Optimal. Leaf size=163 \[ -\frac {3 (-i \text {ArcSin}(a x))^{-n} \text {ArcSin}(a x)^n \text {Gamma}(1+n,-i \text {ArcSin}(a x))}{8 a^4}-\frac {3 (i \text {ArcSin}(a x))^{-n} \text {ArcSin}(a x)^n \text {Gamma}(1+n,i \text {ArcSin}(a x))}{8 a^4}+\frac {3^{-1-n} (-i \text {ArcSin}(a x))^{-n} \text {ArcSin}(a x)^n \text {Gamma}(1+n,-3 i \text {ArcSin}(a x))}{8 a^4}+\frac {3^{-1-n} (i \text {ArcSin}(a x))^{-n} \text {ArcSin}(a x)^n \text {Gamma}(1+n,3 i \text {ArcSin}(a x))}{8 a^4} \]
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Rubi [A]
time = 0.17, antiderivative size = 163, normalized size of antiderivative = 1.00, 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 = {4809, 3393,
3389, 2212} \begin {gather*} -\frac {3 \text {ArcSin}(a x)^n (-i \text {ArcSin}(a x))^{-n} \text {Gamma}(n+1,-i \text {ArcSin}(a x))}{8 a^4}+\frac {3^{-n-1} \text {ArcSin}(a x)^n (-i \text {ArcSin}(a x))^{-n} \text {Gamma}(n+1,-3 i \text {ArcSin}(a x))}{8 a^4}-\frac {3 (i \text {ArcSin}(a x))^{-n} \text {ArcSin}(a x)^n \text {Gamma}(n+1,i \text {ArcSin}(a x))}{8 a^4}+\frac {3^{-n-1} (i \text {ArcSin}(a x))^{-n} \text {ArcSin}(a x)^n \text {Gamma}(n+1,3 i \text {ArcSin}(a x))}{8 a^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3389
Rule 3393
Rule 4809
Rubi steps
\begin {align*} \int \frac {x^3 \sin ^{-1}(a x)^n}{\sqrt {1-a^2 x^2}} \, dx &=\frac {\text {Subst}\left (\int x^n \sin ^3(x) \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=\frac {\text {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 {\text {Subst}\left (\int x^n \sin (3 x) \, dx,x,\sin ^{-1}(a x)\right )}{4 a^4}+\frac {3 \text {Subst}\left (\int x^n \sin (x) \, dx,x,\sin ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {i \text {Subst}\left (\int e^{-3 i x} x^n \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}+\frac {i \text {Subst}\left (\int e^{3 i x} x^n \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}+\frac {(3 i) \text {Subst}\left (\int e^{-i x} x^n \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}-\frac {(3 i) \text {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*}
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Mathematica [A]
time = 0.23, size = 153, normalized size = 0.94 \begin {gather*} -\frac {3^{-1-n} \text {ArcSin}(a x)^n \left (\text {ArcSin}(a x)^2\right )^{-2 n} \left (3^{2+n} (i \text {ArcSin}(a x))^n \left (\text {ArcSin}(a x)^2\right )^n \text {Gamma}(1+n,-i \text {ArcSin}(a x))+(-i \text {ArcSin}(a x))^n \left (3^{2+n} \left (\text {ArcSin}(a x)^2\right )^n \text {Gamma}(1+n,i \text {ArcSin}(a x))-(i \text {ArcSin}(a x))^{2 n} \text {Gamma}(1+n,-3 i \text {ArcSin}(a x))-\left (\text {ArcSin}(a x)^2\right )^n \text {Gamma}(1+n,3 i \text {ArcSin}(a x))\right )\right )}{8 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.39, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \arcsin \left (a x \right )^{n}}{\sqrt {-a^{2} x^{2}+1}}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^{3} \operatorname {asin}^{n}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {x^3\,{\mathrm {asin}\left (a\,x\right )}^n}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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