Optimal. Leaf size=24 \[ \frac{2 x \sinh (x)}{3 \sqrt{\text{sech}(x)}}-\frac{4}{9 \text{sech}^{\frac{3}{2}}(x)} \]
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Rubi [A] time = 0.0868486, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4187, 4189} \[ \frac{2 x \sinh (x)}{3 \sqrt{\text{sech}(x)}}-\frac{4}{9 \text{sech}^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 4187
Rule 4189
Rubi steps
\begin{align*} \int \left (\frac{x}{\text{sech}^{\frac{3}{2}}(x)}-\frac{1}{3} x \sqrt{\text{sech}(x)}\right ) \, dx &=-\left (\frac{1}{3} \int x \sqrt{\text{sech}(x)} \, dx\right )+\int \frac{x}{\text{sech}^{\frac{3}{2}}(x)} \, dx\\ &=-\frac{4}{9 \text{sech}^{\frac{3}{2}}(x)}+\frac{2 x \sinh (x)}{3 \sqrt{\text{sech}(x)}}+\frac{1}{3} \int x \sqrt{\text{sech}(x)} \, dx-\frac{1}{3} \left (\sqrt{\cosh (x)} \sqrt{\text{sech}(x)}\right ) \int \frac{x}{\sqrt{\cosh (x)}} \, dx\\ &=-\frac{4}{9 \text{sech}^{\frac{3}{2}}(x)}+\frac{2 x \sinh (x)}{3 \sqrt{\text{sech}(x)}}\\ \end{align*}
Mathematica [A] time = 0.0880857, size = 17, normalized size = 0.71 \[ \frac{2 (3 x \tanh (x)-2)}{9 \text{sech}^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{x \left ({\rm sech} \left (x\right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{x}{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 \sqrt{\operatorname{sech}\left (x\right )} + \frac{x}{\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}{\operatorname{sech}^{\frac{3}{2}}{\left (x \right )}}\, dx + \int x \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 \sqrt{\operatorname{sech}\left (x\right )} + \frac{x}{\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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