \sinh (c+d x) \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5972 |
| \(\displaystyle \frac {\int (e+f x) \cosh (c+d x)dx-\int (e+f x) \text {sech}(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {i f \int -i \sinh (c+d x)dx}{d}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {f \int \sinh (c+d x)dx}{d}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {f \int -i \sin (i c+i d x)dx}{d}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {i f \int \sin (i c+i d x)dx}{d}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {i f \int \log \left (1-i e^{c+d x}\right )dx}{d}-\frac {i f \int \log \left (1+i e^{c+d x}\right )dx}{d}-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {i f \int e^{-c-d x} \log \left (1-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i f \int e^{-c-d x} \log \left (1+i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 6115 |
| \(\displaystyle -\frac {a \left (\frac {\int (e+f x) \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int -i (e+f x) \tan (i c+i d x)dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \int (e+f x) \tan (i c+i d x)dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}}dx-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \left (\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int \log \left (1+e^{2 (c+d x)}\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \left (\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int e^{-2 (c+d x)} \log \left (1+e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6101 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x) \text {sech}(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {i f \int \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {i f \int \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {i f \int e^{-c-d x} \log \left (1-i e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {i f \int e^{-c-d x} \log \left (1+i e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \left (\frac {b^2 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \left (\frac {b^2 \left (\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \left (\frac {b^2 \left (-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \left (\frac {b^2 \left (-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \left (\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \left (\frac {\int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}}{b}-\frac {a \left (\frac {b^2 \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\frac {2 a (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d^2}-\frac {b (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{d}+\frac {b (e+f x)^2}{2 f}}{a^2+b^2}\right )}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{b}\right )}{b}\) |
-((a*(((-I)*(((-1/2*I)*(e + f*x)^2)/f + (2*I)*(((e + f*x)*Log[1 + E^(2*(c + d*x))])/(2*d) + (f*PolyLog[2, -E^(2*(c + d*x))])/(4*d^2))))/b - (a*(((2* (e + f*x)*ArcTan[E^(c + d*x)])/d - (I*f*PolyLog[2, (-I)*E^(c + d*x)])/d^2 + (I*f*PolyLog[2, I*E^(c + d*x)])/d^2)/b - (a*((b^2*(-1/2*(e + f*x)^2/(b*f ) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + (f*PolyLog [2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^2) + (f*PolyLog[2, -(( b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^2)))/(a^2 + b^2) + ((b*(e + f *x)^2)/(2*f) + (2*a*(e + f*x)*ArcTan[E^(c + d*x)])/d - (b*(e + f*x)*Log[1 + E^(2*(c + d*x))])/d - (I*a*f*PolyLog[2, (-I)*E^(c + d*x)])/d^2 + (I*a*f* PolyLog[2, I*E^(c + d*x)])/d^2 - (b*f*PolyLog[2, -E^(2*(c + d*x))])/(2*d^2 ))/(a^2 + b^2)))/b))/b))/b) + ((-2*(e + f*x)*ArcTan[E^(c + d*x)])/d - (f*C osh[c + d*x])/d^2 + (I*f*PolyLog[2, (-I)*E^(c + d*x)])/d^2 - (I*f*PolyLog[ 2, I*E^(c + d*x)])/d^2 + ((e + f*x)*Sinh[c + d*x])/d)/b
3.5.8.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[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_.)*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_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[csc[(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[((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Int[(c + d*x)^m*Sinh[a + b*x]^n*Tanh[a + b* x]^(p - 2), x] - Int[(c + d*x)^m*Sinh[a + b*x]^(n - 2)*Tanh[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4065 vs. \(2 (592 ) = 1184\).
Time = 3.15 (sec) , antiderivative size = 4066, normalized size of antiderivative = 6.44
2/(a^2+b^2)^(1/2)/d*b^2*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/( a^2+b^2)^(1/2))+2/d^2/b^2*c*a^4*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*arctanh(1/ 2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/d/b^2*a^4*f/(2*a^2+2*b^2)/(a^2+b ^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-2/d /b^2*a^4*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+ a)/(a+(a^2+b^2)^(1/2)))*x+2/d^2/b^2*a^4*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*ln ((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-2/d^2/b^2*a^4*f /(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2 +b^2)^(1/2)))*c+2/d^2/b^2*c*a^2*f/(2*a^2+2*b^2)*(a^2+b^2)^(1/2)*arctanh(1/ 2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/(a^2+b^2)^(1/2)/d^2*b^2*c*f/(2*a ^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/d^2*f/(2*a^2 +2*b^2)*dilog(1+I*exp(d*x+c))*a-2/d^2*f/(2*a^2+2*b^2)*dilog(1-I*exp(d*x+c) )*a+2/d/b^2*e*a*ln(exp(d*x+c))-4/d*b*e/(2*a^2+2*b^2)*arctan(exp(d*x+c))+1/ d*e/(2*a^2+2*b^2)*a*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/d*e/(2*a^2+2*b ^2)*(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/d* e/(2*a^2+2*b^2)*a*ln(1+exp(2*d*x+2*c))+1/d^2*a*f/(2*a^2+2*b^2)*dilog((-b*e xp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/d^2*a*f/(2*a^2+2*b^2) *dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/2/d^2*a*f/( a^2+b^2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/2/d ^2*a*f/(a^2+b^2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1410 vs. \(2 (570) = 1140\).
Time = 0.31 (sec) , antiderivative size = 1410, normalized size of antiderivative = 2.23 \[ \int \frac {(e+f x) \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
-1/2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e - ((a^2*b + b^3)*d*f*x + (a^ 2*b + b^3)*d*e - (a^2*b + b^3)*f)*cosh(d*x + c)^2 - ((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e - (a^2*b + b^3)*f)*sinh(d*x + c)^2 + (a^2*b + b^3)*f - ((a^3 + a*b^2)*d^2*f*x^2 + 2*(a^3 + a*b^2)*d^2*e*x + 4*(a^3 + a*b^2)*c*d*e - 2*(a^3 + a*b^2)*c^2*f)*cosh(d*x + c) + 2*(a^3*f*cosh(d*x + c) + a^3*f*s inh(d*x + c))*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) + 2*(a^3*f*cosh(d*x + c) + a^3*f*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*c osh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 2*((a* b^2*f + I*b^3*f)*cosh(d*x + c) + (a*b^2*f + I*b^3*f)*sinh(d*x + c))*dilog( I*cosh(d*x + c) + I*sinh(d*x + c)) + 2*((a*b^2*f - I*b^3*f)*cosh(d*x + c) + (a*b^2*f - I*b^3*f)*sinh(d*x + c))*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) + 2*((a^3*d*e - a^3*c*f)*cosh(d*x + c) + (a^3*d*e - a^3*c*f)*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^ 2) + 2*a) + 2*((a^3*d*e - a^3*c*f)*cosh(d*x + c) + (a^3*d*e - a^3*c*f)*sin h(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^ 2)/b^2) + 2*a) + 2*((a^3*d*f*x + a^3*c*f)*cosh(d*x + c) + (a^3*d*f*x + a^3 *c*f)*sinh(d*x + c))*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) + 2*((a^3*d*f*x + a ^3*c*f)*cosh(d*x + c) + (a^3*d*f*x + a^3*c*f)*sinh(d*x + c))*log(-(a*co...
\[ \int \frac {(e+f x) \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \sinh ^{2}{\left (c + d x \right )} \tanh {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
\[ \int \frac {(e+f x) \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \sinh \left (d x + c\right )^{2} \tanh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-1/2*(2*a^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^2*b^2 + b^ 4)*d) - 4*b*arctan(e^(-d*x - c))/((a^2 + b^2)*d) + 2*a*log(e^(-2*d*x - 2*c ) + 1)/((a^2 + b^2)*d) + 2*(d*x + c)*a/(b^2*d) - e^(d*x + c)/(b*d) + e^(-d *x - c)/(b*d))*e - 1/4*f*(2*(a*d^2*x^2*e^c - (b*d*x*e^(2*c) - b*e^(2*c))*e ^(d*x) + (b*d*x + b)*e^(-d*x))*e^(-c)/(b^2*d^2) - integrate(-8*(a^4*x*e^(d *x + c) - a^3*b*x)/(a^2*b^3 + b^5 - (a^2*b^3*e^(2*c) + b^5*e^(2*c))*e^(2*d *x) - 2*(a^3*b^2*e^c + a*b^4*e^c)*e^(d*x)), x) + integrate(8*(b*x*e^(d*x + c) - a*x)/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)), x))
Timed out. \[ \int \frac {(e+f x) \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x) \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2\,\mathrm {tanh}\left (c+d\,x\right )\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]