Optimal. Leaf size=62 \[ \frac {2 e^{\text {ArcSin}(a x)}}{5 a}+\frac {2}{5} e^{\text {ArcSin}(a x)} x \sqrt {1-a^2 x^2}+\frac {e^{\text {ArcSin}(a x)} \left (1-a^2 x^2\right )}{5 a} \]
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Rubi [A]
time = 0.14, 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 = {4920, 6820,
6852, 4520, 2225} \begin {gather*} \frac {2}{5} x \sqrt {1-a^2 x^2} e^{\text {ArcSin}(a x)}+\frac {\left (1-a^2 x^2\right ) e^{\text {ArcSin}(a x)}}{5 a}+\frac {2 e^{\text {ArcSin}(a x)}}{5 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 4520
Rule 4920
Rule 6820
Rule 6852
Rubi steps
\begin {align*} \int e^{\sin ^{-1}(a x)} \sqrt {1-a^2 x^2} \, dx &=\frac {\text {Subst}\left (\int e^x \cos (x) \sqrt {1-\sin ^2(x)} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int e^x \cos (x) \sqrt {\cos ^2(x)} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {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 \text {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 = 0.06, size = 31, normalized size = 0.50 \begin {gather*} \frac {e^{\text {ArcSin}(a x)} (5+\cos (2 \text {ArcSin}(a x))+2 \sin (2 \text {ArcSin}(a x)))}{10 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, 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 \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 = 1.87, size = 35, normalized size = 0.56 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 49, normalized size = 0.79 \begin {gather*} \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 {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 )}\,\sqrt {1-a^2\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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