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 = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 12
Rule 2204
Rule 2234
Rule 4474
Rule 4836
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*}
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Mathematica [A] time = 0.04, 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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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x e^{\left (\arcsin \left (a x\right )^{2}\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x e^{\left (\arcsin \left (a x\right )^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\arcsin \left (a x \right )^{2}} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x e^{\left (\arcsin \left (a x\right )^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\mathrm {e}}^{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]
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 \[ \int x e^{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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