Optimal. Leaf size=65 \[ \frac{\sqrt [4]{e} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)-i\right )\right )}{4 a}+\frac{\sqrt [4]{e} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)+i\right )\right )}{4 a} \]
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Rubi [A] time = 0.0502309, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4836, 4473, 2234, 2204} \[ \frac{\sqrt [4]{e} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)-i\right )\right )}{4 a}+\frac{\sqrt [4]{e} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)+i\right )\right )}{4 a} \]
Antiderivative was successfully verified.
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Rule 4836
Rule 4473
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int e^{\sin ^{-1}(a x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cos (x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2} e^{-i x+x^2}+\frac{1}{2} e^{i x+x^2}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^{-i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}+\frac{\operatorname{Subst}\left (\int e^{i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}\\ &=\frac{\sqrt [4]{e} \operatorname{Subst}\left (\int e^{\frac{1}{4} (-i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}+\frac{\sqrt [4]{e} \operatorname{Subst}\left (\int e^{\frac{1}{4} (i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}\\ &=\frac{\sqrt [4]{e} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (-i+2 \sin ^{-1}(a x)\right )\right )}{4 a}+\frac{\sqrt [4]{e} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (i+2 \sin ^{-1}(a x)\right )\right )}{4 a}\\ \end{align*}
Mathematica [A] time = 0.0315391, size = 48, normalized size = 0.74 \[ \frac{\sqrt [4]{e} \sqrt{\pi } \left (\text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)-i\right )\right )+\text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)+i\right )\right )\right )}{4 a} \]
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}}}\, 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 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 (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 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 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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