Optimal. Leaf size=20 \[ \frac{2 x \sinh (x)}{\sqrt{\cosh (x)}}-4 \sqrt{\cosh (x)} \]
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Rubi [A] time = 0.0498668, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {3315} \[ \frac{2 x \sinh (x)}{\sqrt{\cosh (x)}}-4 \sqrt{\cosh (x)} \]
Antiderivative was successfully verified.
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Rule 3315
Rubi steps
\begin{align*} \int \left (\frac{x}{\cosh ^{\frac{3}{2}}(x)}+x \sqrt{\cosh (x)}\right ) \, dx &=\int \frac{x}{\cosh ^{\frac{3}{2}}(x)} \, dx+\int x \sqrt{\cosh (x)} \, dx\\ &=-4 \sqrt{\cosh (x)}+\frac{2 x \sinh (x)}{\sqrt{\cosh (x)}}\\ \end{align*}
Mathematica [B] time = 0.346142, size = 46, normalized size = 2.3 \[ \frac{2 \sinh (x) \left (x-\frac{2 \sinh (x) \cosh (x) \sqrt{\tanh ^2\left (\frac{x}{2}\right )}}{(\cosh (x)-1)^{3/2} \sqrt{\cosh (x)+1}}\right )}{\sqrt{\cosh (x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \cosh \left ( x \right ) \right ) ^{-{\frac{3}{2}}}}+x\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 x \sqrt{\cosh \left (x\right )} + \frac{x}{\cosh \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*} \int \frac{x \left (\cosh ^{2}{\left (x \right )} + 1\right )}{\cosh ^{\frac{3}{2}}{\left (x \right )}}\, dx \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 x \sqrt{\cosh \left (x\right )} + \frac{x}{\cosh \left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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