Optimal. Leaf size=39 \[ \frac {1}{2} e^{\text {ArcSin}(a x)} x+\frac {e^{\text {ArcSin}(a x)} \sqrt {1-a^2 x^2}}{2 a} \]
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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4920, 4518}
\begin {gather*} \frac {\sqrt {1-a^2 x^2} e^{\text {ArcSin}(a x)}}{2 a}+\frac {1}{2} x e^{\text {ArcSin}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 4518
Rule 4920
Rubi steps
\begin {align*} \int e^{\sin ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int e^x \cos (x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {1}{2} e^{\sin ^{-1}(a x)} x+\frac {e^{\sin ^{-1}(a x)} \sqrt {1-a^2 x^2}}{2 a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 31, normalized size = 0.79 \begin {gather*} \frac {e^{\text {ArcSin}(a x)} \left (a x+\sqrt {1-a^2 x^2}\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\arcsin \left (a x \right )}\, dx\]
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 [A]
time = 1.72, size = 26, normalized size = 0.67 \begin {gather*} \frac {{\left (a x + \sqrt {-a^{2} x^{2} + 1}\right )} e^{\left (\arcsin \left (a x\right )\right )}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 32, normalized size = 0.82 \begin {gather*} \begin {cases} \frac {x e^{\operatorname {asin}{\left (a x \right )}}}{2} + \frac {\sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{2 a} & \text {for}\: a \neq 0 \\x & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 31, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, x e^{\left (\arcsin \left (a x\right )\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} e^{\left (\arcsin \left (a x\right )\right )}}{2 \, a} \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 = -0.03 \begin {gather*} \int {\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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