Integrand size = 10, antiderivative size = 41 \[ \int \frac {1}{4-\cos ^2(x)} \, dx=\frac {x}{2 \sqrt {3}}+\frac {\arctan \left (\frac {\cos (x) \sin (x)}{3+2 \sqrt {3}+\sin ^2(x)}\right )}{2 \sqrt {3}} \]
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Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3260, 209} \[ \int \frac {1}{4-\cos ^2(x)} \, dx=\frac {\arctan \left (\frac {\sin (x) \cos (x)}{\sin ^2(x)+2 \sqrt {3}+3}\right )}{2 \sqrt {3}}+\frac {x}{2 \sqrt {3}} \]
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Rule 209
Rule 3260
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{4+3 x^2} \, dx,x,\cot (x)\right ) \\ & = \frac {x}{2 \sqrt {3}}+\frac {\arctan \left (\frac {\cos (x) \sin (x)}{3+2 \sqrt {3}+\sin ^2(x)}\right )}{2 \sqrt {3}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.46 \[ \int \frac {1}{4-\cos ^2(x)} \, dx=\frac {\arctan \left (\frac {2 \tan (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
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Time = 0.15 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.34
method | result | size |
default | \(\frac {\sqrt {3}\, \arctan \left (\frac {2 \tan \left (x \right ) \sqrt {3}}{3}\right )}{6}\) | \(14\) |
risch | \(\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}-4 \sqrt {3}-7\right )}{12}-\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}+4 \sqrt {3}-7\right )}{12}\) | \(40\) |
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76 \[ \int \frac {1}{4-\cos ^2(x)} \, dx=-\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {7 \, \sqrt {3} \cos \left (x\right )^{2} - 4 \, \sqrt {3}}{12 \, \cos \left (x\right ) \sin \left (x\right )}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.49 \[ \int \frac {1}{4-\cos ^2(x)} \, dx=\frac {\sqrt {3} \left (\operatorname {atan}{\left (\frac {\sqrt {3} \tan {\left (\frac {x}{2} \right )}}{3} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{6} + \frac {\sqrt {3} \left (\operatorname {atan}{\left (\sqrt {3} \tan {\left (\frac {x}{2} \right )} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{6} \]
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Time = 0.31 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.32 \[ \int \frac {1}{4-\cos ^2(x)} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \tan \left (x\right )\right ) \]
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Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12 \[ \int \frac {1}{4-\cos ^2(x)} \, dx=\frac {1}{6} \, \sqrt {3} {\left (x + \arctan \left (-\frac {\sqrt {3} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt {3} \cos \left (2 \, x\right ) + \sqrt {3} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )} \]
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Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63 \[ \int \frac {1}{4-\cos ^2(x)} \, dx=\frac {\sqrt {3}\,\left (x-\mathrm {atan}\left (\mathrm {tan}\left (x\right )\right )\right )}{6}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,\mathrm {tan}\left (x\right )}{3}\right )}{6} \]
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