Optimal. Leaf size=48 \[ -\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.154337, antiderivative size = 48, 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} \[ -\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5676
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0328619, size = 48, normalized size = 1. \[ -\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 41, normalized size = 0.9 \begin{align*} -{\frac{2}{3\,a}\sqrt{ax-1}\sqrt{ax+1} \left ({\rm arccosh} \left (ax\right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}}}} \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{1}{\sqrt{-a^{2} c x^{2} + c} \operatorname{arcosh}\left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96189, size = 134, normalized size = 2.79 \begin{align*} \frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{a^{2} x^{2} - 1}}{3 \,{\left (a^{3} c x^{2} - a c\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{\frac{3}{2}}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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