Optimal. Leaf size=65 \[ \frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (-i+2 \text {ArcSin}(a x))\right )}{4 a}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (i+2 \text {ArcSin}(a x))\right )}{4 a} \]
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Rubi [A]
time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4920, 4561,
2266, 2235} \begin {gather*} \frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (2 \text {ArcSin}(a x)-i)\right )}{4 a}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (2 \text {ArcSin}(a x)+i)\right )}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2266
Rule 4561
Rule 4920
Rubi steps
\begin {align*} \int e^{\sin ^{-1}(a x)^2} \, dx &=\frac {\text {Subst}\left (\int e^{x^2} \cos (x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {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 {\text {Subst}\left (\int e^{-i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}+\frac {\text {Subst}\left (\int e^{i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}\\ &=\frac {\sqrt [4]{e} \text {Subst}\left (\int e^{\frac {1}{4} (-i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}+\frac {\sqrt [4]{e} \text {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*}
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Mathematica [A]
time = 0.03, size = 48, normalized size = 0.74 \begin {gather*} \frac {\sqrt [4]{e} \sqrt {\pi } \left (\text {Erfi}\left (\frac {1}{2} (-i+2 \text {ArcSin}(a x))\right )+\text {Erfi}\left (\frac {1}{2} (i+2 \text {ArcSin}(a x))\right )\right )}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\arcsin \left (a x \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int e^{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\mathrm {e}}^{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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