Integrand size = 34, antiderivative size = 1067 \[ \int \frac {(e+f x)^2 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:
2*a^3*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^3+2 *a^3*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^3-a^ 3*(f*x+e)^2*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)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d+2*I*f*(f*x+e )*polylog(2,-I*exp(d*x+c))/b/d^2-2*a^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/( a+(a^2+b^2)^(1/2)))/b^2/(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^2/(a^2+b^2)/d^2-2*I*a^4*f^2*polylog(3,-I*exp(d* x+c))/b^3/(a^2+b^2)/d^3-2*I*a^2*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/b^3/d^2 +2*I*f^2*polylog(3,I*exp(d*x+c))/b/d^3+2*I*a^4*f*(f*x+e)*polylog(2,-I*exp( d*x+c))/b^3/(a^2+b^2)/d^2+a^3*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/b^2/(a^ 2+b^2)/d^2+2*I*a^2*f^2*polylog(3,-I*exp(d*x+c))/b^3/d^3+2*I*a^4*f^2*polylo g(3,I*exp(d*x+c))/b^3/(a^2+b^2)/d^3+2*I*a^2*f*(f*x+e)*polylog(2,I*exp(d*x+ c))/b^3/d^2+1/2*a*f^2*polylog(3,-exp(2*d*x+2*c))/b^2/d^3+2*a^2*(f*x+e)^2*a rctan(exp(d*x+c))/b^3/d-2*I*f^2*polylog(3,-I*exp(d*x+c))/b/d^3-2*I*a^4*f*( f*x+e)*polylog(2,I*exp(d*x+c))/b^3/(a^2+b^2)/d^2+1/3*a*(f*x+e)^3/b^2/f+2*f ^2*sinh(d*x+c)/b/d^3-2*(f*x+e)^2*arctan(exp(d*x+c))/b/d-2*a^4*(f*x+e)^2*ar ctan(exp(d*x+c))/b^3/(a^2+b^2)/d-1/2*a^3*f^2*polylog(3,-exp(2*d*x+2*c))/b^ 2/(a^2+b^2)/d^3-2*I*a^2*f^2*polylog(3,I*exp(d*x+c))/b^3/d^3-2*I*f*(f*x+e)* polylog(2,I*exp(d*x+c))/b/d^2+(f*x+e)^2*sinh(d*x+c)/b/d-a*(f*x+e)^2*ln(1+e xp(2*d*x+2*c))/b^2/d-a*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/b^2/d^2+a^3...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3418\) vs. \(2(1067)=2134\).
Time = 10.57 (sec) , antiderivative size = 3418, normalized size of antiderivative = 3.20 \[ \int \frac {(e+f x)^2 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^2*Sinh[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x] ),x]
Output:
-1/6*(-12*a*d^3*e^2*E^(2*c)*x + 12*a*d^3*e^2*(1 + E^(2*c))*x + 12*a*d^3*e* f*x^2 + 4*a*d^3*f^2*x^3 + 12*b*d^2*e^2*(1 + E^(2*c))*ArcTan[E^(c + d*x)] - 6*a*d^2*e^2*(1 + E^(2*c))*(2*d*x - Log[1 + E^(2*(c + d*x))]) + (12*I)*b*d *e*(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)]) - 6*a*d*e*(1 + E^(2*c))*f*(2*d*x*(d*x - Log[1 + E^(2*(c + d*x))]) - PolyLog[2, -E^(2*(c + d*x))]) + (6*I)*b*(1 + E^(2*c))*f^2*(d^2*x^2*Log[1 - I*E^(c + d*x)] - d^ 2*x^2*Log[1 + I*E^(c + d*x)] - 2*d*x*PolyLog[2, (-I)*E^(c + d*x)] + 2*d*x* PolyLog[2, I*E^(c + d*x)] + 2*PolyLog[3, (-I)*E^(c + d*x)] - 2*PolyLog[3, I*E^(c + d*x)]) - a*(1 + E^(2*c))*f^2*(2*d^2*x^2*(2*d*x - 3*Log[1 + E^(2*( c + d*x))]) - 6*d*x*PolyLog[2, -E^(2*(c + d*x))] + 3*PolyLog[3, -E^(2*(c + d*x))]))/((a^2 + b^2)*d^3*(1 + E^(2*c))) + (a^3*(6*e^2*E^(2*c)*x + 6*e*E^ (2*c)*f*x^2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a^2 + b^2]*e^2*ArcTan[(a + b*E ^(c + d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(3/2)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*ArcTanh[(a + b*E^(c + d*x) )/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2 *E^(2*c)*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2) *d) + (3*e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d - (3*e^2*E ^(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e*f*x*Lo...
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.
| \(\displaystyle \int \frac {(e+f x)^2 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6115 |
| \(\displaystyle \frac {\int (e+f x)^2 \sinh (c+d x) \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5972 |
| \(\displaystyle \frac {\int (e+f x)^2 \cosh (c+d x)dx-\int (e+f x)^2 \text {sech}(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}-\int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}-\int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 i f \int (e+f x) \log \left (1-i e^{c+d x}\right )dx}{d}-\frac {2 i f \int (e+f x) \log \left (1+i e^{c+d x}\right )dx}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 6115 |
| \(\displaystyle -\frac {a \left (\frac {\int (e+f x)^2 \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int -i (e+f x)^2 \tan (i c+i d x)dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \int (e+f x)^2 \tan (i c+i d x)dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}}dx-\frac {i (e+f x)^3}{3 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int (e+f x) \log \left (1+e^{2 (c+d x)}\right )dx}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6101 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^2 \text {sech}(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {2 i f \int (e+f x) \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {2 i f \int (e+f x) \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \left (\frac {b^2 \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \left (\frac {b^2 \left (\int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^3}{3 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}+\frac {(e+f x)^2 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \left (\frac {b^2 \left (-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^3}{3 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {2 \arctan \left (e^{c+d x}\right ) (e+f x)^2}{d}+\frac {\sinh (c+d x) (e+f x)^2}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {i \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{b}-\frac {a \left (\frac {\frac {2 \arctan \left (e^{c+d x}\right ) (e+f x)^2}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (\frac {\left (-\frac {(e+f x)^3}{3 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^2}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^2}{b d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right ) b^2}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {2 \arctan \left (e^{c+d x}\right ) (e+f x)^2}{d}+\frac {\sinh (c+d x) (e+f x)^2}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {i \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{b}-\frac {a \left (\frac {\frac {2 \arctan \left (e^{c+d x}\right ) (e+f x)^2}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (\frac {\left (-\frac {(e+f x)^3}{3 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^2}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^2}{b d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right ) b^2}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}\) |
Input:
Int[((e + f*x)^2*Sinh[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{2} \sinh \left (d x +c \right )^{2} \tanh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^2*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2775 vs. \(2 (970) = 1940\).
Time = 0.17 (sec) , antiderivative size = 2775, normalized size of antiderivative = 2.60 \[ \int \frac {(e+f x)^2 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorit hm="fricas")
Output:
-1/6*(3*(a^2*b + b^3)*d^2*f^2*x^2 + 3*(a^2*b + b^3)*d^2*e^2 + 6*(a^2*b + b ^3)*d*e*f + 6*(a^2*b + b^3)*f^2 - 3*((a^2*b + b^3)*d^2*f^2*x^2 + (a^2*b + b^3)*d^2*e^2 - 2*(a^2*b + b^3)*d*e*f + 2*(a^2*b + b^3)*f^2 + 2*((a^2*b + b ^3)*d^2*e*f - (a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c)^2 - 3*((a^2*b + b^3)*d ^2*f^2*x^2 + (a^2*b + b^3)*d^2*e^2 - 2*(a^2*b + b^3)*d*e*f + 2*(a^2*b + b^ 3)*f^2 + 2*((a^2*b + b^3)*d^2*e*f - (a^2*b + b^3)*d*f^2)*x)*sinh(d*x + c)^ 2 + 6*((a^2*b + b^3)*d^2*e*f + (a^2*b + b^3)*d*f^2)*x - 2*((a^3 + a*b^2)*d ^3*f^2*x^3 + 3*(a^3 + a*b^2)*d^3*e*f*x^2 + 3*(a^3 + a*b^2)*d^3*e^2*x + 6*( a^3 + a*b^2)*c*d^2*e^2 - 6*(a^3 + a*b^2)*c^2*d*e*f + 2*(a^3 + a*b^2)*c^3*f ^2)*cosh(d*x + c) + 12*((a^3*d*f^2*x + a^3*d*e*f)*cosh(d*x + c) + (a^3*d*f ^2*x + a^3*d*e*f)*sinh(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) + 12*((a^3*d*f^2*x + a^3*d*e*f)*cosh(d*x + c) + (a^3*d*f^2*x + a^3*d*e*f)*si nh(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) + 12*((a*b^2*d*f^2*x + I*b^3*d*f^2*x + a*b^2*d*e*f + I*b^3*d*e*f)*cosh(d*x + c) + (a*b^2*d*f^2*x + I*b^3*d*f^2*x + a*b^2*d*e*f + I*b^3*d*e*f)*sinh(d*x + c))*dilog(I*cosh( d*x + c) + I*sinh(d*x + c)) + 12*((a*b^2*d*f^2*x - I*b^3*d*f^2*x + a*b^2*d *e*f - I*b^3*d*e*f)*cosh(d*x + c) + (a*b^2*d*f^2*x - I*b^3*d*f^2*x + a*b^2 *d*e*f - I*b^3*d*e*f)*sinh(d*x + c))*dilog(-I*cosh(d*x + c) - I*sinh(d*...
\[ \int \frac {(e+f x)^2 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \sinh ^{2}{\left (c + d x \right )} \tanh {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**2*sinh(d*x+c)**2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)**2*sinh(c + d*x)**2*tanh(c + d*x)/(a + b*sinh(c + d*x)) , x)
\[ \int \frac {(e+f x)^2 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \sinh \left (d x + c\right )^{2} \tanh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorit hm="maxima")
Output:
-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^2 - 1/6*(2*a*d^3*f^2*x^3*e^c + 6*a*d^3*e*f*x^2*e^c - 3*(b *d^2*f^2*x^2*e^(2*c) + 2*(d^2*e*f - d*f^2)*b*x*e^(2*c) - 2*(d*e*f - f^2)*b *e^(2*c))*e^(d*x) + 3*(b*d^2*f^2*x^2 + 2*(d^2*e*f + d*f^2)*b*x + 2*(d*e*f + f^2)*b)*e^(-d*x))*e^(-c)/(b^2*d^3) + integrate(2*(a^3*b*f^2*x^2 + 2*a^3* b*e*f*x - (a^4*f^2*x^2*e^c + 2*a^4*e*f*x*e^c)*e^(d*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(-2*(a*f^2*x^2 + 2*a*e*f*x - (b*f^2*x^2*e^c + 2*b*e*f* x*e^c)*e^(d*x))/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)), x)
Timed out. \[ \int \frac {(e+f x)^2 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^2*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorit hm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^2 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2\,\mathrm {tanh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((sinh(c + d*x)^2*tanh(c + d*x)*(e + f*x)^2)/(a + b*sinh(c + d*x)),x)
Output:
int((sinh(c + d*x)^2*tanh(c + d*x)*(e + f*x)^2)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^2 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^2*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
( - 12*e**(c + d*x)*atan(e**(c + d*x))*b**4*d**2*e**2 + 3*e**(2*c + 2*d*x) *a**2*b**2*d**2*e**2 + 6*e**(2*c + 2*d*x)*a**2*b**2*d**2*e*f*x + 3*e**(2*c + 2*d*x)*a**2*b**2*d**2*f**2*x**2 - 6*e**(2*c + 2*d*x)*a**2*b**2*d*e*f - 6*e**(2*c + 2*d*x)*a**2*b**2*d*f**2*x + 6*e**(2*c + 2*d*x)*a**2*b**2*f**2 + 3*e**(2*c + 2*d*x)*b**4*d**2*e**2 + 6*e**(2*c + 2*d*x)*b**4*d**2*e*f*x + 3*e**(2*c + 2*d*x)*b**4*d**2*f**2*x**2 - 6*e**(2*c + 2*d*x)*b**4*d*e*f - 6*e**(2*c + 2*d*x)*b**4*d*f**2*x + 6*e**(2*c + 2*d*x)*b**4*f**2 - 24*e**(3 *c + d*x)*int((e**(2*d*x)*x**2)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*a**5*d**3*f**2 - 12*e**(3*c + d*x)*int((e**(2* d*x)*x**2)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*a**3*b**2*d**3*f**2 + 12*e**(3*c + d*x)*int((e**(2*d*x)*x**2)/(e**( 4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*a*b**4*d* *3*f**2 - 48*e**(3*c + d*x)*int((e**(2*d*x)*x)/(e**(4*c + 4*d*x)*b + 2*e** (3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*a**5*d**3*e*f - 24*e**(3*c + d* x)*int((e**(2*d*x)*x)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*a**3*b**2*d**3*e*f + 24*e**(3*c + d*x)*int((e**(2*d*x)*x )/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*a* b**4*d**3*e*f + 12*e**(2*c + d*x)*int((e**(d*x)*x**2)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*a**4*b*d**3*f**2 + 24*e* *(2*c + d*x)*int((e**(d*x)*x**2)/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*...