Optimal. Leaf size=495 \[ \frac {6 \sqrt {a^2-b^2} \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^4}-\frac {6 \sqrt {a^2-b^2} \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^4}-\frac {6 x \sqrt {a^2-b^2} \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^3}+\frac {6 x \sqrt {a^2-b^2} \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^3}+\frac {3 x^2 \sqrt {a^2-b^2} \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {3 x^2 \sqrt {a^2-b^2} \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2}+\frac {x^3 \sqrt {a^2-b^2} \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b^2 d}-\frac {x^3 \sqrt {a^2-b^2} \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b^2 d}-\frac {a x^4}{4 b^2}-\frac {6 \cosh (c+d x)}{b d^4}+\frac {6 x \sinh (c+d x)}{b d^3}-\frac {3 x^2 \cosh (c+d x)}{b d^2}+\frac {x^3 \sinh (c+d x)}{b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.84, antiderivative size = 495, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5566, 30, 3296, 2638, 3320, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac {3 x^2 \sqrt {a^2-b^2} \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {3 x^2 \sqrt {a^2-b^2} \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}\right )}{b^2 d^2}-\frac {6 x \sqrt {a^2-b^2} \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^3}+\frac {6 x \sqrt {a^2-b^2} \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}\right )}{b^2 d^3}+\frac {6 \sqrt {a^2-b^2} \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^4}-\frac {6 \sqrt {a^2-b^2} \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}\right )}{b^2 d^4}+\frac {x^3 \sqrt {a^2-b^2} \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )}{b^2 d}-\frac {x^3 \sqrt {a^2-b^2} \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )}{b^2 d}-\frac {a x^4}{4 b^2}-\frac {3 x^2 \cosh (c+d x)}{b d^2}+\frac {6 x \sinh (c+d x)}{b d^3}-\frac {6 \cosh (c+d x)}{b d^4}+\frac {x^3 \sinh (c+d x)}{b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 2638
Rule 3296
Rule 3320
Rule 5566
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^3 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx &=-\frac {a \int x^3 \, dx}{b^2}+\frac {\int x^3 \cosh (c+d x) \, dx}{b}+\frac {\left (a^2-b^2\right ) \int \frac {x^3}{a+b \cosh (c+d x)} \, dx}{b^2}\\ &=-\frac {a x^4}{4 b^2}+\frac {x^3 \sinh (c+d x)}{b d}+\frac {\left (2 \left (a^2-b^2\right )\right ) \int \frac {e^{c+d x} x^3}{b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^2}-\frac {3 \int x^2 \sinh (c+d x) \, dx}{b d}\\ &=-\frac {a x^4}{4 b^2}-\frac {3 x^2 \cosh (c+d x)}{b d^2}+\frac {x^3 \sinh (c+d x)}{b d}+\frac {\left (2 \sqrt {a^2-b^2}\right ) \int \frac {e^{c+d x} x^3}{2 a-2 \sqrt {a^2-b^2}+2 b e^{c+d x}} \, dx}{b}-\frac {\left (2 \sqrt {a^2-b^2}\right ) \int \frac {e^{c+d x} x^3}{2 a+2 \sqrt {a^2-b^2}+2 b e^{c+d x}} \, dx}{b}+\frac {6 \int x \cosh (c+d x) \, dx}{b d^2}\\ &=-\frac {a x^4}{4 b^2}-\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 {6 x \sinh (c+d x)}{b d^3}+\frac {x^3 \sinh (c+d x)}{b d}-\frac {6 \int \sinh (c+d x) \, dx}{b d^3}-\frac {\left (3 \sqrt {a^2-b^2}\right ) \int x^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 d}+\frac {\left (3 \sqrt {a^2-b^2}\right ) \int x^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 d}\\ &=-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2}+\frac {6 x \sinh (c+d x)}{b d^3}+\frac {x^3 \sinh (c+d x)}{b d}-\frac {\left (6 \sqrt {a^2-b^2}\right ) \int x \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 d^2}+\frac {\left (6 \sqrt {a^2-b^2}\right ) \int x \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 d^2}\\ &=-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^3}+\frac {6 x \sinh (c+d x)}{b d^3}+\frac {x^3 \sinh (c+d x)}{b d}+\frac {\left (6 \sqrt {a^2-b^2}\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 d^3}-\frac {\left (6 \sqrt {a^2-b^2}\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 d^3}\\ &=-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^3}+\frac {6 x \sinh (c+d x)}{b d^3}+\frac {x^3 \sinh (c+d x)}{b d}+\frac {\left (6 \sqrt {a^2-b^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^4}-\frac {\left (6 \sqrt {a^2-b^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^4}\\ &=-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_3\left (-\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 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^3}+\frac {6 \sqrt {a^2-b^2} \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^4}-\frac {6 \sqrt {a^2-b^2} \text {Li}_4\left (-\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}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.51, size = 386, normalized size = 0.78 \[ \frac {4 \sqrt {a^2-b^2} \left (d^3 x^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2-b^2}}+1\right )-d^3 x^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}+a}+1\right )+3 d^2 x^2 \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}-a}\right )-3 d^2 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )-6 d x \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}-a}\right )+6 d x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )+6 \text {Li}_4\left (\frac {b e^{c+d x}}{\sqrt {a^2-b^2}-a}\right )-6 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2-b^2}}\right )\right )-a d^4 x^4+4 b \cosh (d x) \left (d x \sinh (c) \left (d^2 x^2+6\right )-3 \cosh (c) \left (d^2 x^2+2\right )\right )+4 b \sinh (d x) \left (d x \cosh (c) \left (d^2 x^2+6\right )-3 \sinh (c) \left (d^2 x^2+2\right )\right )}{4 b^2 d^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.69, size = 1174, normalized size = 2.37 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \sinh \left (d x + c\right )^{2}}{b \cosh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (\sinh ^{2}\left (d x +c \right )\right )}{a +b \cosh \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \sinh ^{2}{\left (c + d x \right )}}{a + b \cosh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________