Optimal. Leaf size=37 \[ 2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{3 \cos (x)+2}}{\sqrt{2}}\right )-2 \sqrt{3 \cos (x)+2} \]
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Rubi [A] time = 0.0405188, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2721, 50, 63, 207} \[ 2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{3 \cos (x)+2}}{\sqrt{2}}\right )-2 \sqrt{3 \cos (x)+2} \]
Antiderivative was successfully verified.
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Rule 2721
Rule 50
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \sqrt{2+3 \cos (x)} \tan (x) \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{2+x}}{x} \, dx,x,3 \cos (x)\right )\\ &=-2 \sqrt{2+3 \cos (x)}-2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{2+x}} \, dx,x,3 \cos (x)\right )\\ &=-2 \sqrt{2+3 \cos (x)}-4 \operatorname{Subst}\left (\int \frac{1}{-2+x^2} \, dx,x,\sqrt{2+3 \cos (x)}\right )\\ &=2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2+3 \cos (x)}}{\sqrt{2}}\right )-2 \sqrt{2+3 \cos (x)}\\ \end{align*}
Mathematica [A] time = 0.0191128, size = 33, normalized size = 0.89 \[ 2 \sqrt{2} \tanh ^{-1}\left (\sqrt{\frac{3 \cos (x)}{2}+1}\right )-2 \sqrt{3 \cos (x)+2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 31, normalized size = 0.8 \begin{align*} 2\,{\it Artanh} \left ( 1/2\,\sqrt{2+3\,\cos \left ( x \right ) }\sqrt{2} \right ) \sqrt{2}-2\,\sqrt{2+3\,\cos \left ( x \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40633, size = 63, normalized size = 1.7 \begin{align*} -\sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{3 \, \cos \left (x\right ) + 2}}{\sqrt{2} + \sqrt{3 \, \cos \left (x\right ) + 2}}\right ) - 2 \, \sqrt{3 \, \cos \left (x\right ) + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.00829, size = 182, normalized size = 4.92 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{9 \, \cos \left (x\right )^{2} + 4 \,{\left (3 \, \sqrt{2} \cos \left (x\right ) + 4 \, \sqrt{2}\right )} \sqrt{3 \, \cos \left (x\right ) + 2} + 48 \, \cos \left (x\right ) + 32}{\cos \left (x\right )^{2}}\right ) - 2 \, \sqrt{3 \, \cos \left (x\right ) + 2} \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*} \int \sqrt{3 \cos{\left (x \right )} + 2} \tan{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06913, size = 68, normalized size = 1.84 \begin{align*} -\sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \sqrt{3 \, \cos \left (x\right ) + 2} \right |}}{2 \,{\left (\sqrt{2} + \sqrt{3 \, \cos \left (x\right ) + 2}\right )}}\right ) - 2 \, \sqrt{3 \, \cos \left (x\right ) + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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