Optimal. Leaf size=4 \[ \sin ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2249, 216} \[ \sin ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 2249
Rubi steps
\begin {align*} \int \frac {e^x}{\sqrt {1-e^{2 x}}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,e^x\right )\\ &=\sin ^{-1}\left (e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 4, normalized size = 1.00 \[ \sin ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 20, normalized size = 5.00 \[ -2 \, \arctan \left ({\left (\sqrt {-e^{\left (2 \, x\right )} + 1} - 1\right )} e^{\left (-x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 3, normalized size = 0.75 \[ \arcsin \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 4, normalized size = 1.00 \[ \arcsin \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.07, size = 3, normalized size = 0.75 \[ \arcsin \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 3, normalized size = 0.75 \[ \mathrm {asin}\left ({\mathrm {e}}^x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 3, normalized size = 0.75 \[ \operatorname {asin}{\left (e^{x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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