Integrand size = 24, antiderivative size = 432 \[ \int \frac {x^2 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {x^2}{4 b d}-\frac {\left (a^2-b^2\right ) x^3}{3 b^3}-\frac {2 a \cosh (c+d x)}{b^2 d^3}-\frac {a x^2 \cosh (c+d x)}{b^2 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {\left (a^2-b^2\right ) x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d}+\frac {2 \left (a^2-b^2\right ) x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2}+\frac {2 \left (a^2-b^2\right ) x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2}-\frac {2 \left (a^2-b^2\right ) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3}-\frac {2 \left (a^2-b^2\right ) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^3}+\frac {2 a x \sinh (c+d x)}{b^2 d^2}-\frac {x \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {\sinh ^2(c+d x)}{4 b d^3}+\frac {x^2 \sinh ^2(c+d x)}{2 b d} \] Output:
1/4*x^2/b/d-1/3*(a^2-b^2)*x^3/b^3-2*a*cosh(d*x+c)/b^2/d^3-a*x^2*cosh(d*x+c )/b^2/d+(a^2-b^2)*x^2*ln(1+b*exp(d*x+c)/(a-(a^2-b^2)^(1/2)))/b^3/d+(a^2-b^ 2)*x^2*ln(1+b*exp(d*x+c)/(a+(a^2-b^2)^(1/2)))/b^3/d+2*(a^2-b^2)*x*polylog( 2,-b*exp(d*x+c)/(a-(a^2-b^2)^(1/2)))/b^3/d^2+2*(a^2-b^2)*x*polylog(2,-b*ex p(d*x+c)/(a+(a^2-b^2)^(1/2)))/b^3/d^2-2*(a^2-b^2)*polylog(3,-b*exp(d*x+c)/ (a-(a^2-b^2)^(1/2)))/b^3/d^3-2*(a^2-b^2)*polylog(3,-b*exp(d*x+c)/(a+(a^2-b ^2)^(1/2)))/b^3/d^3+2*a*x*sinh(d*x+c)/b^2/d^2-1/2*x*cosh(d*x+c)*sinh(d*x+c )/b/d^2+1/4*sinh(d*x+c)^2/b/d^3+1/2*x^2*sinh(d*x+c)^2/b/d
Time = 3.07 (sec) , antiderivative size = 758, normalized size of antiderivative = 1.75 \[ \int \frac {x^2 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {\frac {8 \left (-a^2+b^2\right ) \left (-2 x^3+\frac {3 b^2 \left (1+e^{2 c}\right ) \left (d^2 x^2 \log \left (1+\frac {\left (a-\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )-2 d x \operatorname {PolyLog}\left (2,\frac {\left (-a+\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )-2 \operatorname {PolyLog}\left (3,\frac {\left (-a+\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )\right )}{\sqrt {a^2-b^2} \left (-a+\sqrt {a^2-b^2}\right ) d^3}+\frac {3 b^2 \left (1+e^{2 c}\right ) \left (d^2 x^2 \log \left (1+\frac {\left (a+\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )-2 d x \operatorname {PolyLog}\left (2,-\frac {\left (a+\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )-2 \operatorname {PolyLog}\left (3,-\frac {\left (a+\sqrt {a^2-b^2}\right ) e^{-c-d x}}{b}\right )\right )}{\sqrt {a^2-b^2} \left (a+\sqrt {a^2-b^2}\right ) d^3}+\frac {3 a \left (1+e^{2 c}\right ) \left (d^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )+2 d x \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )\right )}{\sqrt {a^2-b^2} d^3}-\frac {3 a \left (1+e^{2 c}\right ) \left (d^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )+2 d x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )-2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )\right )}{\sqrt {a^2-b^2} d^3}\right )}{1+e^{2 c}}-\frac {24 a b \cosh (d x) \left (\left (2+d^2 x^2\right ) \cosh (c)-2 d x \sinh (c)\right )}{d^3}+\frac {3 b^2 \cosh (2 d x) \left (\left (1+2 d^2 x^2\right ) \cosh (2 c)-2 d x \sinh (2 c)\right )}{d^3}-\frac {24 a b \left (-2 d x \cosh (c)+\left (2+d^2 x^2\right ) \sinh (c)\right ) \sinh (d x)}{d^3}+\frac {3 b^2 \left (-2 d x \cosh (2 c)+\left (1+2 d^2 x^2\right ) \sinh (2 c)\right ) \sinh (2 d x)}{d^3}+8 \left (a^2-b^2\right ) x^3 \tanh (c)}{24 b^3} \] Input:
Integrate[(x^2*Sinh[c + d*x]^3)/(a + b*Cosh[c + d*x]),x]
Output:
((8*(-a^2 + b^2)*(-2*x^3 + (3*b^2*(1 + E^(2*c))*(d^2*x^2*Log[1 + ((a - Sqr t[a^2 - b^2])*E^(-c - d*x))/b] - 2*d*x*PolyLog[2, ((-a + Sqrt[a^2 - b^2])* E^(-c - d*x))/b] - 2*PolyLog[3, ((-a + Sqrt[a^2 - b^2])*E^(-c - d*x))/b])) /(Sqrt[a^2 - b^2]*(-a + Sqrt[a^2 - b^2])*d^3) + (3*b^2*(1 + E^(2*c))*(d^2* x^2*Log[1 + ((a + Sqrt[a^2 - b^2])*E^(-c - d*x))/b] - 2*d*x*PolyLog[2, -(( (a + Sqrt[a^2 - b^2])*E^(-c - d*x))/b)] - 2*PolyLog[3, -(((a + Sqrt[a^2 - b^2])*E^(-c - d*x))/b)]))/(Sqrt[a^2 - b^2]*(a + Sqrt[a^2 - b^2])*d^3) + (3 *a*(1 + E^(2*c))*(d^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])] + 2*d*x*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 - b^2])] - 2*PolyLog[3, ( b*E^(c + d*x))/(-a + Sqrt[a^2 - b^2])]))/(Sqrt[a^2 - b^2]*d^3) - (3*a*(1 + E^(2*c))*(d^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])] + 2*d*x* PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))] - 2*PolyLog[3, -((b*E ^(c + d*x))/(a + Sqrt[a^2 - b^2]))]))/(Sqrt[a^2 - b^2]*d^3)))/(1 + E^(2*c) ) - (24*a*b*Cosh[d*x]*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c]))/d^3 + (3*b^ 2*Cosh[2*d*x]*((1 + 2*d^2*x^2)*Cosh[2*c] - 2*d*x*Sinh[2*c]))/d^3 - (24*a*b *(-2*d*x*Cosh[c] + (2 + d^2*x^2)*Sinh[c])*Sinh[d*x])/d^3 + (3*b^2*(-2*d*x* Cosh[2*c] + (1 + 2*d^2*x^2)*Sinh[2*c])*Sinh[2*d*x])/d^3 + 8*(a^2 - b^2)*x^ 3*Tanh[c])/(24*b^3)
Result contains complex when optimal does not.
Time = 2.15 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.90, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6100, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 5895, 3042, 25, 3791, 15, 6096, 2620, 3011, 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.
| \(\displaystyle \int \frac {x^2 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6100 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}-\frac {a \int x^2 \sinh (c+d x)dx}{b^2}+\frac {\int x^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}-\frac {a \int -i x^2 \sin (i c+i d x)dx}{b^2}+\frac {\int x^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \int x^2 \sin (i c+i d x)dx}{b^2}+\frac {\int x^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \int x \cosh (c+d x)dx}{d}\right )}{b^2}+\frac {\int x^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \int x \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b^2}+\frac {\int x^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {i \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{b^2}+\frac {\int x^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\int \sinh (c+d x)dx}{d}\right )}{d}\right )}{b^2}+\frac {\int x^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b^2}+\frac {\int x^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}+\frac {i \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b^2}+\frac {\int x^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\int x^2 \cosh (c+d x) \sinh (c+d x)dx}{b}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 5895 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\frac {x^2 \sinh ^2(c+d x)}{2 d}-\frac {\int x \sinh ^2(c+d x)dx}{d}}{b}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\frac {x^2 \sinh ^2(c+d x)}{2 d}-\frac {\int -x \sin (i c+i d x)^2dx}{d}}{b}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\frac {x^2 \sinh ^2(c+d x)}{2 d}+\frac {\int x \sin (i c+i d x)^2dx}{d}}{b}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\frac {\frac {\int xdx}{2}+\frac {\sinh ^2(c+d x)}{4 d^2}-\frac {x \sinh (c+d x) \cosh (c+d x)}{2 d}}{d}+\frac {x^2 \sinh ^2(c+d x)}{2 d}}{b}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^2 \sinh (c+d x)}{a+b \cosh (c+d x)}dx}{b^2}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{b^2}+\frac {\frac {\frac {\sinh ^2(c+d x)}{4 d^2}-\frac {x \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^2}{4}}{d}+\frac {x^2 \sinh ^2(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 6096 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \left (\int \frac {e^{c+d x} x^2}{a+b e^{c+d x}-\sqrt {a^2-b^2}}dx+\int \frac {e^{c+d x} x^2}{a+b e^{c+d x}+\sqrt {a^2-b^2}}dx-\frac {x^3}{3 b}\right )}{b^2}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{b^2}+\frac {\frac {\frac {\sinh ^2(c+d x)}{4 d^2}-\frac {x \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^2}{4}}{d}+\frac {x^2 \sinh ^2(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \left (-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2-b^2}}+1\right )dx}{b d}-\frac {2 \int x \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2-b^2}}+1\right )dx}{b d}+\frac {x^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}+\frac {x^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {x^3}{3 b}\right )}{b^2}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{b^2}+\frac {\frac {\frac {\sinh ^2(c+d x)}{4 d^2}-\frac {x \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^2}{4}}{d}+\frac {x^2 \sinh ^2(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \left (-\frac {2 \left (\frac {\int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )dx}{d}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}-\frac {2 \left (\frac {\int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )dx}{d}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}+\frac {x^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}+\frac {x^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {x^3}{3 b}\right )}{b^2}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{b^2}+\frac {\frac {\frac {\sinh ^2(c+d x)}{4 d^2}-\frac {x \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^2}{4}}{d}+\frac {x^2 \sinh ^2(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \left (-\frac {2 \left (\frac {\int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )de^{c+d x}}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}-\frac {2 \left (\frac {\int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )de^{c+d x}}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}+\frac {x^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}+\frac {x^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {x^3}{3 b}\right )}{b^2}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{b^2}+\frac {\frac {\frac {\sinh ^2(c+d x)}{4 d^2}-\frac {x \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^2}{4}}{d}+\frac {x^2 \sinh ^2(c+d x)}{2 d}}{b}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \left (-\frac {2 \left (\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}-\frac {2 \left (\frac {\operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d^2}-\frac {x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}+\frac {x^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}+\frac {x^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {x^3}{3 b}\right )}{b^2}+\frac {i a \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{b^2}+\frac {\frac {\frac {\sinh ^2(c+d x)}{4 d^2}-\frac {x \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {x^2}{4}}{d}+\frac {x^2 \sinh ^2(c+d x)}{2 d}}{b}\) |
Input:
Int[(x^2*Sinh[c + d*x]^3)/(a + b*Cosh[c + d*x]),x]
Output:
((a^2 - b^2)*(-1/3*x^3/b + (x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^ 2])])/(b*d) + (x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b*d) - (2*(-((x*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/d) + PolyL og[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))]/d^2))/(b*d) - (2*(-((x*Pol yLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/d) + PolyLog[3, -((b*E^ (c + d*x))/(a + Sqrt[a^2 - b^2]))]/d^2))/(b*d)))/b^2 + (I*a*((I*x^2*Cosh[c + d*x])/d - ((2*I)*(-(Cosh[c + d*x]/d^2) + (x*Sinh[c + d*x])/d))/d))/b^2 + ((x^2*Sinh[c + d*x]^2)/(2*d) + (x^2/4 - (x*Cosh[c + d*x]*Sinh[c + d*x])/ (2*d) + Sinh[c + d*x]^2/(4*d^2))/d)/b
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[((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_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 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]
\[\int \frac {x^{2} \sinh \left (d x +c \right )^{3}}{a +b \cosh \left (d x +c \right )}d x\]
Input:
int(x^2*sinh(d*x+c)^3/(a+b*cosh(d*x+c)),x)
Output:
int(x^2*sinh(d*x+c)^3/(a+b*cosh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 1622 vs. \(2 (402) = 804\).
Time = 0.11 (sec) , antiderivative size = 1622, normalized size of antiderivative = 3.75 \[ \int \frac {x^2 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(x^2*sinh(d*x+c)^3/(a+b*cosh(d*x+c)),x, algorithm="fricas")
Output:
1/48*(6*b^2*d^2*x^2 + 3*(2*b^2*d^2*x^2 - 2*b^2*d*x + b^2)*cosh(d*x + c)^4 + 3*(2*b^2*d^2*x^2 - 2*b^2*d*x + b^2)*sinh(d*x + c)^4 + 6*b^2*d*x - 24*(a* b*d^2*x^2 - 2*a*b*d*x + 2*a*b)*cosh(d*x + c)^3 - 12*(2*a*b*d^2*x^2 - 4*a*b *d*x + 4*a*b - (2*b^2*d^2*x^2 - 2*b^2*d*x + b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 16*((a^2 - b^2)*d^3*x^3 + 2*(a^2 - b^2)*c^3)*cosh(d*x + c)^2 - 2*( 8*(a^2 - b^2)*d^3*x^3 + 16*(a^2 - b^2)*c^3 - 9*(2*b^2*d^2*x^2 - 2*b^2*d*x + b^2)*cosh(d*x + c)^2 + 36*(a*b*d^2*x^2 - 2*a*b*d*x + 2*a*b)*cosh(d*x + c ))*sinh(d*x + c)^2 + 3*b^2 - 24*(a*b*d^2*x^2 + 2*a*b*d*x + 2*a*b)*cosh(d*x + c) + 96*((a^2 - b^2)*d*x*cosh(d*x + c)^2 + 2*(a^2 - b^2)*d*x*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*d*x*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))*sqrt((a^2 - b^2 )/b^2) + b)/b + 1) + 96*((a^2 - b^2)*d*x*cosh(d*x + c)^2 + 2*(a^2 - b^2)*d *x*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*d*x*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))*sq rt((a^2 - b^2)/b^2) + b)/b + 1) + 48*((a^2 - b^2)*c^2*cosh(d*x + c)^2 + 2* (a^2 - b^2)*c^2*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*c^2*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) + 48*((a^2 - b^2)*c^2*cosh(d*x + c)^2 + 2*(a^2 - b^2)*c^2*cosh(d*x + c)*sinh(d*x + c) + (a^2 - b^2)*c^2*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) + 48*(((a^2 -...
\[ \int \frac {x^2 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^{2} \sinh ^{3}{\left (c + d x \right )}}{a + b \cosh {\left (c + d x \right )}}\, dx \] Input:
integrate(x**2*sinh(d*x+c)**3/(a+b*cosh(d*x+c)),x)
Output:
Integral(x**2*sinh(c + d*x)**3/(a + b*cosh(c + d*x)), x)
\[ \int \frac {x^2 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x^{2} \sinh \left (d x + c\right )^{3}}{b \cosh \left (d x + c\right ) + a} \,d x } \] Input:
integrate(x^2*sinh(d*x+c)^3/(a+b*cosh(d*x+c)),x, algorithm="maxima")
Output:
1/48*(16*(a^2*d^3*e^(2*c) - b^2*d^3*e^(2*c))*x^3 + 3*(2*b^2*d^2*x^2*e^(4*c ) - 2*b^2*d*x*e^(4*c) + b^2*e^(4*c))*e^(2*d*x) - 24*(a*b*d^2*x^2*e^(3*c) - 2*a*b*d*x*e^(3*c) + 2*a*b*e^(3*c))*e^(d*x) - 24*(a*b*d^2*x^2*e^c + 2*a*b* d*x*e^c + 2*a*b*e^c)*e^(-d*x) + 3*(2*b^2*d^2*x^2 + 2*b^2*d*x + b^2)*e^(-2* d*x))*e^(-2*c)/(b^3*d^3) - 1/8*integrate(16*((a^3*e^c - a*b^2*e^c)*x^2*e^( d*x) + (a^2*b - b^3)*x^2)/(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) + b^4 ), x)
\[ \int \frac {x^2 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x^{2} \sinh \left (d x + c\right )^{3}}{b \cosh \left (d x + c\right ) + a} \,d x } \] Input:
integrate(x^2*sinh(d*x+c)^3/(a+b*cosh(d*x+c)),x, algorithm="giac")
Output:
integrate(x^2*sinh(d*x + c)^3/(b*cosh(d*x + c) + a), x)
Timed out. \[ \int \frac {x^2 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \] Input:
int((x^2*sinh(c + d*x)^3)/(a + b*cosh(c + d*x)),x)
Output:
int((x^2*sinh(c + d*x)^3)/(a + b*cosh(c + d*x)), x)
\[ \int \frac {x^2 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {192 e^{d x +c} a^{3} b -96 a^{4} d^{2} x^{2}-96 a^{4} d x -42 b^{4} d^{2} x^{2}-42 b^{4} d x -24 e^{3 d x +3 c} a \,b^{3} d^{2} x^{2}+48 e^{3 d x +3 c} a \,b^{3} d x -21 b^{4}-16 e^{2 d x +2 c} b^{4} d^{3} x^{3}+672 e^{2 d x +c} \left (\int \frac {x^{2}}{e^{3 d x +2 c} b +2 e^{2 d x +c} a +e^{d x} b}d x \right ) a^{3} b^{2} d^{3}-288 e^{2 d x +c} \left (\int \frac {x^{2}}{e^{3 d x +2 c} b +2 e^{2 d x +c} a +e^{d x} b}d x \right ) a \,b^{4} d^{3}-192 e^{2 d x +2 c} \left (\int \frac {x^{2}}{e^{4 d x +4 c} b +2 e^{3 d x +3 c} a +e^{2 d x +2 c} b}d x \right ) a^{4} b \,d^{3}+288 e^{2 d x +2 c} \left (\int \frac {x^{2}}{e^{4 d x +4 c} b +2 e^{3 d x +3 c} a +e^{2 d x +2 c} b}d x \right ) a^{2} b^{3} d^{3}+16 e^{2 d x +2 c} a^{2} b^{2} d^{3} x^{3}-6 e^{4 d x +4 c} b^{4} d x +96 e^{d x +c} a^{3} b \,d^{2} x^{2}+192 e^{d x +c} a^{3} b d x -120 e^{d x +c} a \,b^{3} d^{2} x^{2}-240 e^{d x +c} a \,b^{3} d x +3 e^{4 d x +4 c} b^{4}-48 e^{3 d x +3 c} a \,b^{3}-240 e^{d x +c} a \,b^{3}+72 a^{2} b^{2}-48 a^{4}+6 e^{4 d x +4 c} b^{4} d^{2} x^{2}+144 a^{2} b^{2} d^{2} x^{2}+144 a^{2} b^{2} d x -96 e^{2 d x +2 c} \left (\int \frac {x^{2}}{e^{4 d x +4 c} b +2 e^{3 d x +3 c} a +e^{2 d x +2 c} b}d x \right ) b^{5} d^{3}-384 e^{2 d x +c} \left (\int \frac {x^{2}}{e^{3 d x +2 c} b +2 e^{2 d x +c} a +e^{d x} b}d x \right ) a^{5} d^{3}}{48 e^{2 d x +2 c} b^{5} d^{3}} \] Input:
int(x^2*sinh(d*x+c)^3/(a+b*cosh(d*x+c)),x)
Output:
(6*e**(4*c + 4*d*x)*b**4*d**2*x**2 - 6*e**(4*c + 4*d*x)*b**4*d*x + 3*e**(4 *c + 4*d*x)*b**4 - 24*e**(3*c + 3*d*x)*a*b**3*d**2*x**2 + 48*e**(3*c + 3*d *x)*a*b**3*d*x - 48*e**(3*c + 3*d*x)*a*b**3 - 192*e**(2*c + 2*d*x)*int(x** 2/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + e**(2*c + 2*d*x)*b),x)*a**4 *b*d**3 + 288*e**(2*c + 2*d*x)*int(x**2/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + e**(2*c + 2*d*x)*b),x)*a**2*b**3*d**3 - 96*e**(2*c + 2*d*x)*int (x**2/(e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + e**(2*c + 2*d*x)*b),x)* b**5*d**3 + 16*e**(2*c + 2*d*x)*a**2*b**2*d**3*x**3 - 16*e**(2*c + 2*d*x)* b**4*d**3*x**3 - 384*e**(c + 2*d*x)*int(x**2/(e**(2*c + 3*d*x)*b + 2*e**(c + 2*d*x)*a + e**(d*x)*b),x)*a**5*d**3 + 672*e**(c + 2*d*x)*int(x**2/(e**( 2*c + 3*d*x)*b + 2*e**(c + 2*d*x)*a + e**(d*x)*b),x)*a**3*b**2*d**3 - 288* e**(c + 2*d*x)*int(x**2/(e**(2*c + 3*d*x)*b + 2*e**(c + 2*d*x)*a + e**(d*x )*b),x)*a*b**4*d**3 + 96*e**(c + d*x)*a**3*b*d**2*x**2 + 192*e**(c + d*x)* a**3*b*d*x + 192*e**(c + d*x)*a**3*b - 120*e**(c + d*x)*a*b**3*d**2*x**2 - 240*e**(c + d*x)*a*b**3*d*x - 240*e**(c + d*x)*a*b**3 - 96*a**4*d**2*x**2 - 96*a**4*d*x - 48*a**4 + 144*a**2*b**2*d**2*x**2 + 144*a**2*b**2*d*x + 7 2*a**2*b**2 - 42*b**4*d**2*x**2 - 42*b**4*d*x - 21*b**4)/(48*e**(2*c + 2*d *x)*b**5*d**3)