Optimal. Leaf size=10 \[ \frac{e^{\sin ^{-1}(a x)}}{a} \]
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Rubi [A] time = 0.214341, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4836, 6688, 6720, 2194} \[ \frac{e^{\sin ^{-1}(a x)}}{a} \]
Antiderivative was successfully verified.
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Rule 4836
Rule 6688
Rule 6720
Rule 2194
Rubi steps
\begin{align*} \int \frac{e^{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^x \cos (x)}{\sqrt{1-\sin ^2(x)}} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \sqrt{\cos ^2(x)} \sec (x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{e^{\sin ^{-1}(a x)}}{a}\\ \end{align*}
Mathematica [A] time = 0.0153145, size = 10, normalized size = 1. \[ \frac{e^{\sin ^{-1}(a x)}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 10, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{\arcsin \left ( ax \right ) }}}{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47392, size = 12, normalized size = 1.2 \begin{align*} \frac{e^{\left (\arcsin \left (a x\right )\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18016, size = 26, normalized size = 2.6 \begin{align*} \frac{e^{\left (\arcsin \left (a x\right )\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.461056, size = 8, normalized size = 0.8 \begin{align*} \begin{cases} \frac{e^{\operatorname{asin}{\left (a x \right )}}}{a} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16343, size = 12, normalized size = 1.2 \begin{align*} \frac{e^{\left (\arcsin \left (a x\right )\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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