Optimal. Leaf size=23 \[ \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a} \]
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Rubi [A] time = 0.0832048, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3319, 3301} \[ \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3301
Rubi steps
\begin{align*} \int \frac{\sqrt{a+a \cosh (x)}}{x} \, dx &=\left (\sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \frac{\cosh \left (\frac{x}{2}\right )}{x} \, dx\\ &=\sqrt{a+a \cosh (x)} \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0072812, size = 23, normalized size = 1. \[ \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a (\cosh (x)+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\sqrt{a+a\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{\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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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{\sqrt{a \left (\cosh{\left (x \right )} + 1\right )}}{x}\, 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 \frac{\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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