Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\sqrt{1-c^2 x^2} x^m}{a+b \cosh ^{-1}(c x)},x\right ) \]
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Rubi [A] time = 0.450933, 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 = {} \[ \int \frac{x^m \sqrt{1-c^2 x^2}}{a+b \cosh ^{-1}(c x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{x^m \sqrt{1-c^2 x^2}}{a+b \cosh ^{-1}(c x)} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{x^m \sqrt{-1+c x} \sqrt{1+c x}}{a+b \cosh ^{-1}(c x)} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.171441, size = 0, normalized size = 0. \[ \int \frac{x^m \sqrt{1-c^2 x^2}}{a+b \cosh ^{-1}(c x)} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.764, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{a+b{\rm arccosh} \left (cx\right )}\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{\sqrt{-c^{2} x^{2} + 1} x^{m}}{b \operatorname{arcosh}\left (c x\right ) + a}\,{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} x^{m}}{b \operatorname{arcosh}\left (c x\right ) + a}, 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{x^{m} \sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}{a + b \operatorname{acosh}{\left (c 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*} \int \frac{\sqrt{-c^{2} x^{2} + 1} x^{m}}{b \operatorname{arcosh}\left (c x\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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