Optimal. Leaf size=25 \[ -\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{\sqrt{3} e} \]
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Rubi [A] time = 0.0153917, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {627, 63, 206} \[ -\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{\sqrt{3} e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2+e x} \sqrt{12-3 e^2 x^2}} \, dx &=\int \frac{1}{\sqrt{6-3 e x} (2+e x)} \, dx\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{4-\frac{x^2}{3}} \, dx,x,\sqrt{6-3 e x}\right )}{3 e}\\ &=-\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{\sqrt{3} e}\\ \end{align*}
Mathematica [A] time = 0.042496, size = 50, normalized size = 2. \[ \frac{\sqrt{e x-2} \sqrt{e x+2} \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )}{e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.125, size = 50, normalized size = 2. \begin{align*} -{\frac{\sqrt{3}}{3\,e}\sqrt{-{e}^{2}{x}^{2}+4}{\it Artanh} \left ({\frac{\sqrt{3}}{6}\sqrt{-3\,ex+6}} \right ){\frac{1}{\sqrt{ex+2}}}{\frac{1}{\sqrt{-ex+2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.89868, size = 158, normalized size = 6.32 \begin{align*} \frac{\sqrt{3} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right )}{6 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{3} \int \frac{1}{\sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4}}\, dx}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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