Optimal. Leaf size=10 \[ \frac {e^{\sin ^{-1}(a x)}}{a} \]
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Rubi [A] time = 0.21, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4836, 6688, 6720, 2194} \[ \frac {e^{\sin ^{-1}(a x)}}{a} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 4836
Rule 6688
Rule 6720
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*}
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Mathematica [A] time = 0.02, size = 10, normalized size = 1.00 \[ \frac {e^{\sin ^{-1}(a x)}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 9, normalized size = 0.90 \[ \frac {e^{\left (\arcsin \left (a x\right )\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 9, normalized size = 0.90 \[ \frac {e^{\left (\arcsin \left (a x\right )\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 10, normalized size = 1.00 \[ \frac {{\mathrm e}^{\arcsin \left (a x \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 9, normalized size = 0.90 \[ \frac {e^{\left (\arcsin \left (a x\right )\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 9, normalized size = 0.90 \[ \frac {{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 8, normalized size = 0.80 \[ \begin {cases} \frac {e^{\operatorname {asin}{\left (a x \right )}}}{a} & \text {for}\: a \neq 0 \\x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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