Optimal. Leaf size=36 \[ \frac {1}{2} e^x \sqrt {3-4 e^{2 x}}+\frac {3}{4} \sin ^{-1}\left (\frac {2 e^x}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 36, 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} \[ \frac {1}{2} e^x \sqrt {3-4 e^{2 x}}+\frac {3}{4} \sin ^{-1}\left (\frac {2 e^x}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 2249
Rubi steps
\begin {align*} \int e^x \sqrt {3-4 e^{2 x}} \, dx &=\operatorname {Subst}\left (\int \sqrt {3-4 x^2} \, dx,x,e^x\right )\\ &=\frac {1}{2} e^x \sqrt {3-4 e^{2 x}}+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-4 x^2}} \, dx,x,e^x\right )\\ &=\frac {1}{2} e^x \sqrt {3-4 e^{2 x}}+\frac {3}{4} \sin ^{-1}\left (\frac {2 e^x}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 1.00 \[ \frac {1}{4} \left (2 e^x \sqrt {3-4 e^{2 x}}+3 \sin ^{-1}\left (\frac {2 e^x}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 34, normalized size = 0.94 \[ \frac {1}{2} \, \sqrt {-4 \, e^{\left (2 \, x\right )} + 3} e^{x} - \frac {3}{4} \, \arctan \left (\frac {1}{2} \, \sqrt {-4 \, e^{\left (2 \, x\right )} + 3} e^{\left (-x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 25, normalized size = 0.69 \[ \frac {1}{2} \, \sqrt {-4 \, e^{\left (2 \, x\right )} + 3} e^{x} + \frac {3}{4} \, \arcsin \left (\frac {2}{3} \, \sqrt {3} e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 26, normalized size = 0.72 \[ \frac {3 \arcsin \left (\frac {2 \sqrt {3}\, {\mathrm e}^{x}}{3}\right )}{4}+\frac {\sqrt {-4 \,{\mathrm e}^{2 x}+3}\, {\mathrm e}^{x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.08, size = 25, normalized size = 0.69 \[ \frac {1}{2} \, \sqrt {-4 \, e^{\left (2 \, x\right )} + 3} e^{x} + \frac {3}{4} \, \arcsin \left (\frac {2}{3} \, \sqrt {3} e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 24, normalized size = 0.67 \[ \frac {3\,\mathrm {asin}\left (\frac {2\,\sqrt {3}\,{\mathrm {e}}^x}{3}\right )}{4}+{\mathrm {e}}^x\,\sqrt {\frac {3}{4}-{\mathrm {e}}^{2\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.56, size = 42, normalized size = 1.17 \[ \begin {cases} \frac {\sqrt {3 - 4 e^{2 x}} e^{x}}{2} + \frac {3 \operatorname {asin}{\left (\frac {2 \sqrt {3} e^{x}}{3} \right )}}{4} & \text {for}\: e^{x} < \log {\left (\frac {\sqrt {3}}{2} \right )} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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