Optimal. Leaf size=61 \[ \frac {\log \left (\sqrt {2} \cos (3 x+2)-\sin (3 x+2)\right )}{6 \sqrt {2}}-\frac {\log \left (\sin (3 x+2)+\sqrt {2} \cos (3 x+2)\right )}{6 \sqrt {2}} \]
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Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {12, 2659, 206} \[ \frac {\log \left (\sqrt {2} \cos (3 x+2)-\sin (3 x+2)\right )}{6 \sqrt {2}}-\frac {\log \left (\sin (3 x+2)+\sqrt {2} \cos (3 x+2)\right )}{6 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 2659
Rubi steps
\begin {align*} \int -\frac {2}{1+3 \cos (4+6 x)} \, dx &=-\left (2 \int \frac {1}{1+3 \cos (4+6 x)} \, dx\right )\\ &=-\left (\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{4-2 x^2} \, dx,x,\tan \left (\frac {1}{2} (4+6 x)\right )\right )\right )\\ &=\frac {\log \left (\sqrt {2} \cos (2+3 x)-\sin (2+3 x)\right )}{6 \sqrt {2}}-\frac {\log \left (\sqrt {2} \cos (2+3 x)+\sin (2+3 x)\right )}{6 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 22, normalized size = 0.36 \[ -\frac {\tanh ^{-1}\left (\frac {\tan (3 x+2)}{\sqrt {2}}\right )}{3 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 74, normalized size = 1.21 \[ \frac {1}{24} \, \sqrt {2} \log \left (-\frac {7 \, \cos \left (6 \, x + 4\right )^{2} + 4 \, {\left (\sqrt {2} \cos \left (6 \, x + 4\right ) + 3 \, \sqrt {2}\right )} \sin \left (6 \, x + 4\right ) - 6 \, \cos \left (6 \, x + 4\right ) - 17}{9 \, \cos \left (6 \, x + 4\right )^{2} + 6 \, \cos \left (6 \, x + 4\right ) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 39, normalized size = 0.64 \[ \frac {1}{12} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \tan \left (3 \, x + 2\right ) \right |}}{{\left | 2 \, \sqrt {2} + 2 \, \tan \left (3 \, x + 2\right ) \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 18, normalized size = 0.30 \[ -\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {2}\, \tan \left (2+3 x \right )}{2}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 53, normalized size = 0.87 \[ \frac {1}{12} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {\sin \left (6 \, x + 4\right )}{\cos \left (6 \, x + 4\right ) + 1}}{\sqrt {2} + \frac {\sin \left (6 \, x + 4\right )}{\cos \left (6 \, x + 4\right ) + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.50, size = 17, normalized size = 0.28 \[ -\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\mathrm {tan}\left (3\,x+2\right )}{2}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 39, normalized size = 0.64 \[ \frac {\sqrt {2} \log {\left (\tan {\left (3 x + 2 \right )} - \sqrt {2} \right )}}{12} - \frac {\sqrt {2} \log {\left (\tan {\left (3 x + 2 \right )} + \sqrt {2} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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