Optimal. Leaf size=59 \[ \frac{2^{-n-3} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 \cosh ^{-1}(a x)\right )}{a^2}+\frac{2^{-n-3} \text{Gamma}\left (n+1,2 \cosh ^{-1}(a x)\right )}{a^2} \]
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Rubi [A] time = 0.087337, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5670, 5448, 12, 3308, 2181} \[ \frac{2^{-n-3} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 \cosh ^{-1}(a x)\right )}{a^2}+\frac{2^{-n-3} \text{Gamma}\left (n+1,2 \cosh ^{-1}(a x)\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 5670
Rule 5448
Rule 12
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x \cosh ^{-1}(a x)^n \, dx &=\frac{\operatorname{Subst}\left (\int x^n \cosh (x) \sinh (x) \, dx,x,\cosh ^{-1}(a x)\right )}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{2} x^n \sinh (2 x) \, dx,x,\cosh ^{-1}(a x)\right )}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int x^n \sinh (2 x) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^2}\\ &=-\frac{\operatorname{Subst}\left (\int e^{-2 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^2}+\frac{\operatorname{Subst}\left (\int e^{2 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^2}\\ &=\frac{2^{-3-n} \left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-2 \cosh ^{-1}(a x)\right )}{a^2}+\frac{2^{-3-n} \Gamma \left (1+n,2 \cosh ^{-1}(a x)\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0447461, size = 58, normalized size = 0.98 \[ \frac{2^{-n-3} \left (-\cosh ^{-1}(a x)\right )^{-n} \left (\left (-\cosh ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,2 \cosh ^{-1}(a x)\right )+\cosh ^{-1}(a x)^n \text{Gamma}\left (n+1,-2 \cosh ^{-1}(a x)\right )\right )}{a^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.031, size = 38, normalized size = 0.6 \begin{align*}{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2+n}}{{a}^{2} \left ( 2+n \right ) }{\mbox{$_1$F$_2$}(1+{\frac{n}{2}};\,{\frac{3}{2}},2+{\frac{n}{2}};\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{arcosh}\left (a x\right )^{n}\,{d x} \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 (x \operatorname{arcosh}\left (a x\right )^{n}, 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 x \operatorname{acosh}^{n}{\left (a x \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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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