Integrand size = 10, antiderivative size = 34 \[ \int \frac {2}{1+\cos ^2(x)} \, dx=\sqrt {2} x-\sqrt {2} \arctan \left (\frac {\cos (x) \sin (x)}{1+\sqrt {2}+\cos ^2(x)}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {12, 3260, 209} \[ \int \frac {2}{1+\cos ^2(x)} \, dx=\sqrt {2} x-\sqrt {2} \arctan \left (\frac {\sin (x) \cos (x)}{\cos ^2(x)+\sqrt {2}+1}\right ) \]
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Rule 12
Rule 209
Rule 3260
Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {1}{1+\cos ^2(x)} \, dx \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\cot (x)\right )\right ) \\ & = \sqrt {2} x-\sqrt {2} \arctan \left (\frac {\cos (x) \sin (x)}{1+\sqrt {2}+\cos ^2(x)}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.44 \[ \int \frac {2}{1+\cos ^2(x)} \, dx=\sqrt {2} \arctan \left (\frac {\tan (x)}{\sqrt {2}}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.38
method | result | size |
default | \(\sqrt {2}\, \arctan \left (\frac {\tan \left (x \right ) \sqrt {2}}{2}\right )\) | \(13\) |
risch | \(\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+2 \sqrt {2}+3\right )}{2}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-2 \sqrt {2}+3\right )}{2}\) | \(40\) |
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Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {2}{1+\cos ^2(x)} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (x\right )^{2} - \sqrt {2}}{4 \, \cos \left (x\right ) \sin \left (x\right )}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.76 \[ \int \frac {2}{1+\cos ^2(x)} \, dx=\sqrt {2} \left (\operatorname {atan}{\left (\sqrt {2} \tan {\left (\frac {x}{2} \right )} - 1 \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) + \sqrt {2} \left (\operatorname {atan}{\left (\sqrt {2} \tan {\left (\frac {x}{2} \right )} + 1 \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \]
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Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.35 \[ \int \frac {2}{1+\cos ^2(x)} \, dx=\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \tan \left (x\right )\right ) \]
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Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.32 \[ \int \frac {2}{1+\cos ^2(x)} \, dx=\sqrt {2} {\left (x + \arctan \left (-\frac {\sqrt {2} \sin \left (2 \, x\right ) - \sin \left (2 \, x\right )}{\sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2} - \cos \left (2 \, x\right ) + 1}\right )\right )} \]
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Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int \frac {2}{1+\cos ^2(x)} \, dx=\sqrt {2}\,\left (x-\mathrm {atan}\left (\mathrm {tan}\left (x\right )\right )\right )+\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {tan}\left (x\right )}{2}\right ) \]
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