Optimal. Leaf size=75 \[ -\frac {\sin ^{-1}(a x)^n \left (-i \sin ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-i \sin ^{-1}(a x)\right )}{2 a^2}-\frac {\left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \Gamma \left (n+1,i \sin ^{-1}(a x)\right )}{2 a^2} \]
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Rubi [A] time = 0.12, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 2181
Rule 3308
Rule 4723
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*}
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Mathematica [A] time = 0.09, size = 70, normalized size = 0.93 \[ -\frac {\sin ^{-1}(a x)^n \left (\sin ^{-1}(a x)^2\right )^{-n} \left (\left (-i \sin ^{-1}(a x)\right )^n \Gamma \left (n+1,i \sin ^{-1}(a x)\right )+\left (i \sin ^{-1}(a x)\right )^n \Gamma \left (n+1,-i \sin ^{-1}(a x)\right )\right )}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{n}}{a^{2} x^{2} - 1}, x\right ) \]
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 \[ \int \frac {x \arcsin \left (a x\right )^{n}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {x \arcsin \left (a x \right )^{n}}{\sqrt {-a^{2} x^{2}+1}}\, dx \]
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 \[ \int \frac {x\,{\mathrm {asin}\left (a\,x\right )}^n}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x \operatorname {asin}^{n}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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