Optimal. Leaf size=109 \[ \frac{i 2^{-n-3} \sin ^{-1}(a x)^n \left (-i \sin ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 i \sin ^{-1}(a x)\right )}{a^3}-\frac{i 2^{-n-3} \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,2 i \sin ^{-1}(a x)\right )}{a^3}+\frac{\sin ^{-1}(a x)^{n+1}}{2 a^3 (n+1)} \]
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Rubi [A] time = 0.20733, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4723, 3312, 3307, 2181} \[ \frac{i 2^{-n-3} \sin ^{-1}(a x)^n \left (-i \sin ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 i \sin ^{-1}(a x)\right )}{a^3}-\frac{i 2^{-n-3} \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,2 i \sin ^{-1}(a x)\right )}{a^3}+\frac{\sin ^{-1}(a x)^{n+1}}{2 a^3 (n+1)} \]
Antiderivative was successfully verified.
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Rule 4723
Rule 3312
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int \frac{x^2 \sin ^{-1}(a x)^n}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int x^n \sin ^2(x) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{x^n}{2}-\frac{1}{2} x^n \cos (2 x)\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac{\sin ^{-1}(a x)^{1+n}}{2 a^3 (1+n)}-\frac{\operatorname{Subst}\left (\int x^n \cos (2 x) \, dx,x,\sin ^{-1}(a x)\right )}{2 a^3}\\ &=\frac{\sin ^{-1}(a x)^{1+n}}{2 a^3 (1+n)}-\frac{\operatorname{Subst}\left (\int e^{-2 i x} x^n \, dx,x,\sin ^{-1}(a x)\right )}{4 a^3}-\frac{\operatorname{Subst}\left (\int e^{2 i x} x^n \, dx,x,\sin ^{-1}(a x)\right )}{4 a^3}\\ &=\frac{\sin ^{-1}(a x)^{1+n}}{2 a^3 (1+n)}+\frac{i 2^{-3-n} \left (-i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \Gamma \left (1+n,-2 i \sin ^{-1}(a x)\right )}{a^3}-\frac{i 2^{-3-n} \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \Gamma \left (1+n,2 i \sin ^{-1}(a x)\right )}{a^3}\\ \end{align*}
Mathematica [A] time = 0.253703, size = 109, normalized size = 1. \[ \frac{2^{-n-3} \sin ^{-1}(a x)^n \left (\sin ^{-1}(a x)^2\right )^{-n} \left (-i (n+1) \left (-i \sin ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,2 i \sin ^{-1}(a x)\right )+i (n+1) \left (i \sin ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,-2 i \sin ^{-1}(a x)\right )+2^{n+2} \sin ^{-1}(a x) \left (\sin ^{-1}(a x)^2\right )^n\right )}{a^3 (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.217, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \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^{2} \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^{2} \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^{2} \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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