Optimal. Leaf size=75 \[ -\frac{\sin ^{-1}(a x)^n \left (-i \sin ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-i \sin ^{-1}(a x)\right )}{2 a^2}-\frac{\left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,i \sin ^{-1}(a x)\right )}{2 a^2} \]
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Rubi [A] time = 0.115914, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4723, 3308, 2181} \[ -\frac{\sin ^{-1}(a x)^n \left (-i \sin ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-i \sin ^{-1}(a x)\right )}{2 a^2}-\frac{\left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,i \sin ^{-1}(a x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 4723
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int \frac{x \sin ^{-1}(a x)^n}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int x^n \sin (x) \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=\frac{i \operatorname{Subst}\left (\int e^{-i x} x^n \, dx,x,\sin ^{-1}(a x)\right )}{2 a^2}-\frac{i \operatorname{Subst}\left (\int e^{i x} x^n \, dx,x,\sin ^{-1}(a x)\right )}{2 a^2}\\ &=-\frac{\left (-i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \Gamma \left (1+n,-i \sin ^{-1}(a x)\right )}{2 a^2}-\frac{\left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \Gamma \left (1+n,i \sin ^{-1}(a x)\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.074226, size = 70, normalized size = 0.93 \[ -\frac{\sin ^{-1}(a x)^n \left (\sin ^{-1}(a x)^2\right )^{-n} \left (\left (-i \sin ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,i \sin ^{-1}(a x)\right )+\left (i \sin ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,-i \sin ^{-1}(a x)\right )\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.097, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \arcsin \left ( ax \right ) \right ) ^{n}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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 (-\frac{\sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{n}}{a^{2} x^{2} - 1}, 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 \frac{x \operatorname{asin}^{n}{\left (a x \right )}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\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*} \int \frac{x \arcsin \left (a x\right )^{n}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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