Optimal. Leaf size=129 \[ \frac{\sqrt [4]{e} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)-i\right )\right )}{16 a^3}+\frac{\sqrt [4]{e} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)+i\right )\right )}{16 a^3}-\frac{e^{9/4} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)-3 i\right )\right )}{16 a^3}-\frac{e^{9/4} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)+3 i\right )\right )}{16 a^3} \]
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Rubi [A] time = 0.129241, antiderivative size = 129, 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} \[ \frac{\sqrt [4]{e} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)-i\right )\right )}{16 a^3}+\frac{\sqrt [4]{e} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)+i\right )\right )}{16 a^3}-\frac{e^{9/4} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)-3 i\right )\right )}{16 a^3}-\frac{e^{9/4} \sqrt{\pi } \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)+3 i\right )\right )}{16 a^3} \]
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^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^{x^2} \cos (x) \sin ^2(x)}{a^2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cos (x) \sin ^2(x) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{8} e^{-i x+x^2}+\frac{1}{8} e^{i x+x^2}-\frac{1}{8} e^{-3 i x+x^2}-\frac{1}{8} e^{3 i x+x^2}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac{\operatorname{Subst}\left (\int e^{-i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}+\frac{\operatorname{Subst}\left (\int e^{i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}-\frac{\operatorname{Subst}\left (\int e^{-3 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}-\frac{\operatorname{Subst}\left (\int e^{3 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}\\ &=\frac{\sqrt [4]{e} \operatorname{Subst}\left (\int e^{\frac{1}{4} (-i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}+\frac{\sqrt [4]{e} \operatorname{Subst}\left (\int e^{\frac{1}{4} (i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}-\frac{e^{9/4} \operatorname{Subst}\left (\int e^{\frac{1}{4} (-3 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}-\frac{e^{9/4} \operatorname{Subst}\left (\int e^{\frac{1}{4} (3 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}\\ &=\frac{\sqrt [4]{e} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (-i+2 \sin ^{-1}(a x)\right )\right )}{16 a^3}+\frac{\sqrt [4]{e} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (i+2 \sin ^{-1}(a x)\right )\right )}{16 a^3}-\frac{e^{9/4} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (-3 i+2 \sin ^{-1}(a x)\right )\right )}{16 a^3}-\frac{e^{9/4} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (3 i+2 \sin ^{-1}(a x)\right )\right )}{16 a^3}\\ \end{align*}
Mathematica [A] time = 0.085105, size = 84, normalized size = 0.65 \[ \frac{\sqrt [4]{e} \sqrt{\pi } \left (\text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)-i\right )\right )+\text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)+i\right )\right )-e^2 \left (\text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)-3 i\right )\right )+\text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a x)+3 i\right )\right )\right )\right )}{16 a^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.008, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{ \left ( \arcsin \left ( ax \right ) \right ) ^{2}}}{x}^{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 x^{2} 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^{2} 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^{2} 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^{2} 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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