Optimal. Leaf size=163 \[ -\frac{3 \sin ^{-1}(a x)^n \left (-i \sin ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-i \sin ^{-1}(a x)\right )}{8 a^4}+\frac{3^{-n-1} \sin ^{-1}(a x)^n \left (-i \sin ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 i \sin ^{-1}(a x)\right )}{8 a^4}-\frac{3 \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,i \sin ^{-1}(a x)\right )}{8 a^4}+\frac{3^{-n-1} \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,3 i \sin ^{-1}(a x)\right )}{8 a^4} \]
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Rubi [A] time = 0.246396, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4723, 3312, 3308, 2181} \[ -\frac{3 \sin ^{-1}(a x)^n \left (-i \sin ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-i \sin ^{-1}(a x)\right )}{8 a^4}+\frac{3^{-n-1} \sin ^{-1}(a x)^n \left (-i \sin ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 i \sin ^{-1}(a x)\right )}{8 a^4}-\frac{3 \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,i \sin ^{-1}(a x)\right )}{8 a^4}+\frac{3^{-n-1} \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,3 i \sin ^{-1}(a x)\right )}{8 a^4} \]
Antiderivative was successfully verified.
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Rule 4723
Rule 3312
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int \frac{x^3 \sin ^{-1}(a x)^n}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int x^n \sin ^3(x) \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{3}{4} x^n \sin (x)-\frac{1}{4} x^n \sin (3 x)\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int x^n \sin (3 x) \, dx,x,\sin ^{-1}(a x)\right )}{4 a^4}+\frac{3 \operatorname{Subst}\left (\int x^n \sin (x) \, dx,x,\sin ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac{i \operatorname{Subst}\left (\int e^{-3 i x} x^n \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}+\frac{i \operatorname{Subst}\left (\int e^{3 i x} x^n \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}+\frac{(3 i) \operatorname{Subst}\left (\int e^{-i x} x^n \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}-\frac{(3 i) \operatorname{Subst}\left (\int e^{i x} x^n \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}\\ &=-\frac{3 \left (-i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \Gamma \left (1+n,-i \sin ^{-1}(a x)\right )}{8 a^4}-\frac{3 \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \Gamma \left (1+n,i \sin ^{-1}(a x)\right )}{8 a^4}+\frac{3^{-1-n} \left (-i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \Gamma \left (1+n,-3 i \sin ^{-1}(a x)\right )}{8 a^4}+\frac{3^{-1-n} \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \Gamma \left (1+n,3 i \sin ^{-1}(a x)\right )}{8 a^4}\\ \end{align*}
Mathematica [A] time = 0.320295, size = 153, normalized size = 0.94 \[ -\frac{3^{-n-1} \sin ^{-1}(a x)^n \left (\sin ^{-1}(a x)^2\right )^{-2 n} \left (\left (-i \sin ^{-1}(a x)\right )^n \left (3^{n+2} \left (\sin ^{-1}(a x)^2\right )^n \text{Gamma}\left (n+1,i \sin ^{-1}(a x)\right )-\left (\sin ^{-1}(a x)^2\right )^n \text{Gamma}\left (n+1,3 i \sin ^{-1}(a x)\right )-\left (i \sin ^{-1}(a x)\right )^{2 n} \text{Gamma}\left (n+1,-3 i \sin ^{-1}(a x)\right )\right )+3^{n+2} \left (i \sin ^{-1}(a x)\right )^n \left (\sin ^{-1}(a x)^2\right )^n \text{Gamma}\left (n+1,-i \sin ^{-1}(a x)\right )\right )}{8 a^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.248, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \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^{3} \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^{3} \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^{3} \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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