Integrand size = 9, antiderivative size = 55 \[ \int \frac {1}{(\cos (x)+2 \sec (x))^2} \, dx=\frac {x}{6 \sqrt {6}}-\frac {\arctan \left (\frac {\cos (x) \sin (x)}{2+\sqrt {6}+\cos ^2(x)}\right )}{6 \sqrt {6}}+\frac {\tan (x)}{6 \left (3+2 \tan ^2(x)\right )} \]
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Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {205, 209} \[ \int \frac {1}{(\cos (x)+2 \sec (x))^2} \, dx=-\frac {\arctan \left (\frac {\sin (x) \cos (x)}{\cos ^2(x)+\sqrt {6}+2}\right )}{6 \sqrt {6}}+\frac {x}{6 \sqrt {6}}+\frac {\tan (x)}{6 \left (2 \tan ^2(x)+3\right )} \]
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Rule 205
Rule 209
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (3+2 x^2\right )^2} \, dx,x,\tan (x)\right ) \\ & = \frac {\tan (x)}{6 \left (3+2 \tan ^2(x)\right )}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{3+2 x^2} \, dx,x,\tan (x)\right ) \\ & = \frac {x}{6 \sqrt {6}}-\frac {\arctan \left (\frac {\cos (x) \sin (x)}{2+\sqrt {6}+\cos ^2(x)}\right )}{6 \sqrt {6}}+\frac {\tan (x)}{6 \left (3+2 \tan ^2(x)\right )} \\ \end{align*}
Time = 1.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(\cos (x)+2 \sec (x))^2} \, dx=\frac {(5+\cos (2 x)) \sec ^4(x) \left (\sqrt {6} \arctan \left (\sqrt {\frac {2}{3}} \tan (x)\right ) (5+\cos (2 x))+6 \sin (2 x)\right )}{144 \left (1+2 \sec ^2(x)\right )^2} \]
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Time = 0.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.53
method | result | size |
default | \(\frac {\tan \left (x \right )}{18+12 \left (\tan ^{2}\left (x \right )\right )}+\frac {\sqrt {6}\, \arctan \left (\frac {\tan \left (x \right ) \sqrt {6}}{3}\right )}{36}\) | \(29\) |
risch | \(\frac {i \left (5 \,{\mathrm e}^{2 i x}+1\right )}{3 \,{\mathrm e}^{4 i x}+30 \,{\mathrm e}^{2 i x}+3}+\frac {i \sqrt {6}\, \ln \left ({\mathrm e}^{2 i x}+2 \sqrt {6}+5\right )}{72}-\frac {i \sqrt {6}\, \ln \left ({\mathrm e}^{2 i x}-2 \sqrt {6}+5\right )}{72}\) | \(68\) |
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Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(\cos (x)+2 \sec (x))^2} \, dx=-\frac {{\left (\sqrt {6} \cos \left (x\right )^{2} + 2 \, \sqrt {6}\right )} \arctan \left (\frac {5 \, \sqrt {6} \cos \left (x\right )^{2} - 2 \, \sqrt {6}}{12 \, \cos \left (x\right ) \sin \left (x\right )}\right ) - 12 \, \cos \left (x\right ) \sin \left (x\right )}{72 \, {\left (\cos \left (x\right )^{2} + 2\right )}} \]
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\[ \int \frac {1}{(\cos (x)+2 \sec (x))^2} \, dx=\int \frac {1}{\left (\cos {\left (x \right )} + 2 \sec {\left (x \right )}\right )^{2}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.51 \[ \int \frac {1}{(\cos (x)+2 \sec (x))^2} \, dx=\frac {1}{36} \, \sqrt {6} \arctan \left (\frac {1}{3} \, \sqrt {6} \tan \left (x\right )\right ) + \frac {\tan \left (x\right )}{6 \, {\left (2 \, \tan \left (x\right )^{2} + 3\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(\cos (x)+2 \sec (x))^2} \, dx=\frac {1}{36} \, \sqrt {6} {\left (x + \arctan \left (-\frac {\sqrt {6} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt {6} \cos \left (2 \, x\right ) + \sqrt {6} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )} + \frac {\tan \left (x\right )}{6 \, {\left (2 \, \tan \left (x\right )^{2} + 3\right )}} \]
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Time = 0.41 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.40 \[ \int \frac {1}{(\cos (x)+2 \sec (x))^2} \, dx=\frac {\sqrt {6}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {6}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{4}+\frac {5\,\sqrt {6}\,\mathrm {tan}\left (\frac {x}{2}\right )}{12}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {6}\,\mathrm {tan}\left (\frac {x}{2}\right )}{4}\right )\right )}{72}+\frac {\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{9}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{9}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^4+\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{3}+1} \]
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