Optimal. Leaf size=42 \[ \frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sin (6 x+4)}{\cos (6 x+4)+2 \sqrt{2}+3}\right )}{3 \sqrt{2}} \]
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Rubi [A] time = 0.036651, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {12, 2657} \[ \frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sin (6 x+4)}{\cos (6 x+4)+2 \sqrt{2}+3}\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2657
Rubi steps
\begin{align*} \int \frac{2}{3+\cos (4+6 x)} \, dx &=2 \int \frac{1}{3+\cos (4+6 x)} \, dx\\ &=\frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sin (4+6 x)}{3+2 \sqrt{2}+\cos (4+6 x)}\right )}{3 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0294782, size = 22, normalized size = 0.52 \[ \frac{\tan ^{-1}\left (\frac{\tan (3 x+2)}{\sqrt{2}}\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 18, normalized size = 0.4 \begin{align*}{\frac{\sqrt{2}}{6}\arctan \left ({\frac{\tan \left ( 2+3\,x \right ) \sqrt{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68609, size = 36, normalized size = 0.86 \begin{align*} \frac{1}{6} \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sin \left (6 \, x + 4\right )}{2 \,{\left (\cos \left (6 \, x + 4\right ) + 1\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22075, size = 101, normalized size = 2.4 \begin{align*} -\frac{1}{12} \, \sqrt{2} \arctan \left (\frac{3 \, \sqrt{2} \cos \left (6 \, x + 4\right ) + \sqrt{2}}{4 \, \sin \left (6 \, x + 4\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.306932, size = 34, normalized size = 0.81 \begin{align*} \frac{\sqrt{2} \left (\operatorname{atan}{\left (\frac{\sqrt{2} \tan{\left (3 x + 2 \right )}}{2} \right )} + \pi \left \lfloor{\frac{3 x - \frac{\pi }{2} + 2}{\pi }}\right \rfloor \right )}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1251, size = 77, normalized size = 1.83 \begin{align*} \frac{1}{6} \, \sqrt{2}{\left (3 \, x + \arctan \left (-\frac{\sqrt{2} \sin \left (6 \, x + 4\right ) - \sin \left (6 \, x + 4\right )}{\sqrt{2} \cos \left (6 \, x + 4\right ) + \sqrt{2} - \cos \left (6 \, x + 4\right ) + 1}\right ) + 2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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