Optimal. Leaf size=59 \[ \frac {2^{-n-3} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-2 \sinh ^{-1}(a x)\right )}{a^2}+\frac {2^{-n-3} \Gamma \left (n+1,2 \sinh ^{-1}(a x)\right )}{a^2} \]
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Rubi [A] time = 0.08, 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 = {5669, 5448, 12, 3308, 2181} \[ \frac {2^{-n-3} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-2 \sinh ^{-1}(a x)\right )}{a^2}+\frac {2^{-n-3} \text {Gamma}\left (n+1,2 \sinh ^{-1}(a x)\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2181
Rule 3308
Rule 5448
Rule 5669
Rubi steps
\begin {align*} \int x \sinh ^{-1}(a x)^n \, dx &=\frac {\operatorname {Subst}\left (\int x^n \cosh (x) \sinh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{2} x^n \sinh (2 x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int x^n \sinh (2 x) \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^2}\\ &=-\frac {\operatorname {Subst}\left (\int e^{-2 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^2}+\frac {\operatorname {Subst}\left (\int e^{2 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^2}\\ &=\frac {2^{-3-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-2 \sinh ^{-1}(a x)\right )}{a^2}+\frac {2^{-3-n} \Gamma \left (1+n,2 \sinh ^{-1}(a x)\right )}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 59, normalized size = 1.00 \[ \frac {2^{-n-3} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-2 \sinh ^{-1}(a x)\right )}{a^2}+\frac {2^{-n-3} \Gamma \left (n+1,2 \sinh ^{-1}(a x)\right )}{a^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {arsinh}\left (a x\right )^{n}, x\right ) \]
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 \[ \int x \operatorname {arsinh}\left (a x\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 38, normalized size = 0.64 \[ \frac {\arcsinh \left (a x \right )^{n +2} \hypergeom \left (\left [\frac {n}{2}+1\right ], \left [\frac {3}{2}, 2+\frac {n}{2}\right ], \arcsinh \left (a x \right )^{2}\right )}{a^{2} \left (n +2\right )} \]
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 \[ \int x \operatorname {arsinh}\left (a x\right )^{n}\,{d x} \]
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 \[ \int x\,{\mathrm {asinh}\left (a\,x\right )}^n \,d x \]
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 \[ \int x \operatorname {asinh}^{n}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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