Integrand size = 34, antiderivative size = 1519 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:
-6*a^3*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^4- 6*a^3*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^4-a ^3*(f*x+e)^3*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)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d-3*a^3*f*(f* x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^2+6*a^ 3*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d ^3+6*a^3*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2 +b^2)/d^3-3*a^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b ^2/(a^2+b^2)/d^2+6*I*a^4*f^3*polylog(4,-I*exp(d*x+c))/b^3/(a^2+b^2)/d^4+6* I*a^2*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/b^3/d^3+6*I*a^4*f^2*(f*x+e)*pol ylog(3,I*exp(d*x+c))/b^3/(a^2+b^2)/d^3+3*I*a^4*f*(f*x+e)^2*polylog(2,-I*ex p(d*x+c))/b^3/(a^2+b^2)/d^2+6*I*a^2*f^3*polylog(4,I*exp(d*x+c))/b^3/d^4-3/ 4*a*f^3*polylog(4,-exp(2*d*x+2*c))/b^2/d^4+2*a^2*(f*x+e)^3*arctan(exp(d*x+ c))/b^3/d-6*I*f^3*polylog(4,I*exp(d*x+c))/b/d^4+6*I*f^3*polylog(4,-I*exp(d *x+c))/b/d^4+3/2*a*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/b^2/d^3-3/2*a*f* (f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/b^2/d^2-2*a^4*(f*x+e)^3*arctan(exp(d* x+c))/b^3/(a^2+b^2)/d+3/4*a^3*f^3*polylog(4,-exp(2*d*x+2*c))/b^2/(a^2+b^2) /d^4-6*I*a^2*f^3*polylog(4,-I*exp(d*x+c))/b^3/d^4-6*I*f^2*(f*x+e)*polylog( 3,-I*exp(d*x+c))/b/d^3-3*I*f*(f*x+e)^2*polylog(2,I*exp(d*x+c))/b/d^2+6*I*f ^2*(f*x+e)*polylog(3,I*exp(d*x+c))/b/d^3+3*I*f*(f*x+e)^2*polylog(2,-I*e...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(4239\) vs. \(2(1519)=3038\).
Time = 11.06 (sec) , antiderivative size = 4239, normalized size of antiderivative = 2.79 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^3*Sinh[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x] ),x]
Output:
-1/4*(-8*a*d^4*e^3*E^(2*c)*x - 12*a*d^4*e^2*E^(2*c)*f*x^2 - 8*a*d^4*e*E^(2 *c)*f^2*x^3 - 2*a*d^4*E^(2*c)*f^3*x^4 + 8*b*d^3*e^3*ArcTan[E^(c + d*x)] + 8*b*d^3*e^3*E^(2*c)*ArcTan[E^(c + d*x)] + (12*I)*b*d^3*e^2*f*x*Log[1 - I*E ^(c + d*x)] + (12*I)*b*d^3*e^2*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (12*I) *b*d^3*e*f^2*x^2*Log[1 - I*E^(c + d*x)] + (12*I)*b*d^3*e*E^(2*c)*f^2*x^2*L og[1 - I*E^(c + d*x)] + (4*I)*b*d^3*f^3*x^3*Log[1 - I*E^(c + d*x)] + (4*I) *b*d^3*E^(2*c)*f^3*x^3*Log[1 - I*E^(c + d*x)] - (12*I)*b*d^3*e^2*f*x*Log[1 + I*E^(c + d*x)] - (12*I)*b*d^3*e^2*E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] - (12*I)*b*d^3*e*f^2*x^2*Log[1 + I*E^(c + d*x)] - (12*I)*b*d^3*e*E^(2*c)*f^2 *x^2*Log[1 + I*E^(c + d*x)] - (4*I)*b*d^3*f^3*x^3*Log[1 + I*E^(c + d*x)] - (4*I)*b*d^3*E^(2*c)*f^3*x^3*Log[1 + I*E^(c + d*x)] + 4*a*d^3*e^3*Log[1 + E^(2*(c + d*x))] + 4*a*d^3*e^3*E^(2*c)*Log[1 + E^(2*(c + d*x))] + 12*a*d^3 *e^2*f*x*Log[1 + E^(2*(c + d*x))] + 12*a*d^3*e^2*E^(2*c)*f*x*Log[1 + E^(2* (c + d*x))] + 12*a*d^3*e*f^2*x^2*Log[1 + E^(2*(c + d*x))] + 12*a*d^3*e*E^( 2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] + 4*a*d^3*f^3*x^3*Log[1 + E^(2*(c + d*x))] + 4*a*d^3*E^(2*c)*f^3*x^3*Log[1 + E^(2*(c + d*x))] - (12*I)*b*d^2*( 1 + E^(2*c))*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)] + (12*I)*b*d^2*(1 + E^(2*c))*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)] + 6*a*d^2*e^2*f*PolyLog [2, -E^(2*(c + d*x))] + 6*a*d^2*e^2*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))] + 12*a*d^2*e*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 12*a*d^2*e*E^(2*c)*f...
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.
| \(\displaystyle \int \frac {(e+f x)^3 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6115 |
| \(\displaystyle \frac {\int (e+f x)^3 \sinh (c+d x) \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 5972 |
| \(\displaystyle \frac {\int (e+f x)^3 \cosh (c+d x)dx-\int (e+f x)^3 \text {sech}(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {3 i f \int -i (e+f x)^2 \sinh (c+d x)dx}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {3 f \int (e+f x)^2 \sinh (c+d x)dx}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {3 f \int -i (e+f x)^2 \sin (i c+i d x)dx}{d}-\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {3 i f \int (e+f x)^2 \sin (i c+i d x)dx}{d}-\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {3 i f \int (e+f x)^2 \log \left (1-i e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+i e^{c+d x}\right )dx}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 6115 |
| \(\displaystyle -\frac {a \left (\frac {\int (e+f x)^3 \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int -i (e+f x)^3 \tan (i c+i d x)dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \int (e+f x)^3 \tan (i c+i d x)dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \int \frac {e^{2 (c+d x)} (e+f x)^3}{1+e^{2 (c+d x)}}dx-\frac {i (e+f x)^4}{4 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \int (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )dx}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6101 |
| \(\displaystyle \frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {3 i f \int (e+f x)^2 \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {3 i f \int (e+f x)^2 \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6107 |
| \(\displaystyle \frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \left (\frac {b^2 \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 6095 |
| \(\displaystyle \frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \left (\frac {b^2 \left (\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}+\frac {(e+f x)^3 \sinh (c+d x)}{d}}{b}-\frac {a \left (-\frac {a \left (-\frac {a \left (\frac {b^2 \left (-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {2 \arctan \left (e^{c+d x}\right ) (e+f x)^3}{d}+\frac {\sinh (c+d x) (e+f x)^3}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \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)^3 \log \left (1+e^{2 (c+d x)}\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}-\frac {a \left (\frac {\frac {2 \arctan \left (e^{c+d x}\right ) (e+f x)^3}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (\frac {\left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \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)^3 \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)^3*Sinh[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{3} \sinh \left (d x +c \right )^{2} \tanh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^3*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^3*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. 4522 vs. \(2 (1378) = 2756\).
Time = 0.29 (sec) , antiderivative size = 4522, normalized size of antiderivative = 2.98 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorit hm="fricas")
Output:
Too large to include
\[ \int \frac {(e+f x)^3 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \sinh ^{2}{\left (c + d x \right )} \tanh {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**3*sinh(d*x+c)**2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)**3*sinh(c + d*x)**2*tanh(c + d*x)/(a + b*sinh(c + d*x)) , x)
\[ \int \frac {(e+f x)^3 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sinh \left (d x + c\right )^{2} \tanh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*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^3 - 1/4*(a*d^4*f^3*x^4*e^c + 4*a*d^4*e*f^2*x^3*e^c + 6*a* d^4*e^2*f*x^2*e^c - 2*(b*d^3*f^3*x^3*e^(2*c) + 3*(d^3*e*f^2 - d^2*f^3)*b*x ^2*e^(2*c) + 3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*b*x*e^(2*c) - 3*(d^2*e^ 2*f - 2*d*e*f^2 + 2*f^3)*b*e^(2*c))*e^(d*x) + 2*(b*d^3*f^3*x^3 + 3*(d^3*e* f^2 + d^2*f^3)*b*x^2 + 3*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*b*x + 3*(d^2* e^2*f + 2*d*e*f^2 + 2*f^3)*b)*e^(-d*x))*e^(-c)/(b^2*d^4) + integrate(2*(a^ 3*b*f^3*x^3 + 3*a^3*b*e*f^2*x^2 + 3*a^3*b*e^2*f*x - (a^4*f^3*x^3*e^c + 3*a ^4*e*f^2*x^2*e^c + 3*a^4*e^2*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^3*x^3 + 3*a*e*f^2*x^2 + 3*a*e^2*f*x - (b*f^3*x^3*e^c + 3*b*e*f^2*x^2*e^c + 3*b*e^2*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. \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^3*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorit hm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2\,\mathrm {tanh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((sinh(c + d*x)^2*tanh(c + d*x)*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)
Output:
int((sinh(c + d*x)^2*tanh(c + d*x)*(e + f*x)^3)/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^3*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)
Output:
( - 8*e**(c + d*x)*atan(e**(c + d*x))*b**4*d**3*e**3 + 2*e**(2*c + 2*d*x)* a**2*b**2*d**3*e**3 + 6*e**(2*c + 2*d*x)*a**2*b**2*d**3*e**2*f*x + 6*e**(2 *c + 2*d*x)*a**2*b**2*d**3*e*f**2*x**2 + 2*e**(2*c + 2*d*x)*a**2*b**2*d**3 *f**3*x**3 - 6*e**(2*c + 2*d*x)*a**2*b**2*d**2*e**2*f - 12*e**(2*c + 2*d*x )*a**2*b**2*d**2*e*f**2*x - 6*e**(2*c + 2*d*x)*a**2*b**2*d**2*f**3*x**2 + 12*e**(2*c + 2*d*x)*a**2*b**2*d*e*f**2 + 12*e**(2*c + 2*d*x)*a**2*b**2*d*f **3*x - 12*e**(2*c + 2*d*x)*a**2*b**2*f**3 + 2*e**(2*c + 2*d*x)*b**4*d**3* e**3 + 6*e**(2*c + 2*d*x)*b**4*d**3*e**2*f*x + 6*e**(2*c + 2*d*x)*b**4*d** 3*e*f**2*x**2 + 2*e**(2*c + 2*d*x)*b**4*d**3*f**3*x**3 - 6*e**(2*c + 2*d*x )*b**4*d**2*e**2*f - 12*e**(2*c + 2*d*x)*b**4*d**2*e*f**2*x - 6*e**(2*c + 2*d*x)*b**4*d**2*f**3*x**2 + 12*e**(2*c + 2*d*x)*b**4*d*e*f**2 + 12*e**(2* c + 2*d*x)*b**4*d*f**3*x - 12*e**(2*c + 2*d*x)*b**4*f**3 - 16*e**(3*c + d* x)*int((e**(2*d*x)*x**3)/(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**4*f**3 - 8*e**(3*c + d*x)*int((e**(2*d*x)*x** 3)/(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**4*f**3 + 8*e**(3*c + d*x)*int((e**(2*d*x)*x**3)/(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**4*f**3 - 48*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**4*e*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*...