Optimal. Leaf size=432 \[ \frac{2 x \left (a^2-b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^3 d^2}+\frac{2 x \left (a^2-b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^3 d^2}-\frac{2 \left (a^2-b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^3 d^3}-\frac{2 \left (a^2-b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^3 d^3}+\frac{x^2 \left (a^2-b^2\right ) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}+1\right )}{b^3 d}+\frac{x^2 \left (a^2-b^2\right ) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}+1\right )}{b^3 d}-\frac{x^3 \left (a^2-b^2\right )}{3 b^3}+\frac{2 a x \sinh (c+d x)}{b^2 d^2}-\frac{2 a \cosh (c+d x)}{b^2 d^3}-\frac{a x^2 \cosh (c+d x)}{b^2 d}+\frac{\sinh ^2(c+d x)}{4 b d^3}-\frac{x \sinh (c+d x) \cosh (c+d x)}{2 b d^2}+\frac{x^2 \sinh ^2(c+d x)}{2 b d}+\frac{x^2}{4 b d} \]
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Rubi [A] time = 0.563287, antiderivative size = 432, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {5566, 3296, 2638, 5372, 3310, 30, 5562, 2190, 2531, 2282, 6589} \[ \frac{2 x \left (a^2-b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^3 d^2}+\frac{2 x \left (a^2-b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^3 d^2}-\frac{2 \left (a^2-b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^3 d^3}-\frac{2 \left (a^2-b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^3 d^3}+\frac{x^2 \left (a^2-b^2\right ) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}+1\right )}{b^3 d}+\frac{x^2 \left (a^2-b^2\right ) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}+1\right )}{b^3 d}-\frac{x^3 \left (a^2-b^2\right )}{3 b^3}+\frac{2 a x \sinh (c+d x)}{b^2 d^2}-\frac{2 a \cosh (c+d x)}{b^2 d^3}-\frac{a x^2 \cosh (c+d x)}{b^2 d}+\frac{\sinh ^2(c+d x)}{4 b d^3}-\frac{x \sinh (c+d x) \cosh (c+d x)}{2 b d^2}+\frac{x^2 \sinh ^2(c+d x)}{2 b d}+\frac{x^2}{4 b d} \]
Antiderivative was successfully verified.
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Rule 5566
Rule 3296
Rule 2638
Rule 5372
Rule 3310
Rule 30
Rule 5562
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^2 \sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx &=-\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}+\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{\left (a^2-b^2\right ) x^3}{3 b^3}-\frac{a x^2 \cosh (c+d x)}{b^2 d}+\frac{x^2 \sinh ^2(c+d x)}{2 b d}+\frac{\left (a^2-b^2\right ) \int \frac{e^{c+d x} x^2}{a-\sqrt{a^2-b^2}+b e^{c+d x}} \, dx}{b^2}+\frac{\left (a^2-b^2\right ) \int \frac{e^{c+d x} x^2}{a+\sqrt{a^2-b^2}+b e^{c+d x}} \, dx}{b^2}+\frac{(2 a) \int x \cosh (c+d x) \, dx}{b^2 d}-\frac{\int x \sinh ^2(c+d x) \, dx}{b d}\\ &=-\frac{\left (a^2-b^2\right ) x^3}{3 b^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 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}-\frac{(2 a) \int \sinh (c+d x) \, dx}{b^2 d^2}+\frac{\int x \, dx}{2 b d}-\frac{\left (2 \left (a^2-b^2\right )\right ) \int x \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right ) \, dx}{b^3 d}-\frac{\left (2 \left (a^2-b^2\right )\right ) \int x \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right ) \, dx}{b^3 d}\\ &=\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 \text{Li}_2\left (-\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 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^3 d^2}+\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}-\frac{\left (2 \left (a^2-b^2\right )\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right ) \, dx}{b^3 d^2}-\frac{\left (2 \left (a^2-b^2\right )\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right ) \, dx}{b^3 d^2}\\ &=\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 \text{Li}_2\left (-\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 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^3 d^2}+\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}-\frac{\left (2 \left (a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{b x}{-a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^3}-\frac{\left (2 \left (a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^3}\\ &=\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 \text{Li}_2\left (-\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 \text{Li}_2\left (-\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 ) \text{Li}_3\left (-\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 ) \text{Li}_3\left (-\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}\\ \end{align*}
Mathematica [A] time = 8.91343, size = 831, normalized size = 1.92 \[ \frac{8 \left (a^2-b^2\right ) \tanh (c) x^3-\frac{24 a b \cosh (d x) \left (\left (d^2 x^2+2\right ) \cosh (c)-2 d x \sinh (c)\right )}{d^3}+\frac{3 b^2 \cosh (2 d x) \left (\left (2 d^2 x^2+1\right ) \cosh (2 c)-2 d x \sinh (2 c)\right )}{d^3}-\frac{24 a b \left (\left (d^2 x^2+2\right ) \sinh (c)-2 d x \cosh (c)\right ) \sinh (d x)}{d^3}+\frac{3 b^2 \left (\left (2 d^2 x^2+1\right ) \sinh (2 c)-2 d x \cosh (2 c)\right ) \sinh (2 d x)}{d^3}+4 \left (b^2-a^2\right ) \left (-2 x^3+\frac{3 b^2 \left (d^2 \log \left (\frac{\left (a-\sqrt{a^2-b^2}\right ) (\cosh (c+d x)-\sinh (c+d x))}{b}+1\right ) x^2-2 d \text{PolyLog}\left (2,\frac{\left (\sqrt{a^2-b^2}-a\right ) (\cosh (c+d x)-\sinh (c+d x))}{b}\right ) x-2 \text{PolyLog}\left (3,\frac{\left (\sqrt{a^2-b^2}-a\right ) (\cosh (c+d x)-\sinh (c+d x))}{b}\right )\right ) (\cosh (2 c)+\sinh (2 c)+1)}{\sqrt{a^2-b^2} \left (\sqrt{a^2-b^2}-a\right ) d^3}+\frac{3 b^2 \left (d^2 \log \left (\frac{\left (a+\sqrt{a^2-b^2}\right ) (\cosh (c+d x)-\sinh (c+d x))}{b}+1\right ) x^2-2 d \text{PolyLog}\left (2,\frac{\left (a+\sqrt{a^2-b^2}\right ) (\sinh (c+d x)-\cosh (c+d x))}{b}\right ) x-2 \text{PolyLog}\left (3,\frac{\left (a+\sqrt{a^2-b^2}\right ) (\sinh (c+d x)-\cosh (c+d x))}{b}\right )\right ) (\cosh (2 c)+\sinh (2 c)+1)}{\sqrt{a^2-b^2} \left (a+\sqrt{a^2-b^2}\right ) d^3}+\frac{3 a \left (d^2 \log \left (\frac{b (\cosh (c+d x)+\sinh (c+d x))}{a-\sqrt{a^2-b^2}}+1\right ) x^2+2 d \text{PolyLog}\left (2,\frac{b (\cosh (c+d x)+\sinh (c+d x))}{\sqrt{a^2-b^2}-a}\right ) x-2 \text{PolyLog}\left (3,\frac{b (\cosh (c+d x)+\sinh (c+d x))}{\sqrt{a^2-b^2}-a}\right )\right ) (\cosh (2 c)+\sinh (2 c)+1)}{\sqrt{a^2-b^2} d^3}-\frac{3 a \left (d^2 \log \left (\frac{b (\cosh (c+d x)+\sinh (c+d x))}{a+\sqrt{a^2-b^2}}+1\right ) x^2+2 d \text{PolyLog}\left (2,-\frac{b (\cosh (c+d x)+\sinh (c+d x))}{a+\sqrt{a^2-b^2}}\right ) x-2 \text{PolyLog}\left (3,-\frac{b (\cosh (c+d x)+\sinh (c+d x))}{a+\sqrt{a^2-b^2}}\right )\right ) (\cosh (2 c)+\sinh (2 c)+1)}{\sqrt{a^2-b^2} d^3}\right ) (1-\tanh (c))}{24 b^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.17, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{a+b\cosh \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (16 \,{\left (a^{2} d^{3} e^{\left (2 \, c\right )} - b^{2} d^{3} e^{\left (2 \, c\right )}\right )} x^{3} + 3 \,{\left (2 \, b^{2} d^{2} x^{2} e^{\left (4 \, c\right )} - 2 \, b^{2} d x e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 24 \,{\left (a b d^{2} x^{2} e^{\left (3 \, c\right )} - 2 \, a b d x e^{\left (3 \, c\right )} + 2 \, a b e^{\left (3 \, c\right )}\right )} e^{\left (d x\right )} - 24 \,{\left (a b d^{2} x^{2} e^{c} + 2 \, a b d x e^{c} + 2 \, a b e^{c}\right )} e^{\left (-d x\right )} + 3 \,{\left (2 \, b^{2} d^{2} x^{2} + 2 \, b^{2} d x + b^{2}\right )} e^{\left (-2 \, d x\right )}\right )} e^{\left (-2 \, c\right )}}{48 \, b^{3} d^{3}} - \frac{1}{8} \, \int \frac{16 \,{\left ({\left (a^{3} e^{c} - a b^{2} e^{c}\right )} x^{2} e^{\left (d x\right )} +{\left (a^{2} b - b^{3}\right )} x^{2}\right )}}{b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{3} e^{\left (d x + c\right )} + b^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.24674, size = 3811, normalized size = 8.82 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \sinh \left (d x + c\right )^{3}}{b \cosh \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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