Optimal. Leaf size=49 \[ \frac{\cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\cosh ^{-1}(a x)\right )}{2 a}+\frac{\text{Gamma}\left (n+1,\cosh ^{-1}(a x)\right )}{2 a} \]
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Rubi [A] time = 0.0492336, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5658, 3308, 2181} \[ \frac{\cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\cosh ^{-1}(a x)\right )}{2 a}+\frac{\text{Gamma}\left (n+1,\cosh ^{-1}(a x)\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 5658
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int \cosh ^{-1}(a x)^n \, dx &=\frac{\operatorname{Subst}\left (\int x^n \sinh (x) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int e^{-x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{2 a}+\frac{\operatorname{Subst}\left (\int e^x x^n \, dx,x,\cosh ^{-1}(a x)\right )}{2 a}\\ &=\frac{\left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-\cosh ^{-1}(a x)\right )}{2 a}+\frac{\Gamma \left (1+n,\cosh ^{-1}(a x)\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0241498, size = 43, normalized size = 0.88 \[ \frac{\cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\cosh ^{-1}(a x)\right )+\text{Gamma}\left (n+1,\cosh ^{-1}(a x)\right )}{2 a} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.041, size = 40, normalized size = 0.8 \begin{align*}{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2+n}}{a \left ( 2+n \right ) }{\mbox{$_1$F$_2$}(1+{\frac{n}{2}};\,{\frac{3}{2}},2+{\frac{n}{2}};\,{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{4}})}} \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 \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 (\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 \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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