Optimal. Leaf size=46 \[ \frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^4}{4 a \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.156155, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {5713, 5676} \[ \frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^4}{4 a \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5676
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)^3}{\sqrt{c-a^2 c x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^4}{4 a \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0314924, size = 46, normalized size = 1. \[ \frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^4}{4 a \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 55, normalized size = 1.2 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}}{4\,ca \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}\sqrt{ax-1}\sqrt{ax+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} c x^{2} + c} \operatorname{arcosh}\left (a x\right )^{3}}{a^{2} c x^{2} - c}, x\right ) \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}^{3}{\left (a x \right )}}{\sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \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*} \int \frac{\operatorname{arcosh}\left (a x\right )^{3}}{\sqrt{-a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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