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 = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 1.32, size = 31, normalized size = 0.74 \[ -\frac {1}{12} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (6 \, x + 4\right ) + \sqrt {2}}{4 \, \sin \left (6 \, x + 4\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 57, normalized size = 1.36 \[ \frac {1}{6} \, \sqrt {2} {\left (3 \, x + \arctan \left (-\frac {\sqrt {2} \sin \left (6 \, x + 4\right ) - \sin \left (6 \, x + 4\right )}{\sqrt {2} \cos \left (6 \, x + 4\right ) + \sqrt {2} - \cos \left (6 \, x + 4\right ) + 1}\right ) + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 18, normalized size = 0.43 \[ \frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \tan \left (2+3 x \right )}{2}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 27, normalized size = 0.64 \[ \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sin \left (6 \, x + 4\right )}{2 \, {\left (\cos \left (6 \, x + 4\right ) + 1\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.54, size = 36, normalized size = 0.86 \[ \frac {\sqrt {2}\,\left (3\,x-\mathrm {atan}\left (\mathrm {tan}\left (3\,x+2\right )\right )\right )}{6}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {tan}\left (3\,x+2\right )}{2}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 34, normalized size = 0.81 \[ \frac {\sqrt {2} \left (\operatorname {atan}{\left (\frac {\sqrt {2} \tan {\left (3 x + 2 \right )}}{2} \right )} + \pi \left \lfloor {\frac {3 x - \frac {\pi }{2} + 2}{\pi }}\right \rfloor \right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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