Optimal. Leaf size=109 \[ \frac {\sin ^{-1}(a x)^{n+1}}{2 a^3 (n+1)}+\frac {i 2^{-n-3} \sin ^{-1}(a x)^n \left (-i \sin ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-2 i \sin ^{-1}(a x)\right )}{a^3}-\frac {i 2^{-n-3} \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \Gamma \left (n+1,2 i \sin ^{-1}(a x)\right )}{a^3} \]
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Rubi [A] time = 0.21, antiderivative size = 109, normalized size of antiderivative = 1.00, 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 2181
Rule 3307
Rule 3312
Rule 4723
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*}
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Mathematica [A] time = 0.27, size = 109, normalized size = 1.00 \[ \frac {2^{-n-3} \sin ^{-1}(a x)^n \left (\sin ^{-1}(a x)^2\right )^{-n} \left (2^{n+2} \sin ^{-1}(a x) \left (\sin ^{-1}(a x)^2\right )^n-i (n+1) \left (-i \sin ^{-1}(a x)\right )^n \Gamma \left (n+1,2 i \sin ^{-1}(a x)\right )+i (n+1) \left (i \sin ^{-1}(a x)\right )^n \Gamma \left (n+1,-2 i \sin ^{-1}(a x)\right )\right )}{a^3 (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} x^{2} \arcsin \left (a x\right )^{n}}{a^{2} x^{2} - 1}, x\right ) \]
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 \[ \int \frac {x^{2} \arcsin \left (a x\right )^{n}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \arcsin \left (a x \right )^{n}}{\sqrt {-a^{2} x^{2}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,{\mathrm {asin}\left (a\,x\right )}^n}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{2} \operatorname {asin}^{n}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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