Integrand size = 24, antiderivative size = 495 \[ \int \frac {x^3 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=-\frac {a x^4}{4 b^2}-\frac {6 \cosh (c+d x)}{b d^4}-\frac {3 x^2 \cosh (c+d x)}{b d^2}+\frac {\sqrt {a^2-b^2} x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2-b^2} x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d}+\frac {3 \sqrt {a^2-b^2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {3 \sqrt {a^2-b^2} x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {6 \sqrt {a^2-b^2} x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^3}+\frac {6 \sqrt {a^2-b^2} x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^3}+\frac {6 \sqrt {a^2-b^2} \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^4}-\frac {6 \sqrt {a^2-b^2} \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^4}+\frac {6 x \sinh (c+d x)}{b d^3}+\frac {x^3 \sinh (c+d x)}{b d} \] Output:
-1/4*a*x^4/b^2-6*cosh(d*x+c)/b/d^4-3*x^2*cosh(d*x+c)/b/d^2+(a^2-b^2)^(1/2) *x^3*ln(1+b*exp(d*x+c)/(a-(a^2-b^2)^(1/2)))/b^2/d-(a^2-b^2)^(1/2)*x^3*ln(1 +b*exp(d*x+c)/(a+(a^2-b^2)^(1/2)))/b^2/d+3*(a^2-b^2)^(1/2)*x^2*polylog(2,- b*exp(d*x+c)/(a-(a^2-b^2)^(1/2)))/b^2/d^2-3*(a^2-b^2)^(1/2)*x^2*polylog(2, -b*exp(d*x+c)/(a+(a^2-b^2)^(1/2)))/b^2/d^2-6*(a^2-b^2)^(1/2)*x*polylog(3,- b*exp(d*x+c)/(a-(a^2-b^2)^(1/2)))/b^2/d^3+6*(a^2-b^2)^(1/2)*x*polylog(3,-b *exp(d*x+c)/(a+(a^2-b^2)^(1/2)))/b^2/d^3+6*(a^2-b^2)^(1/2)*polylog(4,-b*ex p(d*x+c)/(a-(a^2-b^2)^(1/2)))/b^2/d^4-6*(a^2-b^2)^(1/2)*polylog(4,-b*exp(d *x+c)/(a+(a^2-b^2)^(1/2)))/b^2/d^4+6*x*sinh(d*x+c)/b/d^3+x^3*sinh(d*x+c)/b /d
Time = 0.65 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.78 \[ \int \frac {x^3 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {-a d^4 x^4+4 \sqrt {a^2-b^2} \left (d^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )-d^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )+3 d^2 x^2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )-3 d^2 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )-6 d x \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )+6 d x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {b e^{c+d x}}{-a+\sqrt {a^2-b^2}}\right )-6 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )\right )+4 b \cosh (d x) \left (-3 \left (2+d^2 x^2\right ) \cosh (c)+d x \left (6+d^2 x^2\right ) \sinh (c)\right )+4 b \left (d x \left (6+d^2 x^2\right ) \cosh (c)-3 \left (2+d^2 x^2\right ) \sinh (c)\right ) \sinh (d x)}{4 b^2 d^4} \] Input:
Integrate[(x^3*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]),x]
Output:
(-(a*d^4*x^4) + 4*Sqrt[a^2 - b^2]*(d^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sq rt[a^2 - b^2])] - 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])] - 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*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])] - 6*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))]) + 4*b*Cosh [d*x]*(-3*(2 + d^2*x^2)*Cosh[c] + d*x*(6 + d^2*x^2)*Sinh[c]) + 4*b*(d*x*(6 + d^2*x^2)*Cosh[c] - 3*(2 + d^2*x^2)*Sinh[c])*Sinh[d*x])/(4*b^2*d^4)
Result contains complex when optimal does not.
Time = 2.35 (sec) , antiderivative size = 463, normalized size of antiderivative = 0.94, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6100, 15, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 3801, 2694, 27, 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.
\(\displaystyle \int \frac {x^3 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx\) |
\(\Big \downarrow \) 6100 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \cosh (c+d x)}dx}{b^2}-\frac {a \int x^3dx}{b^2}+\frac {\int x^3 \cosh (c+d x)dx}{b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \cosh (c+d x)}dx}{b^2}+\frac {\int x^3 \cosh (c+d x)dx}{b}-\frac {a x^4}{4 b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\int x^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}-\frac {a x^4}{4 b^2}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}-\frac {3 i \int -i x^2 \sinh (c+d x)dx}{d}}{b}-\frac {a x^4}{4 b^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}-\frac {3 \int x^2 \sinh (c+d x)dx}{d}}{b}-\frac {a x^4}{4 b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}-\frac {3 \int -i x^2 \sin (i c+i d x)dx}{d}}{b}-\frac {a x^4}{4 b^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \int x^2 \sin (i c+i d x)dx}{d}}{b}-\frac {a x^4}{4 b^2}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \int x \cosh (c+d x)dx}{d}\right )}{d}}{b}-\frac {a x^4}{4 b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \int x \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}}{b}-\frac {a x^4}{4 b^2}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {i \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}}{b}-\frac {a x^4}{4 b^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\int \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}}{b}-\frac {a x^4}{4 b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{d}}{b}-\frac {a x^4}{4 b^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}+\frac {i \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{d}}{b}-\frac {a x^4}{4 b^2}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx}{b^2}-\frac {a x^4}{4 b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3801 |
\(\displaystyle \frac {2 \left (a^2-b^2\right ) \int \frac {e^{c+d x} x^3}{2 e^{c+d x} a+b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a x^4}{4 b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 2694 |
\(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \int \frac {e^{c+d x} x^3}{2 \left (a+b e^{c+d x}-\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}-\frac {b \int \frac {e^{c+d x} x^3}{2 \left (a+b e^{c+d x}+\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^4}{4 b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \int \frac {e^{c+d x} x^3}{a+b e^{c+d x}-\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}-\frac {b \int \frac {e^{c+d x} x^3}{a+b e^{c+d x}+\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^4}{4 b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}-\frac {3 \int x^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2-b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {3 \int x^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2-b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^4}{4 b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}-\frac {3 \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )dx}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {3 \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )dx}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^4}{4 b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )dx}{d}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )dx}{d}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^4}{4 b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {\int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {\int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^4}{4 b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {2 \left (a^2-b^2\right ) \left (\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b d}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {\operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d^2}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {b \left (\frac {x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b d}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {\operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d^2}\right )}{d}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b^2}-\frac {a x^4}{4 b^2}+\frac {\frac {x^3 \sinh (c+d x)}{d}+\frac {3 i \left (\frac {i x^2 \cosh (c+d x)}{d}-\frac {2 i \left (\frac {x \sinh (c+d x)}{d}-\frac {\cosh (c+d x)}{d^2}\right )}{d}\right )}{d}}{b}\) |
Input:
Int[(x^3*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]),x]
Output:
-1/4*(a*x^4)/b^2 + (2*(a^2 - b^2)*((b*((x^3*Log[1 + (b*E^(c + d*x))/(a - S qrt[a^2 - b^2])])/(b*d) - (3*(-((x^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqr t[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)))/(2*Sqrt[a^2 - b^2]) - (b*((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)))/(2*Sqrt[a^2 - b^2])))/b^2 + ((x^3*Sinh[c + d*x])/d + ((3*I)*((I*x^2 *Cosh[c + d*x])/d - ((2*I)*(-(Cosh[c + d*x]/d^2) + (x*Sinh[c + d*x])/d))/d ))/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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && 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_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Comple x[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Simp[2 Int[((c + d*x)^m*(E^((-I)*e + f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*((-I) *e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c , d, e, f, fz}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 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]
\[\int \frac {x^{3} \sinh \left (d x +c \right )^{2}}{a +b \cosh \left (d x +c \right )}d x\]
Input:
int(x^3*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x)
Output:
int(x^3*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 1174 vs. \(2 (451) = 902\).
Time = 0.14 (sec) , antiderivative size = 1174, normalized size of antiderivative = 2.37 \[ \int \frac {x^3 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(x^3*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x, algorithm="fricas")
Output:
-1/4*(a*d^4*x^4*cosh(d*x + c) + 2*b*d^3*x^3 + 6*b*d^2*x^2 + 12*b*d*x - 2*( b*d^3*x^3 - 3*b*d^2*x^2 + 6*b*d*x - 6*b)*cosh(d*x + c)^2 - 2*(b*d^3*x^3 - 3*b*d^2*x^2 + 6*b*d*x - 6*b)*sinh(d*x + c)^2 - 12*(b*d^2*x^2*cosh(d*x + c) + b*d^2*x^2*sinh(d*x + c))*sqrt((a^2 - b^2)/b^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) + 12*(b*d^2*x^2*cosh(d*x + c) + b*d^2*x^2*sinh(d*x + c))*s qrt((a^2 - b^2)/b^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) - 4*(b*c^3*co sh(d*x + c) + b*c^3*sinh(d*x + c))*sqrt((a^2 - b^2)/b^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) + 4*(b*c^3*cos h(d*x + c) + b*c^3*sinh(d*x + c))*sqrt((a^2 - b^2)/b^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) - 4*((b*d^3*x^3 + b*c^3)*cosh(d*x + c) + (b*d^3*x^3 + b*c^3)*sinh(d*x + c))*sqrt((a^2 - b ^2)/b^2)*log((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sin h(d*x + c))*sqrt((a^2 - b^2)/b^2) + b)/b) + 4*((b*d^3*x^3 + b*c^3)*cosh(d* x + c) + (b*d^3*x^3 + b*c^3)*sinh(d*x + c))*sqrt((a^2 - b^2)/b^2)*log((a*c osh(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) - 24*(b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^ 2 - b^2)/b^2)*polylog(4, -(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) + 24*(b*cosh(d*x + c...
\[ \int \frac {x^3 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^{3} \sinh ^{2}{\left (c + d x \right )}}{a + b \cosh {\left (c + d x \right )}}\, dx \] Input:
integrate(x**3*sinh(d*x+c)**2/(a+b*cosh(d*x+c)),x)
Output:
Integral(x**3*sinh(c + d*x)**2/(a + b*cosh(c + d*x)), x)
Exception generated. \[ \int \frac {x^3 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a-b>0)', see `assume?` for more details)Is
\[ \int \frac {x^3 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int { \frac {x^{3} \sinh \left (d x + c\right )^{2}}{b \cosh \left (d x + c\right ) + a} \,d x } \] Input:
integrate(x^3*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x, algorithm="giac")
Output:
integrate(x^3*sinh(d*x + c)^2/(b*cosh(d*x + c) + a), x)
Timed out. \[ \int \frac {x^3 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\int \frac {x^3\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \] Input:
int((x^3*sinh(c + d*x)^2)/(a + b*cosh(c + d*x)),x)
Output:
int((x^3*sinh(c + d*x)^2)/(a + b*cosh(c + d*x)), x)
\[ \int \frac {x^3 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx=\frac {2 e^{2 d x +2 c} b^{2} d^{3} x^{3}-6 e^{2 d x +2 c} b^{2} d^{2} x^{2}+12 e^{2 d x +2 c} b^{2} d x -12 e^{2 d x +2 c} b^{2}-16 e^{d x +c} \left (\int \frac {x^{3}}{e^{2 d x +2 c} b +2 e^{d x +c} a +b}d x \right ) a^{3} d^{4}+16 e^{d x +c} \left (\int \frac {x^{3}}{e^{2 d x +2 c} b +2 e^{d x +c} a +b}d x \right ) a \,b^{2} d^{4}-e^{d x +c} a b \,d^{4} x^{4}-8 e^{d x} \left (\int \frac {x^{3}}{e^{3 d x +2 c} b +2 e^{2 d x +c} a +e^{d x} b}d x \right ) a^{2} b \,d^{4}+8 e^{d x} \left (\int \frac {x^{3}}{e^{3 d x +2 c} b +2 e^{2 d x +c} a +e^{d x} b}d x \right ) b^{3} d^{4}-8 a^{2} d^{3} x^{3}-24 a^{2} d^{2} x^{2}-48 a^{2} d x -48 a^{2}+6 b^{2} d^{3} x^{3}+18 b^{2} d^{2} x^{2}+36 b^{2} d x +36 b^{2}}{4 e^{d x +c} b^{3} d^{4}} \] Input:
int(x^3*sinh(d*x+c)^2/(a+b*cosh(d*x+c)),x)
Output:
(2*e**(2*c + 2*d*x)*b**2*d**3*x**3 - 6*e**(2*c + 2*d*x)*b**2*d**2*x**2 + 1 2*e**(2*c + 2*d*x)*b**2*d*x - 12*e**(2*c + 2*d*x)*b**2 - 16*e**(c + d*x)*i nt(x**3/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a + b),x)*a**3*d**4 + 16*e**( c + d*x)*int(x**3/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a + b),x)*a*b**2*d* *4 - e**(c + d*x)*a*b*d**4*x**4 - 8*e**(d*x)*int(x**3/(e**(2*c + 3*d*x)*b + 2*e**(c + 2*d*x)*a + e**(d*x)*b),x)*a**2*b*d**4 + 8*e**(d*x)*int(x**3/(e **(2*c + 3*d*x)*b + 2*e**(c + 2*d*x)*a + e**(d*x)*b),x)*b**3*d**4 - 8*a**2 *d**3*x**3 - 24*a**2*d**2*x**2 - 48*a**2*d*x - 48*a**2 + 6*b**2*d**3*x**3 + 18*b**2*d**2*x**2 + 36*b**2*d*x + 36*b**2)/(4*e**(c + d*x)*b**3*d**4)