Optimal. Leaf size=29 \[ \frac {1}{2} e^x \sqrt {1-e^{2 x}}+\frac {1}{2} \sin ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2249, 195, 216} \[ \frac {1}{2} e^x \sqrt {1-e^{2 x}}+\frac {1}{2} \sin ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 2249
Rubi steps
\begin {align*} \int e^x \sqrt {1-e^{2 x}} \, dx &=\operatorname {Subst}\left (\int \sqrt {1-x^2} \, dx,x,e^x\right )\\ &=\frac {1}{2} e^x \sqrt {1-e^{2 x}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,e^x\right )\\ &=\frac {1}{2} e^x \sqrt {1-e^{2 x}}+\frac {1}{2} \sin ^{-1}\left (e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.90 \[ \frac {1}{2} \left (e^x \sqrt {1-e^{2 x}}+\sin ^{-1}\left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 35, normalized size = 1.21 \[ \frac {1}{2} \, \sqrt {-e^{\left (2 \, x\right )} + 1} e^{x} - \arctan \left ({\left (\sqrt {-e^{\left (2 \, x\right )} + 1} - 1\right )} e^{\left (-x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 20, normalized size = 0.69 \[ \frac {1}{2} \, \sqrt {-e^{\left (2 \, x\right )} + 1} e^{x} + \frac {1}{2} \, \arcsin \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 21, normalized size = 0.72 \[ \frac {\arcsin \left ({\mathrm e}^{x}\right )}{2}+\frac {\sqrt {-{\mathrm e}^{2 x}+1}\, {\mathrm e}^{x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.77, size = 20, normalized size = 0.69 \[ \frac {1}{2} \, \sqrt {-e^{\left (2 \, x\right )} + 1} e^{x} + \frac {1}{2} \, \arcsin \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.35, size = 20, normalized size = 0.69 \[ \frac {\mathrm {asin}\left ({\mathrm {e}}^x\right )}{2}+\frac {{\mathrm {e}}^x\,\sqrt {1-{\mathrm {e}}^{2\,x}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.35, size = 24, normalized size = 0.83 \[ \begin {cases} \frac {\sqrt {1 - e^{2 x}} e^{x}}{2} + \frac {\operatorname {asin}{\left (e^{x} \right )}}{2} & \text {for}\: e^{x} < 0 \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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