Optimal. Leaf size=32 \[ \text{Unintegrable}\left (\frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^3}{\sqrt{1-c^2 x^2}},x\right ) \]
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Rubi [A] time = 0.458437, 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 = {} \[ \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^3}{\sqrt{1-c^2 x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^3}{\sqrt{1-c^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 3.49909, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^3}{\sqrt{1-c^2 x^2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.379, size = 0, normalized size = 0. \begin{align*} \int{ \left ( fx \right ) ^{m} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{3}{\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 )}^{3} \left (f x\right )^{m}}{\sqrt{-c^{2} x^{2} + 1}}\,{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{{\left (b^{3} \operatorname{arcosh}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname{arcosh}\left (c x\right ) + a^{3}\right )} \sqrt{-c^{2} x^{2} + 1} \left (f x\right )^{m}}{c^{2} x^{2} - 1}, x\right ) \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*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{3} \left (f x\right )^{m}}{\sqrt{-c^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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