Optimal. Leaf size=32 \[ 2 x \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-4 \sqrt{a \cosh (x)+a} \]
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Rubi [A] time = 0.0508111, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3319, 3296, 2638} \[ 2 x \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-4 \sqrt{a \cosh (x)+a} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x \sqrt{a+a \cosh (x)} \, dx &=\left (\sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int x \cosh \left (\frac{x}{2}\right ) \, dx\\ &=2 x \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )-\left (2 \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \sinh \left (\frac{x}{2}\right ) \, dx\\ &=-4 \sqrt{a+a \cosh (x)}+2 x \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0201894, size = 22, normalized size = 0.69 \[ 2 \left (x \tanh \left (\frac{x}{2}\right )-2\right ) \sqrt{a (\cosh (x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 38, normalized size = 1.2 \begin{align*}{\frac{\sqrt{2} \left ( x{{\rm e}^{x}}-x-2\,{{\rm e}^{x}}-2 \right ) }{{{\rm e}^{x}}+1}\sqrt{a \left ({{\rm e}^{x}}+1 \right ) ^{2}{{\rm e}^{-x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68724, size = 59, normalized size = 1.84 \begin{align*} -{\left (\sqrt{2} \sqrt{a} x -{\left (\sqrt{2} \sqrt{a} x - 2 \, \sqrt{2} \sqrt{a}\right )} e^{x} + 2 \, \sqrt{2} \sqrt{a}\right )} e^{\left (-\frac{1}{2} \, x\right )} \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 x \sqrt{a \left (\cosh{\left (x \right )} + 1\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 \sqrt{a \cosh \left (x\right ) + a} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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