Optimal. Leaf size=55 \[ \frac{3}{2} a \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{1}{2} a \text{Chi}\left (\frac{3 x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a} \]
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Rubi [A] time = 0.128222, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3319, 3312, 3301} \[ \frac{3}{2} a \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{1}{2} a \text{Chi}\left (\frac{3 x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int \frac{(a+a \cosh (x))^{3/2}}{x} \, dx &=\left (2 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \frac{\cosh ^3\left (\frac{x}{2}\right )}{x} \, dx\\ &=\left (2 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \left (\frac{3 \cosh \left (\frac{x}{2}\right )}{4 x}+\frac{\cosh \left (\frac{3 x}{2}\right )}{4 x}\right ) \, dx\\ &=\frac{1}{2} \left (a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \frac{\cosh \left (\frac{3 x}{2}\right )}{x} \, dx+\frac{1}{2} \left (3 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \frac{\cosh \left (\frac{x}{2}\right )}{x} \, dx\\ &=\frac{3}{2} a \sqrt{a+a \cosh (x)} \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right )+\frac{1}{2} a \sqrt{a+a \cosh (x)} \text{Chi}\left (\frac{3 x}{2}\right ) \text{sech}\left (\frac{x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0196958, size = 36, normalized size = 0.65 \[ \frac{1}{2} a \left (3 \text{Chi}\left (\frac{x}{2}\right )+\text{Chi}\left (\frac{3 x}{2}\right )\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a (\cosh (x)+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+a\cosh \left ( x \right ) \right ) ^{{\frac{3}{2}}}}\, 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{{\left (a \cosh \left (x\right ) + a\right )}^{\frac{3}{2}}}{x}\,{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(-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 [A] time = 1.17945, size = 54, normalized size = 0.98 \begin{align*} \frac{1}{4} \, \sqrt{2}{\left (a^{\frac{3}{2}}{\rm Ei}\left (\frac{3}{2} \, x\right ) + 3 \, a^{\frac{3}{2}}{\rm Ei}\left (\frac{1}{2} \, x\right ) + 3 \, a^{\frac{3}{2}}{\rm Ei}\left (-\frac{1}{2} \, x\right ) + a^{\frac{3}{2}}{\rm Ei}\left (-\frac{3}{2} \, x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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