Optimal. Leaf size=49 \[ \frac{e \sqrt{\pi } \text{Erf}\left (1-i \sin ^{-1}(a x)\right )}{8 a^2}+\frac{e \sqrt{\pi } \text{Erf}\left (1+i \sin ^{-1}(a x)\right )}{8 a^2} \]
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Rubi [A] time = 0.06189, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4836, 12, 4474, 2234, 2204} \[ \frac{e \sqrt{\pi } \text{Erf}\left (1-i \sin ^{-1}(a x)\right )}{8 a^2}+\frac{e \sqrt{\pi } \text{Erf}\left (1+i \sin ^{-1}(a x)\right )}{8 a^2} \]
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 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^{x^2} \cos (x) \sin (x)}{a} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cos (x) \sin (x) \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{4} i e^{-2 i x+x^2}-\frac{1}{4} i e^{2 i x+x^2}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=\frac{i \operatorname{Subst}\left (\int e^{-2 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^2}-\frac{i \operatorname{Subst}\left (\int e^{2 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^2}\\ &=\frac{(i e) \operatorname{Subst}\left (\int e^{\frac{1}{4} (-2 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^2}-\frac{(i e) \operatorname{Subst}\left (\int e^{\frac{1}{4} (2 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^2}\\ &=\frac{e \sqrt{\pi } \text{erf}\left (1-i \sin ^{-1}(a x)\right )}{8 a^2}+\frac{e \sqrt{\pi } \text{erf}\left (1+i \sin ^{-1}(a x)\right )}{8 a^2}\\ \end{align*}
Mathematica [A] time = 0.0264709, size = 36, normalized size = 0.73 \[ \frac{e \sqrt{\pi } \left (\text{Erf}\left (1-i \sin ^{-1}(a x)\right )+\text{Erf}\left (1+i \sin ^{-1}(a x)\right )\right )}{8 a^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.006, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{ \left ( \arcsin \left ( ax \right ) \right ) ^{2}}}x\, 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 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 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 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 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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