Optimal. Leaf size=24 \[ \frac{4}{3 \sqrt{\cosh (x)}}+\frac{2 x \sinh (x)}{3 \cosh ^{\frac{3}{2}}(x)} \]
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Rubi [A] time = 0.0511449, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {3315} \[ \frac{4}{3 \sqrt{\cosh (x)}}+\frac{2 x \sinh (x)}{3 \cosh ^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 3315
Rubi steps
\begin{align*} \int \left (\frac{x}{\cosh ^{\frac{5}{2}}(x)}-\frac{x}{3 \sqrt{\cosh (x)}}\right ) \, dx &=-\left (\frac{1}{3} \int \frac{x}{\sqrt{\cosh (x)}} \, dx\right )+\int \frac{x}{\cosh ^{\frac{5}{2}}(x)} \, dx\\ &=\frac{4}{3 \sqrt{\cosh (x)}}+\frac{2 x \sinh (x)}{3 \cosh ^{\frac{3}{2}}(x)}\\ \end{align*}
Mathematica [A] time = 0.0750161, size = 16, normalized size = 0.67 \[ \frac{2 (x \tanh (x)+2)}{3 \sqrt{\cosh (x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \cosh \left ( x \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{x}{3}{\frac{1}{\sqrt{\cosh \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{x}{3 \, \sqrt{\cosh \left (x\right )}} + \frac{x}{\cosh \left (x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.79833, size = 374, normalized size = 15.58 \begin{align*} \frac{4 \,{\left ({\left (x + 2\right )} \cosh \left (x\right )^{3} + 3 \,{\left (x + 2\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} +{\left (x + 2\right )} \sinh \left (x\right )^{3} -{\left (x - 2\right )} \cosh \left (x\right ) +{\left (3 \,{\left (x + 2\right )} \cosh \left (x\right )^{2} - x + 2\right )} \sinh \left (x\right )\right )} \sqrt{\cosh \left (x\right )}}{3 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \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}{3 \, \sqrt{\cosh \left (x\right )}} + \frac{x}{\cosh \left (x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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