Integrand size = 21, antiderivative size = 48 \[ \int \frac {1}{2 \cos ^2(2+3 x)+\sin ^2(2+3 x)} \, dx=\frac {x}{\sqrt {2}}-\frac {\arctan \left (\frac {\cos (2+3 x) \sin (2+3 x)}{1+\sqrt {2}+\cos ^2(2+3 x)}\right )}{3 \sqrt {2}} \]
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Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {209} \[ \int \frac {1}{2 \cos ^2(2+3 x)+\sin ^2(2+3 x)} \, dx=\frac {x}{\sqrt {2}}-\frac {\arctan \left (\frac {\sin (3 x+2) \cos (3 x+2)}{\cos ^2(3 x+2)+\sqrt {2}+1}\right )}{3 \sqrt {2}} \]
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Rule 209
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\tan (2+3 x)\right ) \\ & = \frac {x}{\sqrt {2}}-\frac {\arctan \left (\frac {\cos (2+3 x) \sin (2+3 x)}{1+\sqrt {2}+\cos ^2(2+3 x)}\right )}{3 \sqrt {2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.46 \[ \int \frac {1}{2 \cos ^2(2+3 x)+\sin ^2(2+3 x)} \, dx=\frac {\arctan \left (\frac {\tan (2+3 x)}{\sqrt {2}}\right )}{3 \sqrt {2}} \]
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Time = 1.43 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.38
method | result | size |
derivativedivides | \(\frac {\sqrt {2}\, \arctan \left (\frac {\tan \left (2+3 x \right ) \sqrt {2}}{2}\right )}{6}\) | \(18\) |
default | \(\frac {\sqrt {2}\, \arctan \left (\frac {\tan \left (2+3 x \right ) \sqrt {2}}{2}\right )}{6}\) | \(18\) |
risch | \(\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i \left (2+3 x \right )}+3+2 \sqrt {2}\right )}{12}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i \left (2+3 x \right )}+3-2 \sqrt {2}\right )}{12}\) | \(48\) |
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Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.90 \[ \int \frac {1}{2 \cos ^2(2+3 x)+\sin ^2(2+3 x)} \, dx=-\frac {1}{12} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (3 \, x + 2\right )^{2} - \sqrt {2}}{4 \, \cos \left (3 \, x + 2\right ) \sin \left (3 \, x + 2\right )}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.58 \[ \int \frac {1}{2 \cos ^2(2+3 x)+\sin ^2(2+3 x)} \, dx=\frac {\sqrt {2} \left (\operatorname {atan}{\left (\sqrt {2} \tan {\left (\frac {3 x}{2} + 1 \right )} - 1 \right )} + \pi \left \lfloor {\frac {\frac {3 x}{2} - \frac {\pi }{2} + 1}{\pi }}\right \rfloor \right )}{6} + \frac {\sqrt {2} \left (\operatorname {atan}{\left (\sqrt {2} \tan {\left (\frac {3 x}{2} + 1 \right )} + 1 \right )} + \pi \left \lfloor {\frac {\frac {3 x}{2} - \frac {\pi }{2} + 1}{\pi }}\right \rfloor \right )}{6} \]
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Time = 0.31 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.35 \[ \int \frac {1}{2 \cos ^2(2+3 x)+\sin ^2(2+3 x)} \, dx=\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \tan \left (3 \, x + 2\right )\right ) \]
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Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19 \[ \int \frac {1}{2 \cos ^2(2+3 x)+\sin ^2(2+3 x)} \, dx=\frac {1}{6} \, \sqrt {2} {\left (3 \, x + \arctan \left (-\frac {\sqrt {2} \sin \left (6 \, x + 4\right ) - \sin \left (6 \, x + 4\right )}{\sqrt {2} \cos \left (6 \, x + 4\right ) + \sqrt {2} - \cos \left (6 \, x + 4\right ) + 1}\right ) + 2\right )} \]
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Time = 26.48 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.75 \[ \int \frac {1}{2 \cos ^2(2+3 x)+\sin ^2(2+3 x)} \, dx=\frac {\sqrt {2}\,\left (3\,x-\mathrm {atan}\left (\mathrm {tan}\left (3\,x+2\right )\right )\right )}{6}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {tan}\left (3\,x+2\right )}{2}\right )}{6} \]
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