Optimal. Leaf size=49 \[ \frac {\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-\sinh ^{-1}(a x)\right )}{2 a^2}+\frac {\Gamma \left (n+1,\sinh ^{-1}(a x)\right )}{2 a^2} \]
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Rubi [A] time = 0.12, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5779, 3308, 2181} \[ \frac {\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{2 a^2}+\frac {\text {Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3308
Rule 5779
Rubi steps
\begin {align*} \int \frac {x \sinh ^{-1}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int x^n \sinh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=-\frac {\operatorname {Subst}\left (\int e^{-x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^2}+\frac {\operatorname {Subst}\left (\int e^x x^n \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^2}\\ &=\frac {\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )}{2 a^2}+\frac {\Gamma \left (1+n,\sinh ^{-1}(a x)\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 43, normalized size = 0.88 \[ \frac {\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-\sinh ^{-1}(a x)\right )+\Gamma \left (n+1,\sinh ^{-1}(a x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}}, x\right ) \]
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 \[ \int \frac {x \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {x \arcsinh \left (a x \right )^{n}}{\sqrt {a^{2} x^{2}+1}}\, dx \]
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 \[ \int \frac {x \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x\,{\mathrm {asinh}\left (a\,x\right )}^n}{\sqrt {a^2\,x^2+1}} \,d x \]
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 \[ \int \frac {x \operatorname {asinh}^{n}{\left (a x \right )}}{\sqrt {a^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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