Integrand size = 24, antiderivative size = 586 \[ \int \frac {x^3 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {3 x}{8 b d^3}+\frac {x^3}{4 b d}-\frac {\left (a^2-b^2\right ) x^4}{4 b^3}-\frac {6 a x \cosh (c+d x)}{b^2 d^3}-\frac {a x^3 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {3 \left (a^2-b^2\right ) x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {3 \left (a^2-b^2\right ) x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}-\frac {6 \left (a^2-b^2\right ) x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3}-\frac {6 \left (a^2-b^2\right ) x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^3}+\frac {6 \left (a^2-b^2\right ) \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^4}+\frac {6 \left (a^2-b^2\right ) \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^4}+\frac {6 a \sinh (c+d x)}{b^2 d^4}+\frac {3 a x^2 \sinh (c+d x)}{b^2 d^2}-\frac {3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 x^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 x \sinh ^2(c+d x)}{4 b d^3}+\frac {x^3 \sinh ^2(c+d x)}{2 b d} \]
3/8*x/b/d^3+1/4*x^3/b/d-1/4*(a^2-b^2)*x^4/b^3-6*a*x*cosh(d*x+c)/b^2/d^3-a* x^3*cosh(d*x+c)/b^2/d+(a^2-b^2)*x^3*ln(1+b*exp(d*x+c)/(a-(a^2-b^2)^(1/2))) /b^3/d+(a^2-b^2)*x^3*ln(1+b*exp(d*x+c)/(a+(a^2-b^2)^(1/2)))/b^3/d+3*(a^2-b ^2)*x^2*polylog(2,-b*exp(d*x+c)/(a-(a^2-b^2)^(1/2)))/b^3/d^2+3*(a^2-b^2)*x ^2*polylog(2,-b*exp(d*x+c)/(a+(a^2-b^2)^(1/2)))/b^3/d^2-6*(a^2-b^2)*x*poly log(3,-b*exp(d*x+c)/(a-(a^2-b^2)^(1/2)))/b^3/d^3-6*(a^2-b^2)*x*polylog(3,- b*exp(d*x+c)/(a+(a^2-b^2)^(1/2)))/b^3/d^3+6*(a^2-b^2)*polylog(4,-b*exp(d*x +c)/(a-(a^2-b^2)^(1/2)))/b^3/d^4+6*(a^2-b^2)*polylog(4,-b*exp(d*x+c)/(a+(a ^2-b^2)^(1/2)))/b^3/d^4+6*a*sinh(d*x+c)/b^2/d^4+3*a*x^2*sinh(d*x+c)/b^2/d^ 2-3/8*cosh(d*x+c)*sinh(d*x+c)/b/d^4-3/4*x^2*cosh(d*x+c)*sinh(d*x+c)/b/d^2+ 3/4*x*sinh(d*x+c)^2/b/d^3+1/2*x^3*sinh(d*x+c)^2/b/d
Time = 4.73 (sec) , antiderivative size = 951, normalized size of antiderivative = 1.62 \[ \int \frac {x^3 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {\frac {8 \left (-a^2+b^2\right ) \left (-x^4+\frac {2 b^2 \left (1+e^{2 c}\right ) \left (d^3 x^3 \log \left (1+\frac {\left (a-\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )-3 d^2 x^2 \operatorname {PolyLog}\left (2,\frac {\left (-a+\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )-6 d x \operatorname {PolyLog}\left (3,\frac {\left (-a+\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )-6 \operatorname {PolyLog}\left (4,\frac {\left (-a+\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )\right )}{\sqrt {a^2-b^2} \left (-a+\sqrt {a^2-b^2}\right ) d^4}+\frac {2 b^2 \left (1+e^{2 c}\right ) \left (d^3 x^3 \log \left (1+\frac {\left (a+\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )-3 d^2 x^2 \operatorname {PolyLog}\left (2,-\frac {\left (a+\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )-6 d x \operatorname {PolyLog}\left (3,-\frac {\left (a+\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )-6 \operatorname {PolyLog}\left (4,-\frac {\left (a+\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )\right )}{\sqrt {a^2-b^2} \left (a+\sqrt {a^2-b^2}\right ) d^4}+\frac {2 a \left (1+e^{2 c}\right ) \left (d^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )+3 d^2 x^2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )-6 d x \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )\right )}{\sqrt {a^2-b^2} d^4}-\frac {2 a \left (1+e^{2 c}\right ) \left (d^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )+3 d^2 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )-6 d x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )+6 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )\right )}{\sqrt {a^2-b^2} d^4}\right )}{1+e^{2 c}}-\frac {16 a b \cosh (d x) \left (d x \left (6+d^2 x^2\right ) \cosh (c)-3 \left (2+d^2 x^2\right ) \sinh (c)\right )}{d^4}+\frac {b^2 \cosh (2 d x) \left (2 d x \left (3+2 d^2 x^2\right ) \cosh (2 c)-3 \left (1+2 d^2 x^2\right ) \sinh (2 c)\right )}{d^4}-\frac {16 a b \left (-3 \left (2+d^2 x^2\right ) \cosh (c)+d x \left (6+d^2 x^2\right ) \sinh (c)\right ) \sinh (d x)}{d^4}+\frac {b^2 \left (-3 \left (1+2 d^2 x^2\right ) \cosh (2 c)+2 d x \left (3+2 d^2 x^2\right ) \sinh (2 c)\right ) \sinh (2 d x)}{d^4}+4 (a-b) (a+b) x^4 \tanh (c)}{16 b^3} \]
((8*(-a^2 + b^2)*(-x^4 + (2*b^2*(1 + E^(2*c))*(d^3*x^3*Log[1 + ((a - Sqrt[ a^2 - b^2])*E^(-c - d*x))/b] - 3*d^2*x^2*PolyLog[2, ((-a + Sqrt[a^2 - b^2] )*E^(-c - d*x))/b] - 6*d*x*PolyLog[3, ((-a + Sqrt[a^2 - b^2])*E^(-c - d*x) )/b] - 6*PolyLog[4, ((-a + Sqrt[a^2 - b^2])*E^(-c - d*x))/b]))/(Sqrt[a^2 - b^2]*(-a + Sqrt[a^2 - b^2])*d^4) + (2*b^2*(1 + E^(2*c))*(d^3*x^3*Log[1 + ((a + Sqrt[a^2 - b^2])*E^(-c - d*x))/b] - 3*d^2*x^2*PolyLog[2, -(((a + Sqr t[a^2 - b^2])*E^(-c - d*x))/b)] - 6*d*x*PolyLog[3, -(((a + Sqrt[a^2 - b^2] )*E^(-c - d*x))/b)] - 6*PolyLog[4, -(((a + Sqrt[a^2 - b^2])*E^(-c - d*x))/ b)]))/(Sqrt[a^2 - b^2]*(a + Sqrt[a^2 - b^2])*d^4) + (2*a*(1 + E^(2*c))*(d^ 3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])] + 3*d^2*x^2*PolyLog[2 , (b*E^(c + d*x))/(-a + Sqrt[a^2 - b^2])] - 6*d*x*PolyLog[3, (b*E^(c + d*x ))/(-a + Sqrt[a^2 - b^2])] + 6*PolyLog[4, (b*E^(c + d*x))/(-a + Sqrt[a^2 - b^2])]))/(Sqrt[a^2 - b^2]*d^4) - (2*a*(1 + E^(2*c))*(d^3*x^3*Log[1 + (b*E ^(c + d*x))/(a + Sqrt[a^2 - b^2])] + 3*d^2*x^2*PolyLog[2, -((b*E^(c + d*x) )/(a + Sqrt[a^2 - b^2]))] - 6*d*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a ^2 - b^2]))] + 6*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))]))/(S qrt[a^2 - b^2]*d^4)))/(1 + E^(2*c)) - (16*a*b*Cosh[d*x]*(d*x*(6 + d^2*x^2) *Cosh[c] - 3*(2 + d^2*x^2)*Sinh[c]))/d^4 + (b^2*Cosh[2*d*x]*(2*d*x*(3 + 2* d^2*x^2)*Cosh[2*c] - 3*(1 + 2*d^2*x^2)*Sinh[2*c]))/d^4 - (16*a*b*(-3*(2 + d^2*x^2)*Cosh[c] + d*x*(6 + d^2*x^2)*Sinh[c])*Sinh[d*x])/d^4 + (b^2*(-3...
Result contains complex when optimal does not.
Time = 3.12 (sec) , antiderivative size = 540, normalized size of antiderivative = 0.92, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {6100, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117, 5895, 3042, 25, 3792, 15, 25, 3042, 25, 3115, 24, 6096, 2620, 3011, 7163, 2720, 7143}
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.
| \(\displaystyle \int \frac {x^3 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6100 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}-\frac {a \int x^3 \sinh (c+d x)dx}{b^2}+\frac {\int x^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}-\frac {a \int -i x^3 \sin (i c+i d x)dx}{b^2}+\frac {\int x^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \int x^3 \sin (i c+i d x)dx}{b^2}+\frac {\int x^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \int x^2 \cosh (c+d x)dx}{d}\right )}{b^2}+\frac {\int x^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \int x^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b^2}+\frac {\int x^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}-\frac {2 i \int -i x \sinh (c+d x)dx}{d}\right )}{d}\right )}{b^2}+\frac {\int x^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}-\frac {2 \int x \sinh (c+d x)dx}{d}\right )}{d}\right )}{b^2}+\frac {\int x^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}-\frac {2 \int -i x \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b^2}+\frac {\int x^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \int x \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b^2}+\frac {\int x^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )}{b^2}+\frac {\int x^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )}{d}\right )}{b^2}+\frac {\int x^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\int x^3 \cosh (c+d x) \sinh (c+d x)dx}{b}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 5895 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\frac {x^3 \sinh ^2(c+d x)}{2 d}-\frac {3 \int x^2 \sinh ^2(c+d x)dx}{2 d}}{b}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\frac {x^3 \sinh ^2(c+d x)}{2 d}-\frac {3 \int -x^2 \sin (i c+i d x)^2dx}{2 d}}{b}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\frac {x^3 \sinh ^2(c+d x)}{2 d}+\frac {3 \int x^2 \sin (i c+i d x)^2dx}{2 d}}{b}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\frac {3 \left (\frac {\int -\sinh ^2(c+d x)dx}{2 d^2}+\frac {\int x^2dx}{2}+\frac {x \sinh ^2(c+d x)}{2 d^2}-\frac {x^2 \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}+\frac {x^3 \sinh ^2(c+d x)}{2 d}}{b}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\frac {3 \left (\frac {\int -\sinh ^2(c+d x)dx}{2 d^2}+\frac {x \sinh ^2(c+d x)}{2 d^2}-\frac {x^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^3}{6}\right )}{2 d}+\frac {x^3 \sinh ^2(c+d x)}{2 d}}{b}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\frac {3 \left (-\frac {\int \sinh ^2(c+d x)dx}{2 d^2}+\frac {x \sinh ^2(c+d x)}{2 d^2}-\frac {x^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^3}{6}\right )}{2 d}+\frac {x^3 \sinh ^2(c+d x)}{2 d}}{b}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\frac {x^3 \sinh ^2(c+d x)}{2 d}+\frac {3 \left (-\frac {\int -\sin (i c+i d x)^2dx}{2 d^2}+\frac {x \sinh ^2(c+d x)}{2 d^2}-\frac {x^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^3}{6}\right )}{2 d}}{b}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\frac {x^3 \sinh ^2(c+d x)}{2 d}+\frac {3 \left (\frac {\int \sin (i c+i d x)^2dx}{2 d^2}+\frac {x \sinh ^2(c+d x)}{2 d^2}-\frac {x^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^3}{6}\right )}{2 d}}{b}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\frac {3 \left (\frac {\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}}{2 d^2}+\frac {x \sinh ^2(c+d x)}{2 d^2}-\frac {x^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^3}{6}\right )}{2 d}+\frac {x^3 \sinh ^2(c+d x)}{2 d}}{b}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}+\frac {\frac {3 \left (\frac {x \sinh ^2(c+d x)}{2 d^2}+\frac {\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}}{2 d^2}-\frac {x^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^3}{6}\right )}{2 d}+\frac {x^3 \sinh ^2(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 6096 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \left (\int \frac {e^{c+d x} x^3}{a+b e^{c+d x}-\sqrt {a^2-b^2}}dx+\int \frac {e^{c+d x} x^3}{a+b e^{c+d x}+\sqrt {a^2-b^2}}dx-\frac {x^4}{4 b}\right )}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}+\frac {\frac {3 \left (\frac {x \sinh ^2(c+d x)}{2 d^2}+\frac {\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}}{2 d^2}-\frac {x^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^3}{6}\right )}{2 d}+\frac {x^3 \sinh ^2(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \left (-\frac {3 \int x^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2-b^2}}+1\right )dx}{b d}-\frac {3 \int x^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2-b^2}}+1\right )dx}{b d}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {x^4}{4 b}\right )}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}+\frac {\frac {3 \left (\frac {x \sinh ^2(c+d x)}{2 d^2}+\frac {\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}}{2 d^2}-\frac {x^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^3}{6}\right )}{2 d}+\frac {x^3 \sinh ^2(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \left (-\frac {3 \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )dx}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}-\frac {3 \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )dx}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {x^4}{4 b}\right )}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}+\frac {\frac {3 \left (\frac {x \sinh ^2(c+d x)}{2 d^2}+\frac {\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}}{2 d^2}-\frac {x^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^3}{6}\right )}{2 d}+\frac {x^3 \sinh ^2(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \left (-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )dx}{d}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )dx}{d}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {x^4}{4 b}\right )}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}+\frac {\frac {3 \left (\frac {x \sinh ^2(c+d x)}{2 d^2}+\frac {\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}}{2 d^2}-\frac {x^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^3}{6}\right )}{2 d}+\frac {x^3 \sinh ^2(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \left (-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {\int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {\int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {x^4}{4 b}\right )}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}+\frac {\frac {3 \left (\frac {x \sinh ^2(c+d x)}{2 d^2}+\frac {\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}}{2 d^2}-\frac {x^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^3}{6}\right )}{2 d}+\frac {x^3 \sinh ^2(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \left (-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {\operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d^2}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {\operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d^2}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}+\frac {x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {x^4}{4 b}\right )}{b^2}+\frac {i a \left (\frac {i x^3 \cosh (c+d x)}{d}-\frac {3 i \left (\frac {x^2 \sinh (c+d x)}{d}+\frac {2 i \left (\frac {i x \cosh (c+d x)}{d}-\frac {i \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}+\frac {\frac {3 \left (\frac {x \sinh ^2(c+d x)}{2 d^2}+\frac {\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}}{2 d^2}-\frac {x^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^3}{6}\right )}{2 d}+\frac {x^3 \sinh ^2(c+d x)}{2 d}}{b}\) |
((a^2 - b^2)*(-1/4*x^4/b + (x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^ 2])])/(b*d) + (x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b*d) - (3*(-((x^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/d) + (2* ((x*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/d - PolyLog[4, - ((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))]/d^2))/d))/(b*d) - (3*(-((x^2*Poly Log[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/d) + (2*((x*PolyLog[3, - ((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/d - PolyLog[4, -((b*E^(c + d*x)) /(a + Sqrt[a^2 - b^2]))]/d^2))/d))/(b*d)))/b^2 + (I*a*((I*x^3*Cosh[c + d*x ])/d - ((3*I)*((x^2*Sinh[c + d*x])/d + ((2*I)*((I*x*Cosh[c + d*x])/d - (I* Sinh[c + d*x])/d^2))/d))/d))/b^2 + ((x^3*Sinh[c + d*x]^2)/(2*d) + (3*(x^3/ 6 - (x^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + (x*Sinh[c + d*x]^2)/(2*d^2) + (x/2 - (Cosh[c + d*x]*Sinh[c + d*x])/(2*d))/(2*d^2)))/(2*d))/b
3.3.34.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.) ]^(p_.), x_Symbol] :> Simp[x^(m - n + 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] - Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*Sinh[a + b*x^n]^ (p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_ .)*(x_)]*(b_.) + (a_)), 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_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_))/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[-a/b^2 Int[(e + f*x)^m*Sin h[c + d*x]^(n - 2), x], x] + (Simp[1/b Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2)*Cosh[c + d*x], x], x] + Simp[(a^2 - b^2)/b^2 Int[(e + f*x)^m*(Sinh[c + d*x]^(n - 2)/(a + b*Cosh[c + d*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
\[\int \frac {x^{3} \sinh \left (d x +c \right )^{3}}{a +b \cosh \left (d x +c \right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 2025 vs. \(2 (546) = 1092\).
Time = 0.30 (sec) , antiderivative size = 2025, normalized size of antiderivative = 3.46 \[ \int \frac {x^3 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\text {Too large to display} \]
1/32*(4*b^2*d^3*x^3 + 6*b^2*d^2*x^2 + (4*b^2*d^3*x^3 - 6*b^2*d^2*x^2 + 6*b ^2*d*x - 3*b^2)*cosh(d*x + c)^4 + (4*b^2*d^3*x^3 - 6*b^2*d^2*x^2 + 6*b^2*d *x - 3*b^2)*sinh(d*x + c)^4 + 6*b^2*d*x - 16*(a*b*d^3*x^3 - 3*a*b*d^2*x^2 + 6*a*b*d*x - 6*a*b)*cosh(d*x + c)^3 - 4*(4*a*b*d^3*x^3 - 12*a*b*d^2*x^2 + 24*a*b*d*x - 24*a*b - (4*b^2*d^3*x^3 - 6*b^2*d^2*x^2 + 6*b^2*d*x - 3*b^2) *cosh(d*x + c))*sinh(d*x + c)^3 - 8*((a^2 - b^2)*d^4*x^4 - 2*(a^2 - b^2)*c ^4)*cosh(d*x + c)^2 - 2*(4*(a^2 - b^2)*d^4*x^4 - 8*(a^2 - b^2)*c^4 - 3*(4* b^2*d^3*x^3 - 6*b^2*d^2*x^2 + 6*b^2*d*x - 3*b^2)*cosh(d*x + c)^2 + 24*(a*b *d^3*x^3 - 3*a*b*d^2*x^2 + 6*a*b*d*x - 6*a*b)*cosh(d*x + c))*sinh(d*x + c) ^2 + 3*b^2 - 16*(a*b*d^3*x^3 + 3*a*b*d^2*x^2 + 6*a*b*d*x + 6*a*b)*cosh(d*x + c) + 96*((a^2 - b^2)*d^2*x^2*cosh(d*x + c)^2 + 2*(a^2 - b^2)*d^2*x^2*co sh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*d^2*x^2*sinh(d*x + c)^2)*dilog(-(a *cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqr t((a^2 - b^2)/b^2) + b)/b + 1) + 96*((a^2 - b^2)*d^2*x^2*cosh(d*x + c)^2 + 2*(a^2 - b^2)*d^2*x^2*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*d^2*x^2*s inh(d*x + c)^2)*dilog(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b + 1) - 32*((a^2 - b^2)* c^3*cosh(d*x + c)^2 + 2*(a^2 - b^2)*c^3*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*c^3*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2 *b*sqrt((a^2 - b^2)/b^2) + 2*a) - 32*((a^2 - b^2)*c^3*cosh(d*x + c)^2 +...
\[ \int \frac {x^3 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^{3} \sinh ^{3}{\left (c + d x \right )}}{a + b \cosh {\left (c + d x \right )}}\, dx \]
\[ \int \frac {x^3 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x^{3} \sinh \left (d x + c\right )^{3}}{b \cosh \left (d x + c\right ) + a} \,d x } \]
1/32*(8*(a^2*d^4*e^(2*c) - b^2*d^4*e^(2*c))*x^4 + (4*b^2*d^3*x^3*e^(4*c) - 6*b^2*d^2*x^2*e^(4*c) + 6*b^2*d*x*e^(4*c) - 3*b^2*e^(4*c))*e^(2*d*x) - 16 *(a*b*d^3*x^3*e^(3*c) - 3*a*b*d^2*x^2*e^(3*c) + 6*a*b*d*x*e^(3*c) - 6*a*b* e^(3*c))*e^(d*x) - 16*(a*b*d^3*x^3*e^c + 3*a*b*d^2*x^2*e^c + 6*a*b*d*x*e^c + 6*a*b*e^c)*e^(-d*x) + (4*b^2*d^3*x^3 + 6*b^2*d^2*x^2 + 6*b^2*d*x + 3*b^ 2)*e^(-2*d*x))*e^(-2*c)/(b^3*d^4) - 1/8*integrate(16*((a^3*e^c - a*b^2*e^c )*x^3*e^(d*x) + (a^2*b - b^3)*x^3)/(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) + b^4), x)
\[ \int \frac {x^3 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x^{3} \sinh \left (d x + c\right )^{3}}{b \cosh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {x^3 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \]