Optimal. Leaf size=62 \[ \frac{2}{5} x \sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac{\left (1-a^2 x^2\right ) e^{\sin ^{-1}(a x)}}{5 a}+\frac{2 e^{\sin ^{-1}(a x)}}{5 a} \]
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Rubi [A] time = 0.20668, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4836, 6688, 6720, 4435, 2194} \[ \frac{2}{5} x \sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac{\left (1-a^2 x^2\right ) e^{\sin ^{-1}(a x)}}{5 a}+\frac{2 e^{\sin ^{-1}(a x)}}{5 a} \]
Antiderivative was successfully verified.
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Rule 4836
Rule 6688
Rule 6720
Rule 4435
Rule 2194
Rubi steps
\begin{align*} \int e^{\sin ^{-1}(a x)} \sqrt{1-a^2 x^2} \, dx &=\frac{\operatorname{Subst}\left (\int e^x \cos (x) \sqrt{1-\sin ^2(x)} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \cos (x) \sqrt{\cos ^2(x)} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \cos ^2(x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{2}{5} e^{\sin ^{-1}(a x)} x \sqrt{1-a^2 x^2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )}{5 a}+\frac{2 \operatorname{Subst}\left (\int e^x \, dx,x,\sin ^{-1}(a x)\right )}{5 a}\\ &=\frac{2 e^{\sin ^{-1}(a x)}}{5 a}+\frac{2}{5} e^{\sin ^{-1}(a x)} x \sqrt{1-a^2 x^2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )}{5 a}\\ \end{align*}
Mathematica [A] time = 0.0732716, size = 31, normalized size = 0.5 \[ \frac{e^{\sin ^{-1}(a x)} \left (2 \sin \left (2 \sin ^{-1}(a x)\right )+\cos \left (2 \sin ^{-1}(a x)\right )+5\right )}{10 a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.08, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{\arcsin \left ( ax \right ) }}\sqrt{-{a}^{2}{x}^{2}+1}\, 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 \sqrt{-a^{2} x^{2} + 1} e^{\left (\arcsin \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03805, size = 88, normalized size = 1.42 \begin{align*} -\frac{{\left (a^{2} x^{2} - 2 \, \sqrt{-a^{2} x^{2} + 1} a x - 3\right )} e^{\left (\arcsin \left (a x\right )\right )}}{5 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.542048, size = 49, normalized size = 0.79 \begin{align*} \begin{cases} - \frac{a x^{2} e^{\operatorname{asin}{\left (a x \right )}}}{5} + \frac{2 x \sqrt{- a^{2} x^{2} + 1} e^{\operatorname{asin}{\left (a x \right )}}}{5} + \frac{3 e^{\operatorname{asin}{\left (a x \right )}}}{5 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 = 1.22359, size = 68, normalized size = 1.1 \begin{align*} \frac{2}{5} \, \sqrt{-a^{2} x^{2} + 1} x e^{\left (\arcsin \left (a x\right )\right )} - \frac{{\left (a^{2} x^{2} - 1\right )} e^{\left (\arcsin \left (a x\right )\right )}}{5 \, a} + \frac{2 \, e^{\left (\arcsin \left (a x\right )\right )}}{5 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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