Optimal. Leaf size=76 \[ -\frac{16}{27} i \sqrt{i \sinh (x)} \sqrt{\text{csch}(x)} \text{EllipticF}\left (\frac{\pi }{4}-\frac{i x}{2},2\right )+\frac{2 x^2 \cosh (x)}{3 \sqrt{\text{csch}(x)}}-\frac{8 x}{9 \text{csch}^{\frac{3}{2}}(x)}+\frac{16 \cosh (x)}{27 \sqrt{\text{csch}(x)}} \]
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Rubi [A] time = 0.207747, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4188, 4189, 3769, 3771, 2641} \[ \frac{2 x^2 \cosh (x)}{3 \sqrt{\text{csch}(x)}}-\frac{8 x}{9 \text{csch}^{\frac{3}{2}}(x)}+\frac{16 \cosh (x)}{27 \sqrt{\text{csch}(x)}}-\frac{16}{27} i \sqrt{i \sinh (x)} \sqrt{\text{csch}(x)} F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \]
Antiderivative was successfully verified.
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Rule 4188
Rule 4189
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \left (\frac{x^2}{\text{csch}^{\frac{3}{2}}(x)}+\frac{1}{3} x^2 \sqrt{\text{csch}(x)}\right ) \, dx &=\frac{1}{3} \int x^2 \sqrt{\text{csch}(x)} \, dx+\int \frac{x^2}{\text{csch}^{\frac{3}{2}}(x)} \, dx\\ &=-\frac{8 x}{9 \text{csch}^{\frac{3}{2}}(x)}+\frac{2 x^2 \cosh (x)}{3 \sqrt{\text{csch}(x)}}-\frac{1}{3} \int x^2 \sqrt{\text{csch}(x)} \, dx+\frac{8}{9} \int \frac{1}{\text{csch}^{\frac{3}{2}}(x)} \, dx+\frac{1}{3} \left (\sqrt{\text{csch}(x)} \sqrt{-\sinh (x)}\right ) \int \frac{x^2}{\sqrt{-\sinh (x)}} \, dx\\ &=-\frac{8 x}{9 \text{csch}^{\frac{3}{2}}(x)}+\frac{16 \cosh (x)}{27 \sqrt{\text{csch}(x)}}+\frac{2 x^2 \cosh (x)}{3 \sqrt{\text{csch}(x)}}-\frac{8}{27} \int \sqrt{\text{csch}(x)} \, dx\\ &=-\frac{8 x}{9 \text{csch}^{\frac{3}{2}}(x)}+\frac{16 \cosh (x)}{27 \sqrt{\text{csch}(x)}}+\frac{2 x^2 \cosh (x)}{3 \sqrt{\text{csch}(x)}}-\frac{1}{27} \left (8 \sqrt{\text{csch}(x)} \sqrt{i \sinh (x)}\right ) \int \frac{1}{\sqrt{i \sinh (x)}} \, dx\\ &=-\frac{8 x}{9 \text{csch}^{\frac{3}{2}}(x)}+\frac{16 \cosh (x)}{27 \sqrt{\text{csch}(x)}}+\frac{2 x^2 \cosh (x)}{3 \sqrt{\text{csch}(x)}}-\frac{16}{27} i \sqrt{\text{csch}(x)} F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{i \sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.131453, size = 63, normalized size = 0.83 \[ \frac{1}{27} \sqrt{\text{csch}(x)} \left (-16 i \sqrt{i \sinh (x)} \text{EllipticF}\left (\frac{1}{4} (\pi -2 i x),2\right )+9 x^2 \sinh (2 x)+12 x+8 \sinh (2 x)-12 x \cosh (2 x)\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({\rm csch} \left (x\right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{{x}^{2}}{3}\sqrt{{\rm csch} \left (x\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*} \int \frac{1}{3} \, x^{2} \sqrt{\operatorname{csch}\left (x\right )} + \frac{x^{2}}{\operatorname{csch}\left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{3 x^{2}}{\operatorname{csch}^{\frac{3}{2}}{\left (x \right )}}\, dx + \int x^{2} \sqrt{\operatorname{csch}{\left (x \right )}}\, dx}{3} \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{1}{3} \, x^{2} \sqrt{\operatorname{csch}\left (x\right )} + \frac{x^{2}}{\operatorname{csch}\left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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