Optimal. Leaf size=94 \[ \frac{5 a \sqrt{a x-1} \sqrt{a x+1} \text{Unintegrable}\left (\frac{x \cosh ^{-1}(a x)^{3/2}}{1-a^2 x^2},x\right )}{2 c \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^{5/2}}{c \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.227844, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\cosh ^{-1}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^{5/2}}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \cosh ^{-1}(a x)^{5/2}}{c \sqrt{c-a^2 c x^2}}+\frac{\left (5 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)^{3/2}}{1-a^2 x^2} \, dx}{2 c \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.52244, size = 0, normalized size = 0. \[ \int \frac{\cosh ^{-1}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.299, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{{\frac{5}{2}}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (a x\right )^{\frac{5}{2}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{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(-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 = 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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