Optimal. Leaf size=101 \[ \frac{e \sqrt{\pi } \text{Erf}\left (1-i \sin ^{-1}(a x)\right )}{16 a^4}-\frac{e^4 \sqrt{\pi } \text{Erf}\left (2-i \sin ^{-1}(a x)\right )}{32 a^4}+\frac{e \sqrt{\pi } \text{Erf}\left (1+i \sin ^{-1}(a x)\right )}{16 a^4}-\frac{e^4 \sqrt{\pi } \text{Erf}\left (2+i \sin ^{-1}(a x)\right )}{32 a^4} \]
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Rubi [A] time = 0.123416, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4836, 12, 4474, 2234, 2204} \[ \frac{e \sqrt{\pi } \text{Erf}\left (1-i \sin ^{-1}(a x)\right )}{16 a^4}-\frac{e^4 \sqrt{\pi } \text{Erf}\left (2-i \sin ^{-1}(a x)\right )}{32 a^4}+\frac{e \sqrt{\pi } \text{Erf}\left (1+i \sin ^{-1}(a x)\right )}{16 a^4}-\frac{e^4 \sqrt{\pi } \text{Erf}\left (2+i \sin ^{-1}(a x)\right )}{32 a^4} \]
Antiderivative was successfully verified.
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Rule 4836
Rule 12
Rule 4474
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int e^{\sin ^{-1}(a x)^2} x^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^{x^2} \cos (x) \sin ^3(x)}{a^3} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cos (x) \sin ^3(x) \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{8} i e^{-2 i x+x^2}-\frac{1}{8} i e^{2 i x+x^2}-\frac{1}{16} i e^{-4 i x+x^2}+\frac{1}{16} i e^{4 i x+x^2}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=-\frac{i \operatorname{Subst}\left (\int e^{-4 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^4}+\frac{i \operatorname{Subst}\left (\int e^{4 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^4}+\frac{i \operatorname{Subst}\left (\int e^{-2 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}-\frac{i \operatorname{Subst}\left (\int e^{2 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}\\ &=\frac{(i e) \operatorname{Subst}\left (\int e^{\frac{1}{4} (-2 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}-\frac{(i e) \operatorname{Subst}\left (\int e^{\frac{1}{4} (2 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}-\frac{\left (i e^4\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (-4 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^4}+\frac{\left (i e^4\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (4 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^4}\\ &=\frac{e \sqrt{\pi } \text{erf}\left (1-i \sin ^{-1}(a x)\right )}{16 a^4}-\frac{e^4 \sqrt{\pi } \text{erf}\left (2-i \sin ^{-1}(a x)\right )}{32 a^4}+\frac{e \sqrt{\pi } \text{erf}\left (1+i \sin ^{-1}(a x)\right )}{16 a^4}-\frac{e^4 \sqrt{\pi } \text{erf}\left (2+i \sin ^{-1}(a x)\right )}{32 a^4}\\ \end{align*}
Mathematica [A] time = 0.0779962, size = 67, normalized size = 0.66 \[ \frac{e \sqrt{\pi } \left (2 \left (\text{Erf}\left (1-i \sin ^{-1}(a x)\right )+\text{Erf}\left (1+i \sin ^{-1}(a x)\right )\right )-e^3 \left (\text{Erf}\left (2-i \sin ^{-1}(a x)\right )+\text{Erf}\left (2+i \sin ^{-1}(a x)\right )\right )\right )}{32 a^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.01, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{ \left ( \arcsin \left ( ax \right ) \right ) ^{2}}}{x}^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} e^{\left (\arcsin \left (a x\right )^{2}\right )}\,{d x} \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 (x^{3} e^{\left (\arcsin \left (a x\right )^{2}\right )}, 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 x^{3} e^{\operatorname{asin}^{2}{\left (a x \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 x^{3} e^{\left (\arcsin \left (a x\right )^{2}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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