Optimal. Leaf size=59 \[ \frac {2^{-n-3} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-2 \cosh ^{-1}(a x)\right )}{a^2}+\frac {2^{-n-3} \Gamma \left (n+1,2 \cosh ^{-1}(a x)\right )}{a^2} \]
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Rubi [A] time = 0.09, antiderivative size = 59, normalized size of antiderivative = 1.00, 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 12
Rule 2181
Rule 3308
Rule 5448
Rule 5670
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*}
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Mathematica [A] time = 0.05, size = 58, normalized size = 0.98 \[ \frac {2^{-n-3} \left (-\cosh ^{-1}(a x)\right )^{-n} \left (\left (-\cosh ^{-1}(a x)\right )^n \Gamma \left (n+1,2 \cosh ^{-1}(a x)\right )+\cosh ^{-1}(a x)^n \Gamma \left (n+1,-2 \cosh ^{-1}(a x)\right )\right )}{a^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {arcosh}\left (a x\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {arcosh}\left (a x\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 38, normalized size = 0.64 \[ \frac {\mathrm {arccosh}\left (a x \right )^{2+n} \hypergeom \left (\left [1+\frac {n}{2}\right ], \left [\frac {3}{2}, 2+\frac {n}{2}\right ], \mathrm {arccosh}\left (a x \right )^{2}\right )}{a^{2} \left (2+n \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {arcosh}\left (a x\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\mathrm {acosh}\left (a\,x\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {acosh}^{n}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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