Optimal. Leaf size=109 \[ \frac{3}{16} a \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{9}{16} a \text{Chi}\left (\frac{3 x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}}{x^2}-\frac{3 a \sinh \left (\frac{x}{2}\right ) \cosh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}}{2 x} \]
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Rubi [A] time = 0.172993, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3319, 3314, 3301, 3312} \[ \frac{3}{16} a \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{9}{16} a \text{Chi}\left (\frac{3 x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}}{x^2}-\frac{3 a \sinh \left (\frac{x}{2}\right ) \cosh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}}{2 x} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3314
Rule 3301
Rule 3312
Rubi steps
\begin{align*} \int \frac{(a+a \cosh (x))^{3/2}}{x^3} \, 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^3} \, dx\\ &=-\frac{a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}}{x^2}-\frac{3 a \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )}{2 x}-\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{1}{4} \left (9 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\\ &=-\frac{a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}}{x^2}-\frac{3}{2} a \sqrt{a+a \cosh (x)} \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right )-\frac{3 a \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )}{2 x}+\frac{1}{4} \left (9 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{a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}}{x^2}-\frac{3}{2} a \sqrt{a+a \cosh (x)} \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right )-\frac{3 a \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )}{2 x}+\frac{1}{16} \left (9 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}{16} \left (27 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{a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}}{x^2}+\frac{3}{16} a \sqrt{a+a \cosh (x)} \text{Chi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right )+\frac{9}{16} a \sqrt{a+a \cosh (x)} \text{Chi}\left (\frac{3 x}{2}\right ) \text{sech}\left (\frac{x}{2}\right )-\frac{3 a \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )}{2 x}\\ \end{align*}
Mathematica [A] time = 0.0646642, size = 69, normalized size = 0.63 \[ \frac{(a (\cosh (x)+1))^{3/2} \left (3 x^2 \text{Chi}\left (\frac{x}{2}\right ) \text{sech}^3\left (\frac{x}{2}\right )+9 x^2 \text{Chi}\left (\frac{3 x}{2}\right ) \text{sech}^3\left (\frac{x}{2}\right )-8 \left (3 x \tanh \left (\frac{x}{2}\right )+2\right )\right )}{32 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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^{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(-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 = 1.30933, size = 180, normalized size = 1.65 \begin{align*} \frac{\sqrt{2}{\left (9 \, a^{\frac{3}{2}} x^{2}{\rm Ei}\left (\frac{3}{2} \, x\right ) + 3 \, a^{\frac{3}{2}} x^{2}{\rm Ei}\left (\frac{1}{2} \, x\right ) + 3 \, a^{\frac{3}{2}} x^{2}{\rm Ei}\left (-\frac{1}{2} \, x\right ) + 9 \, a^{\frac{3}{2}} x^{2}{\rm Ei}\left (-\frac{3}{2} \, x\right ) - 6 \, a^{\frac{3}{2}} x e^{\left (\frac{3}{2} \, x\right )} - 6 \, a^{\frac{3}{2}} x e^{\left (\frac{1}{2} \, x\right )} + 6 \, a^{\frac{3}{2}} x e^{\left (-\frac{1}{2} \, x\right )} + 6 \, a^{\frac{3}{2}} x e^{\left (-\frac{3}{2} \, x\right )} - 4 \, a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, x\right )} - 12 \, a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, x\right )} - 12 \, a^{\frac{3}{2}} e^{\left (-\frac{1}{2} \, x\right )} - 4 \, a^{\frac{3}{2}} e^{\left (-\frac{3}{2} \, x\right )}\right )}}{32 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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