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 = 0.21, antiderivative size = 62, normalized size of antiderivative = 1.00, 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 2194
Rule 4435
Rule 4836
Rule 6688
Rule 6720
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*}
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Mathematica [A] time = 0.09, size = 31, normalized size = 0.50 \[ \frac {e^{\sin ^{-1}(a x)} \left (2 \sin \left (2 \sin ^{-1}(a x)\right )+\cos \left (2 \sin ^{-1}(a x)\right )+5\right )}{10 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 35, normalized size = 0.56 \[ -\frac {{\left (a^{2} x^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} a x - 3\right )} e^{\left (\arcsin \left (a x\right )\right )}}{5 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\arcsin \left (a x \right )} \sqrt {-a^{2} x^{2}+1}\, dx \]
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 \[ \int \sqrt {-a^{2} x^{2} + 1} e^{\left (\arcsin \left (a x\right )\right )}\,{d x} \]
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 \[ \int {\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}\,\sqrt {1-a^2\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 49, normalized size = 0.79 \[ \begin {cases} - \frac {a x^{2} e^{\operatorname {asin}{\left (a x \right )}}}{5} + \frac {2 x \sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{5} + \frac {3 e^{\operatorname {asin}{\left (a x \right )}}}{5 a} & \text {for}\: a \neq 0 \\x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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