Optimal. Leaf size=33 \[ \frac{1}{2} e^x \sqrt{9-e^{2 x}}+\frac{9}{2} \sin ^{-1}\left (\frac{e^x}{3}\right ) \]
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Rubi [A] time = 0.0255834, antiderivative size = 33, normalized size of antiderivative = 1., 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} \[ \frac{1}{2} e^x \sqrt{9-e^{2 x}}+\frac{9}{2} \sin ^{-1}\left (\frac{e^x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 2249
Rule 195
Rule 216
Rubi steps
\begin{align*} \int e^x \sqrt{9-e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \sqrt{9-x^2} \, dx,x,e^x\right )\\ &=\frac{1}{2} e^x \sqrt{9-e^{2 x}}+\frac{9}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{9-x^2}} \, dx,x,e^x\right )\\ &=\frac{1}{2} e^x \sqrt{9-e^{2 x}}+\frac{9}{2} \sin ^{-1}\left (\frac{e^x}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.011567, size = 32, normalized size = 0.97 \[ \frac{1}{2} \left (e^x \sqrt{9-e^{2 x}}+9 \sin ^{-1}\left (\frac{e^x}{3}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 23, normalized size = 0.7 \begin{align*}{\frac{{{\rm e}^{x}}}{2}\sqrt{9- \left ({{\rm e}^{x}} \right ) ^{2}}}+{\frac{9}{2}\arcsin \left ({\frac{{{\rm e}^{x}}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4395, size = 30, normalized size = 0.91 \begin{align*} \frac{1}{2} \, \sqrt{-e^{\left (2 \, x\right )} + 9} e^{x} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.713558, size = 97, normalized size = 2.94 \begin{align*} \frac{1}{2} \, \sqrt{-e^{\left (2 \, x\right )} + 9} e^{x} - 9 \, \arctan \left ({\left (\sqrt{-e^{\left (2 \, x\right )} + 9} - 3\right )} e^{\left (-x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.35304, size = 29, normalized size = 0.88 \begin{align*} \begin{cases} \frac{\sqrt{9 - e^{2 x}} e^{x}}{2} + \frac{9 \operatorname{asin}{\left (\frac{e^{x}}{3} \right )}}{2} & \text{for}\: e^{x} < \log{\left (3 \right )} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28686, size = 30, normalized size = 0.91 \begin{align*} \frac{1}{2} \, \sqrt{-e^{\left (2 \, x\right )} + 9} e^{x} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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