Optimal. Leaf size=66 \[ -\frac{16}{27} i \sqrt{\cosh (x)} \sqrt{\text{sech}(x)} \text{EllipticF}\left (\frac{i x}{2},2\right )+\frac{2 x^2 \sinh (x)}{3 \sqrt{\text{sech}(x)}}-\frac{8 x}{9 \text{sech}^{\frac{3}{2}}(x)}+\frac{16 \sinh (x)}{27 \sqrt{\text{sech}(x)}} \]
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Rubi [A] time = 0.169032, antiderivative size = 66, 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} \[ \frac{2 x^2 \sinh (x)}{3 \sqrt{\text{sech}(x)}}-\frac{8 x}{9 \text{sech}^{\frac{3}{2}}(x)}+\frac{16 \sinh (x)}{27 \sqrt{\text{sech}(x)}}-\frac{16}{27} i \sqrt{\cosh (x)} \sqrt{\text{sech}(x)} F\left (\left .\frac{i x}{2}\right |2\right ) \]
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{sech}^{\frac{3}{2}}(x)}-\frac{1}{3} x^2 \sqrt{\text{sech}(x)}\right ) \, dx &=-\left (\frac{1}{3} \int x^2 \sqrt{\text{sech}(x)} \, dx\right )+\int \frac{x^2}{\text{sech}^{\frac{3}{2}}(x)} \, dx\\ &=-\frac{8 x}{9 \text{sech}^{\frac{3}{2}}(x)}+\frac{2 x^2 \sinh (x)}{3 \sqrt{\text{sech}(x)}}+\frac{1}{3} \int x^2 \sqrt{\text{sech}(x)} \, dx+\frac{8}{9} \int \frac{1}{\text{sech}^{\frac{3}{2}}(x)} \, dx-\frac{1}{3} \left (\sqrt{\cosh (x)} \sqrt{\text{sech}(x)}\right ) \int \frac{x^2}{\sqrt{\cosh (x)}} \, dx\\ &=-\frac{8 x}{9 \text{sech}^{\frac{3}{2}}(x)}+\frac{16 \sinh (x)}{27 \sqrt{\text{sech}(x)}}+\frac{2 x^2 \sinh (x)}{3 \sqrt{\text{sech}(x)}}+\frac{8}{27} \int \sqrt{\text{sech}(x)} \, dx\\ &=-\frac{8 x}{9 \text{sech}^{\frac{3}{2}}(x)}+\frac{16 \sinh (x)}{27 \sqrt{\text{sech}(x)}}+\frac{2 x^2 \sinh (x)}{3 \sqrt{\text{sech}(x)}}+\frac{1}{27} \left (8 \sqrt{\cosh (x)} \sqrt{\text{sech}(x)}\right ) \int \frac{1}{\sqrt{\cosh (x)}} \, dx\\ &=-\frac{8 x}{9 \text{sech}^{\frac{3}{2}}(x)}-\frac{16}{27} i \sqrt{\cosh (x)} F\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{\text{sech}(x)}+\frac{16 \sinh (x)}{27 \sqrt{\text{sech}(x)}}+\frac{2 x^2 \sinh (x)}{3 \sqrt{\text{sech}(x)}}\\ \end{align*}
Mathematica [A] time = 0.100945, size = 55, normalized size = 0.83 \[ \frac{1}{27} \sqrt{\text{sech}(x)} \left (-16 i \sqrt{\cosh (x)} \text{EllipticF}\left (\frac{i x}{2},2\right )+9 x^2 \sinh (2 x)-12 x+8 \sinh (2 x)-12 x \cosh (2 x)\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.074, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({\rm sech} \left (x\right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{{x}^{2}}{3}\sqrt{{\rm sech} \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{sech}\left (x\right )} + \frac{x^{2}}{\operatorname{sech}\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{sech}^{\frac{3}{2}}{\left (x \right )}}\, dx + \int x^{2} \sqrt{\operatorname{sech}{\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{sech}\left (x\right )} + \frac{x^{2}}{\operatorname{sech}\left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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