Optimal. Leaf size=370 \[ \frac{2 x \sqrt{a^2-b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{2 x \sqrt{a^2-b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^2}-\frac{2 \sqrt{a^2-b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^3}+\frac{2 \sqrt{a^2-b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^3}+\frac{x^2 \sqrt{a^2-b^2} \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}+1\right )}{b^2 d}-\frac{x^2 \sqrt{a^2-b^2} \log \left (\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}+1\right )}{b^2 d}-\frac{a x^3}{3 b^2}+\frac{2 \sinh (c+d x)}{b d^3}-\frac{2 x \cosh (c+d x)}{b d^2}+\frac{x^2 \sinh (c+d x)}{b d} \]
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Rubi [A] time = 0.704118, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5566, 30, 3296, 2637, 3320, 2264, 2190, 2531, 2282, 6589} \[ \frac{2 x \sqrt{a^2-b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{2 x \sqrt{a^2-b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^2}-\frac{2 \sqrt{a^2-b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^3}+\frac{2 \sqrt{a^2-b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^3}+\frac{x^2 \sqrt{a^2-b^2} \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}+1\right )}{b^2 d}-\frac{x^2 \sqrt{a^2-b^2} \log \left (\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}+1\right )}{b^2 d}-\frac{a x^3}{3 b^2}+\frac{2 \sinh (c+d x)}{b d^3}-\frac{2 x \cosh (c+d x)}{b d^2}+\frac{x^2 \sinh (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 5566
Rule 30
Rule 3296
Rule 2637
Rule 3320
Rule 2264
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^2 \sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx &=-\frac{a \int x^2 \, dx}{b^2}+\frac{\int x^2 \cosh (c+d x) \, dx}{b}+\frac{\left (a^2-b^2\right ) \int \frac{x^2}{a+b \cosh (c+d x)} \, dx}{b^2}\\ &=-\frac{a x^3}{3 b^2}+\frac{x^2 \sinh (c+d x)}{b d}+\frac{\left (2 \left (a^2-b^2\right )\right ) \int \frac{e^{c+d x} x^2}{b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^2}-\frac{2 \int x \sinh (c+d x) \, dx}{b d}\\ &=-\frac{a x^3}{3 b^2}-\frac{2 x \cosh (c+d x)}{b d^2}+\frac{x^2 \sinh (c+d x)}{b d}+\frac{\left (2 \sqrt{a^2-b^2}\right ) \int \frac{e^{c+d x} x^2}{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^2}{2 a+2 \sqrt{a^2-b^2}+2 b e^{c+d x}} \, dx}{b}+\frac{2 \int \cosh (c+d x) \, dx}{b d^2}\\ &=-\frac{a x^3}{3 b^2}-\frac{2 x \cosh (c+d x)}{b d^2}+\frac{\sqrt{a^2-b^2} x^2 \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^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d}+\frac{2 \sinh (c+d x)}{b d^3}+\frac{x^2 \sinh (c+d x)}{b d}-\frac{\left (2 \sqrt{a^2-b^2}\right ) \int x \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 (2 \sqrt{a^2-b^2}\right ) \int x \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^3}{3 b^2}-\frac{2 x \cosh (c+d x)}{b d^2}+\frac{\sqrt{a^2-b^2} x^2 \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^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d}+\frac{2 \sqrt{a^2-b^2} x \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{2 \sqrt{a^2-b^2} x \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d^2}+\frac{2 \sinh (c+d x)}{b d^3}+\frac{x^2 \sinh (c+d x)}{b d}-\frac{\left (2 \sqrt{a^2-b^2}\right ) \int \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 (2 \sqrt{a^2-b^2}\right ) \int \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^3}{3 b^2}-\frac{2 x \cosh (c+d x)}{b d^2}+\frac{\sqrt{a^2-b^2} x^2 \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^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d}+\frac{2 \sqrt{a^2-b^2} x \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{2 \sqrt{a^2-b^2} x \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d^2}+\frac{2 \sinh (c+d x)}{b d^3}+\frac{x^2 \sinh (c+d x)}{b d}-\frac{\left (2 \sqrt{a^2-b^2}\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^2 d^3}+\frac{\left (2 \sqrt{a^2-b^2}\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^2 d^3}\\ &=-\frac{a x^3}{3 b^2}-\frac{2 x \cosh (c+d x)}{b d^2}+\frac{\sqrt{a^2-b^2} x^2 \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^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d}+\frac{2 \sqrt{a^2-b^2} x \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{2 \sqrt{a^2-b^2} x \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2-b^2}}\right )}{b^2 d^2}-\frac{2 \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{2 \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{2 \sinh (c+d x)}{b d^3}+\frac{x^2 \sinh (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 1.19684, size = 293, normalized size = 0.79 \[ \frac{3 \sqrt{a^2-b^2} \left (2 d x \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2-b^2}-a}\right )-2 d x \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )-2 \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2-b^2}-a}\right )+2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}\right )+d^2 x^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2-b^2}}+1\right )-d^2 x^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2-b^2}+a}+1\right )\right )-a d^3 x^3+3 b \cosh (d x) \left (\sinh (c) \left (d^2 x^2+2\right )-2 d x \cosh (c)\right )+3 b \sinh (d x) \left (\cosh (c) \left (d^2 x^2+2\right )-2 d x \sinh (c)\right )}{3 b^2 d^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.109, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.13667, size = 2303, normalized size = 6.22 \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 )^{2}}{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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