Optimal. Leaf size=67 \[ \frac{1}{8} \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{\sqrt{a \cosh (x)+a}}{2 x^2}-\frac{\tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}}{4 x} \]
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Rubi [A] time = 0.10618, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3319, 3297, 3301} \[ \frac{1}{8} \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{\sqrt{a \cosh (x)+a}}{2 x^2}-\frac{\tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}}{4 x} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3297
Rule 3301
Rubi steps
\begin{align*} \int \frac{\sqrt{a+a \cosh (x)}}{x^3} \, dx &=\left (\sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \frac{\cosh \left (\frac{x}{2}\right )}{x^3} \, dx\\ &=-\frac{\sqrt{a+a \cosh (x)}}{2 x^2}+\frac{1}{4} \left (\sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \frac{\sinh \left (\frac{x}{2}\right )}{x^2} \, dx\\ &=-\frac{\sqrt{a+a \cosh (x)}}{2 x^2}-\frac{\sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )}{4 x}+\frac{1}{8} \left (\sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \frac{\cosh \left (\frac{x}{2}\right )}{x} \, dx\\ &=-\frac{\sqrt{a+a \cosh (x)}}{2 x^2}+\frac{1}{8} \sqrt{a+a \cosh (x)} \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right )-\frac{\sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )}{4 x}\\ \end{align*}
Mathematica [A] time = 0.0704671, size = 44, normalized size = 0.66 \[ \frac{\sqrt{a (\cosh (x)+1)} \left (x^2 \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right )-2 x \tanh \left (\frac{x}{2}\right )-4\right )}{8 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\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^{3}}\,{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^{3}}\, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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