Integrand size = 12, antiderivative size = 45 \[ \int \frac {1}{5-\cos (x)+2 \sin (x)} \, dx=\frac {x}{2 \sqrt {5}}+\frac {\arctan \left (\frac {2 \cos (x)+\sin (x)}{5+2 \sqrt {5}-\cos (x)+2 \sin (x)}\right )}{\sqrt {5}} \]
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Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3203, 632, 210} \[ \int \frac {1}{5-\cos (x)+2 \sin (x)} \, dx=\frac {\arctan \left (\frac {\sin (x)+2 \cos (x)}{2 \sin (x)-\cos (x)+2 \sqrt {5}+5}\right )}{\sqrt {5}}+\frac {x}{2 \sqrt {5}} \]
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Rule 210
Rule 632
Rule 3203
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{4+4 x+6 x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right ) \\ & = -\left (4 \text {Subst}\left (\int \frac {1}{-80-x^2} \, dx,x,4+12 \tan \left (\frac {x}{2}\right )\right )\right ) \\ & = \frac {x}{2 \sqrt {5}}+\frac {\arctan \left (\frac {2 \cos (x)+\sin (x)}{5+2 \sqrt {5}-\cos (x)+2 \sin (x)}\right )}{\sqrt {5}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.51 \[ \int \frac {1}{5-\cos (x)+2 \sin (x)} \, dx=\frac {\arctan \left (\frac {1+3 \tan \left (\frac {x}{2}\right )}{\sqrt {5}}\right )}{\sqrt {5}} \]
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Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.44
method | result | size |
default | \(\frac {\sqrt {5}\, \arctan \left (\frac {\left (6 \tan \left (\frac {x}{2}\right )+2\right ) \sqrt {5}}{10}\right )}{5}\) | \(20\) |
risch | \(\frac {i \sqrt {5}\, \ln \left ({\mathrm e}^{i x}-1+2 i+\frac {4 i \sqrt {5}}{5}-\frac {2 \sqrt {5}}{5}\right )}{10}-\frac {i \sqrt {5}\, \ln \left ({\mathrm e}^{i x}-1+2 i-\frac {4 i \sqrt {5}}{5}+\frac {2 \sqrt {5}}{5}\right )}{10}\) | \(56\) |
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Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80 \[ \int \frac {1}{5-\cos (x)+2 \sin (x)} \, dx=\frac {1}{10} \, \sqrt {5} \arctan \left (-\frac {\sqrt {5} \cos \left (x\right ) - 2 \, \sqrt {5} \sin \left (x\right ) - \sqrt {5}}{2 \, {\left (2 \, \cos \left (x\right ) + \sin \left (x\right )\right )}}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int \frac {1}{5-\cos (x)+2 \sin (x)} \, dx=\frac {\sqrt {5} \left (\operatorname {atan}{\left (\frac {3 \sqrt {5} \tan {\left (\frac {x}{2} \right )}}{5} + \frac {\sqrt {5}}{5} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{5} \]
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.51 \[ \int \frac {1}{5-\cos (x)+2 \sin (x)} \, dx=\frac {1}{5} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (\frac {3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {1}{5-\cos (x)+2 \sin (x)} \, dx=\frac {1}{10} \, \sqrt {5} {\left (x + 2 \, \arctan \left (-\frac {\sqrt {5} \sin \left (x\right ) - \cos \left (x\right ) - 3 \, \sin \left (x\right ) - 1}{\sqrt {5} \cos \left (x\right ) + \sqrt {5} - 3 \, \cos \left (x\right ) + \sin \left (x\right ) + 3}\right )\right )} \]
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Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.47 \[ \int \frac {1}{5-\cos (x)+2 \sin (x)} \, dx=\frac {\sqrt {20}\,\mathrm {atan}\left (\frac {3\,\sqrt {20}\,\mathrm {tan}\left (\frac {x}{2}\right )}{10}+\frac {\sqrt {20}}{10}\right )}{10} \]
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