Integrand size = 26, antiderivative size = 894 \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 d}+\frac {(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 \left (a^2+b^2\right )^2 d}-\frac {a^4 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 \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 )}{\left (a^2+b^2\right )^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {i a^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^3 d^2}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac {i a^4 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d^2}+\frac {i a^4 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 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 )}{2 b^3 d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}-\frac {i a^4 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d^2}-\frac {i a^4 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 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 )}{\left (a^2+b^2\right )^2 d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {a^2 f \text {sech}(c+d x)}{2 b^3 d^2}-\frac {f \text {sech}(c+d x)}{2 b d^2}-\frac {a^4 f \text {sech}(c+d x)}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}^2(c+d x)}{2 b^2 d}-\frac {a^3 (e+f x) \text {sech}^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a f \tanh (c+d x)}{2 b^2 d^2}+\frac {a^3 f \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 d}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^4 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 \left (a^2+b^2\right ) d} \]
-1/2*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/b/d-1/2*I*f*polylog(2,-I*exp(d*x+c))/ b/d^2+1/2*a^3*f*polylog(2,-exp(2*d*x+2*c))/(a^2+b^2)^2/d^2+1/2*a^2*f*sech( d*x+c)/b^3/d^2+1/2*a*(f*x+e)*sech(d*x+c)^2/b^2/d-1/2*a*f*tanh(d*x+c)/b^2/d ^2-2*a^4*(f*x+e)*arctan(exp(d*x+c))/b/(a^2+b^2)^2/d-1/2*a^4*f*sech(d*x+c)/ b^3/(a^2+b^2)/d^2-1/2*a^3*(f*x+e)*sech(d*x+c)^2/b^2/(a^2+b^2)/d+1/2*a^3*f* tanh(d*x+c)/b^2/(a^2+b^2)/d^2+1/2*a^2*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/b^3/ d-1/2*I*a^2*f*polylog(2,-I*exp(d*x+c))/b^3/d^2+1/2*I*f*polylog(2,I*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*x+e)*arctan(e xp(d*x+c))/b/d+a^3*(f*x+e)*ln(1+exp(2*d*x+2*c))/(a^2+b^2)^2/d-a^3*(f*x+e)* ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d-a^3*(f*x+e)*ln(1+b*ex p(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d-a^3*f*polylog(2,-b*exp(d*x+c)/ (a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^2-a^3*f*polylog(2,-b*exp(d*x+c)/(a+(a^2 +b^2)^(1/2)))/(a^2+b^2)^2/d^2-1/2*f*sech(d*x+c)/b/d^2+1/2*I*a^4*f*polylog( 2,-I*exp(d*x+c))/b^3/(a^2+b^2)/d^2+I*a^4*f*polylog(2,-I*exp(d*x+c))/b/(a^2 +b^2)^2/d^2+1/2*I*a^2*f*polylog(2,I*exp(d*x+c))/b^3/d^2-1/2*a^4*(f*x+e)*se ch(d*x+c)*tanh(d*x+c)/b^3/(a^2+b^2)/d-I*a^4*f*polylog(2,I*exp(d*x+c))/b/(a ^2+b^2)^2/d^2-1/2*I*a^4*f*polylog(2,I*exp(d*x+c))/b^3/(a^2+b^2)/d^2+a^2*(f *x+e)*arctan(exp(d*x+c))/b^3/d
Time = 9.10 (sec) , antiderivative size = 834, normalized size of antiderivative = 0.93 \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3 \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {-2 a^3 d e (c+d x)+2 a^3 c f (c+d x)-a^3 f (c+d x)^2+6 a^2 b d e \arctan \left (e^{c+d x}\right )+2 b^3 d e \arctan \left (e^{c+d x}\right )-6 a^2 b c f \arctan \left (e^{c+d x}\right )-2 b^3 c f \arctan \left (e^{c+d x}\right )+3 i a^2 b f (c+d x) \log \left (1-i e^{c+d x}\right )+i b^3 f (c+d x) \log \left (1-i e^{c+d x}\right )-3 i a^2 b f (c+d x) \log \left (1+i e^{c+d x}\right )-i b^3 f (c+d x) \log \left (1+i e^{c+d x}\right )+2 a^3 d e \log \left (1+e^{2 (c+d x)}\right )-2 a^3 c f \log \left (1+e^{2 (c+d x)}\right )+2 a^3 f (c+d x) \log \left (1+e^{2 (c+d x)}\right )-i b \left (3 a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+i b \left (3 a^2+b^2\right ) f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+a^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {\text {sech}(c+d x) (-b f-a f \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2}+\frac {\text {sech}^2(c+d x) (a d e-a c f+a f (c+d x)-b d e \sinh (c+d x)+b c f \sinh (c+d x)-b f (c+d x) \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2} \]
-1/2*(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)*Log[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]))]))/((a^2 + b^2)^2*d^2) + (-2*a^3*d*e*(c + d*x) + 2*a^3*c*f*(c + d*x) - a^3*f*(c + d*x)^2 + 6*a^2*b*d*e*ArcTan[E^(c + d*x)] + 2*b^3*d*e*Ar cTan[E^(c + d*x)] - 6*a^2*b*c*f*ArcTan[E^(c + d*x)] - 2*b^3*c*f*ArcTan[E^( c + d*x)] + (3*I)*a^2*b*f*(c + d*x)*Log[1 - I*E^(c + d*x)] + I*b^3*f*(c + d*x)*Log[1 - I*E^(c + d*x)] - (3*I)*a^2*b*f*(c + d*x)*Log[1 + I*E^(c + d*x )] - I*b^3*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + 2*a^3*d*e*Log[1 + E^(2*(c + d*x))] - 2*a^3*c*f*Log[1 + E^(2*(c + d*x))] + 2*a^3*f*(c + d*x)*Log[1 + E^(2*(c + d*x))] - I*b*(3*a^2 + b^2)*f*PolyLog[2, (-I)*E^(c + d*x)] + I*b* (3*a^2 + b^2)*f*PolyLog[2, I*E^(c + d*x)] + a^3*f*PolyLog[2, -E^(2*(c + d* x))])/(2*(a^2 + b^2)^2*d^2) + (Sech[c + d*x]*(-(b*f) - a*f*Sinh[c + d*x])) /(2*(a^2 + b^2)*d^2) + (Sech[c + d*x]^2*(a*d*e - a*c*f + a*f*(c + d*x) - b *d*e*Sinh[c + d*x] + b*c*f*Sinh[c + d*x] - b*f*(c + d*x)*Sinh[c + d*x])...
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.
| \(\displaystyle \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6101 |
| \(\displaystyle \frac {\int (e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5978 |
| \(\displaystyle \frac {\int (e+f x) \text {sech}(c+d x)dx-\int (e+f x) \text {sech}^3(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx-\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{b}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )^3dx-\frac {i f \int \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {i f \int \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {i f \int e^{-c-d x} \log \left (1-i e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {i f \int e^{-c-d x} \log \left (1+i e^{c+d x}\right )de^{c+d x}}{d^2}-\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )^3dx+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )^3dx+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}}{b}\) |
| \(\Big \downarrow \) 4673 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {1}{2} \int (e+f x) \text {sech}(c+d x)dx+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {1}{2} \int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {1}{2} \left (\frac {i f \int \log \left (1-i e^{c+d x}\right )dx}{d}-\frac {i f \int \log \left (1+i e^{c+d x}\right )dx}{d}-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {1}{2} \left (\frac {i f \int e^{-c-d x} \log \left (1-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i f \int e^{-c-d x} \log \left (1+i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 6117 |
| \(\displaystyle -\frac {a \left (\frac {\int (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 5974 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \int \text {sech}^2(c+d x)dx}{2 d}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}+\frac {f \int \csc \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{2 d}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}+\frac {i f \int 1d(-i \tanh (c+d x))}{2 d^2}}{b}\right )}{b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 6117 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\int (e+f x) \text {sech}^3(c+d x)dx}{b}-\frac {a \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4673 |
| \(\displaystyle -\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {1}{2} \int (e+f x) \text {sech}(c+d x)dx+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}+\frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {1}{2} \int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {1}{2} \left (-\frac {i f \int \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {i f \int \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {1}{2} \left (-\frac {i f \int e^{-c-d x} \log \left (1-i e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {i f \int e^{-c-d x} \log \left (1+i e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}\right )}{b}+\frac {\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \left (\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \left (-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {b^2 \left (\frac {b^2 \left (-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {b^2 \left (\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {f \text {sech}(c+d x)}{2 d^2}-\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\frac {f \tanh (c+d x)}{2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {b^2 \left (\frac {\int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int \left (a (e+f x) \text {sech}^3(c+d x)-b (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}\right )}{b}+\frac {\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{b}\right )}{b}\right )}{b}\) |
3.5.17.3.1 Defintions of rubi rules used
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[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 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_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S imp[b^2*((n - 2)/(n - 1)) Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
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[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]*Tanh[(a_.) + (b_.)* (x_)]^(p_), x_Symbol] :> Int[(c + d*x)^m*Sech[a + b*x]*Tanh[a + b*x]^(p - 2 ), x] - Int[(c + d*x)^m*Sech[a + b*x]^3*Tanh[a + b*x]^(p - 2), x] /; FreeQ[ {a, b, c, d, m}, x] && IGtQ[p/2, 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_.)*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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2283 vs. \(2 (820 ) = 1640\).
Time = 2.71 (sec) , antiderivative size = 2284, normalized size of antiderivative = 2.55
3/(a^2+b^2)^(3/2)/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))*a^2+I*b^3/d^2/(a^2+b^2)*f/(2*a^2+2*b^2)*dilog(1-I*exp( d*x+c))+6*b/d/(a^2+b^2)*a^2*e/(2*a^2+2*b^2)*arctan(exp(d*x+c))-1/(a^2+b^2) ^(3/2)/d*b^4*e/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1 /2))+1/(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))-1/(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))+1/(a^2+b^2)^(3/2)/d^2*b^4*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))-3/(a^2+b^2)^( 3/2)/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 ))*a^2+2*b^3/d/(a^2+b^2)*e/(2*a^2+2*b^2)*arctan(exp(d*x+c))+3*I/d^2/(a^2+b ^2)*a^2*f/(2*a^2+2*b^2)*dilog(1-I*exp(d*x+c))*b-3*I*b/d^2/(a^2+b^2)*a^2*f/ (2*a^2+2*b^2)*dilog(1+I*exp(d*x+c))+2/(a^2+b^2)^(3/2)/d^2*c*a^4*f/(2*a^2+2 *b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+I*b^3/d/(a^2+b^2)* f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*x-I*b^3/d^2/(a^2+b^2)*f/(2*a^2+2*b^2)*l n(1+I*exp(d*x+c))*c-I*b^3/d/(a^2+b^2)*f/(2*a^2+2*b^2)*ln(1+I*exp(d*x+c))*x -6*b/d^2/(a^2+b^2)*c*a^2*f/(2*a^2+2*b^2)*arctan(exp(d*x+c))+I*b^3/d^2/(a^2 +b^2)*f/(2*a^2+2*b^2)*ln(1-I*exp(d*x+c))*c-2/d^2*c*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))*a^2-2/(a^2+b^2) ^(3/2)/d*e*a^4/(2*a^2+2*b^2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1 /2))+(-b*d*f*x*exp(3*d*x+3*c)+2*a*d*f*x*exp(2*d*x+2*c)-b*d*e*exp(3*d*x+...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4729 vs. \(2 (793) = 1586\).
Time = 0.37 (sec) , antiderivative size = 4729, normalized size of antiderivative = 5.29 \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
-1/2*(2*((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)^3 + 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)^3 - 2*(2*(a^3 + a*b^2)*d*f*x + 2*(a^3 + a*b^2)*d*e + (a^3 + a*b^2)*f)*cosh(d*x + c)^2 - 2*(2*(a^3 + a*b^2)*d*f*x + 2*(a^3 + a*b^2)*d*e + (a^3 + a*b^2)*f - 3*((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))*sinh(d*x + c)^2 - 2*(a^3 + a*b^2)*f - 2*((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^3*f *cosh(d*x + c)^4 + 4*a^3*f*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*f*sinh(d*x + c)^4 + 2*a^3*f*cosh(d*x + c)^2 + a^3*f + 2*(3*a^3*f*cosh(d*x + c)^2 + a^ 3*f)*sinh(d*x + c)^2 + 4*(a^3*f*cosh(d*x + c)^3 + a^3*f*cosh(d*x + c))*sin h(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 )^4 + 4*a^3*f*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*f*sinh(d*x + c)^4 + 2*a^ 3*f*cosh(d*x + c)^2 + a^3*f + 2*(3*a^3*f*cosh(d*x + c)^2 + a^3*f)*sinh(d*x + c)^2 + 4*(a^3*f*cosh(d*x + c)^3 + a^3*f*cosh(d*x + c))*sinh(d*x + c))*d ilog((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 + I*(3*a^2*b + b^3)*f)*c osh(d*x + c)^4 + 4*(2*a^3*f + I*(3*a^2*b + b^3)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^3*f + I*(3*a^2*b + b^3)*f)*sinh(d*x + c)^4 + 2*a^3*f + 2*(2* a^3*f + I*(3*a^2*b + b^3)*f)*cosh(d*x + c)^2 + 2*(2*a^3*f + 3*(2*a^3*f ...
\[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \tanh ^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
\[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \tanh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-(a^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^4 + 2*a^2*b^2 + b^4)*d) - a^3*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d) + (3*a ^2*b + b^3)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) + (b*e^(-d*x - c) - 2*a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((a^2 + b^2 + 2*(a^2 + b ^2)*e^(-2*d*x - 2*c) + (a^2 + b^2)*e^(-4*d*x - 4*c))*d))*e - f*(((b*d*x*e^ (3*c) + b*e^(3*c))*e^(3*d*x) - (2*a*d*x*e^(2*c) + a*e^(2*c))*e^(2*d*x) - ( b*d*x*e^c - b*e^c)*e^(d*x) - a)/(a^2*d^2 + b^2*d^2 + (a^2*d^2*e^(4*c) + b^ 2*d^2*e^(4*c))*e^(4*d*x) + 2*(a^2*d^2*e^(2*c) + b^2*d^2*e^(2*c))*e^(2*d*x) ) - integrate(-2*(a^4*x*e^(d*x + c) - a^3*b*x)/(a^4*b + 2*a^2*b^3 + b^5 - (a^4*b*e^(2*c) + 2*a^2*b^3*e^(2*c) + b^5*e^(2*c))*e^(2*d*x) - 2*(a^5*e^c + 2*a^3*b^2*e^c + a*b^4*e^c)*e^(d*x)), x) - integrate(-(2*a^3*x - (3*a^2*b* e^c + b^3*e^c)*x*e^(d*x))/(a^4 + 2*a^2*b^2 + b^4 + (a^4*e^(2*c) + 2*a^2*b^ 2*e^(2*c) + b^4*e^(2*c))*e^(2*d*x)), x))
Timed out. \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]