Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{1}{\left (c-a^2 c x^2\right )^{3/2} \sqrt{\cosh ^{-1}(a x)}},x\right ) \]
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Rubi [A] time = 0.198037, 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 = {} \[ \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2} \sqrt{\cosh ^{-1}(a x)}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2} \sqrt{\cosh ^{-1}(a x)}} \, dx &=-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{1}{(-1+a x)^{3/2} (1+a x)^{3/2} \sqrt{\cosh ^{-1}(a x)}} \, dx}{c \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.7483, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2} \sqrt{\cosh ^{-1}(a x)}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.3, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{\rm arccosh} \left (ax\right )}}}}\, 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{1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \sqrt{\operatorname{arcosh}\left (a x\right )}}\,{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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}} \sqrt{\operatorname{acosh}{\left (a x \right )}}}\, dx \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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