Optimal. Leaf size=42 \[ \frac {1}{2} \text {Shi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {\sqrt {a \cosh (x)+a}}{x} \]
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Rubi [A] time = 0.09, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3319, 3297, 3298} \[ \frac {1}{2} \text {Shi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {\sqrt {a \cosh (x)+a}}{x} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3319
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \cosh (x)}}{x^2} \, dx &=\left (\sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\cosh \left (\frac {x}{2}\right )}{x^2} \, dx\\ &=-\frac {\sqrt {a+a \cosh (x)}}{x}+\frac {1}{2} \left (\sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\sinh \left (\frac {x}{2}\right )}{x} \, dx\\ &=-\frac {\sqrt {a+a \cosh (x)}}{x}+\frac {1}{2} \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right ) \text {Shi}\left (\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 33, normalized size = 0.79 \[ \frac {\sqrt {a (\cosh (x)+1)} \left (x \text {Shi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )-2\right )}{2 x} \]
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 \[ \int \frac {\sqrt {a \cosh \relax (x) + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a +a \cosh \relax (x )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \cosh \relax (x) + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a+a\,\mathrm {cosh}\relax (x)}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {a \left (\cosh {\relax (x )} + 1\right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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