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 = 0.05, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2638
Rule 3296
Rule 3319
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*}
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Mathematica [A] time = 0.02, 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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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \cosh \relax (x) + a} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 38, normalized size = 1.19 \[ \frac {\sqrt {2}\, \sqrt {a \left ({\mathrm e}^{x}+1\right )^{2} {\mathrm e}^{-x}}\, \left (x \,{\mathrm e}^{x}-x -2 \,{\mathrm e}^{x}-2\right )}{{\mathrm e}^{x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 44, normalized size = 1.38 \[ -{\left (\sqrt {2} \sqrt {a} x - {\left (\sqrt {2} \sqrt {a} x - 2 \, \sqrt {2} \sqrt {a}\right )} e^{x} + 2 \, \sqrt {2} \sqrt {a}\right )} e^{\left (-\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 39, normalized size = 1.22 \[ -\frac {\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}\,\left (2\,x+4\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^x+4\right )}{{\mathrm {e}}^x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {a \left (\cosh {\relax (x )} + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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