Optimal. Leaf size=50 \[ \frac{2 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \cosh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}} \]
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Rubi [A] time = 0.167808, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5713, 5676} \[ \frac{2 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \cosh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5676
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\sqrt{a^2-x^2}} \, dx &=\frac{\left (\sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}\right ) \int \frac{\cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx}{\sqrt{a^2-x^2}}\\ &=\frac{2 a \sqrt{-1+\frac{x}{a}} \sqrt{1+\frac{x}{a}} \cosh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}}\\ \end{align*}
Mathematica [A] time = 0.0510657, size = 50, normalized size = 1. \[ \frac{2 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \cosh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 44, normalized size = 0.9 \begin{align*}{\frac{2\,a}{5} \left ({\rm arccosh} \left ({\frac{x}{a}}\right ) \right ) ^{{\frac{5}{2}}}\sqrt{{\frac{-a+x}{a}}}\sqrt{{\frac{a+x}{a}}}{\frac{1}{\sqrt{- \left ( -a+x \right ) \left ( a+x \right ) }}}} \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{\operatorname{arcosh}\left (\frac{x}{a}\right )^{\frac{3}{2}}}{\sqrt{a^{2} - x^{2}}}\,{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{\operatorname{acosh}^{\frac{3}{2}}{\left (\frac{x}{a} \right )}}{\sqrt{- \left (- a + x\right ) \left (a + x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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