Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt{1-c^2 x^2}},x\right ) \]
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Rubi [A] time = 0.445807, 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 = {} \[ \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt{1-c^2 x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt{1-c^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 1.29318, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt{1-c^2 x^2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.232, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{n}}{{x}^{2}}{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}\, 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{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt{-c^{2} x^{2} + 1} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} x^{2} + 1}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{n}}{c^{2} x^{4} - x^{2}}, x\right ) \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{\left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{n}}{x^{2} \sqrt{- \left (c x - 1\right ) \left (c x + 1\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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