Integrand size = 16, antiderivative size = 27 \[ \int \frac {1}{4-3 \cos ^2(x)+5 \sin ^2(x)} \, dx=\frac {x}{3}+\frac {1}{3} \arctan \left (\frac {2 \cos (x) \sin (x)}{1+2 \sin ^2(x)}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {209} \[ \int \frac {1}{4-3 \cos ^2(x)+5 \sin ^2(x)} \, dx=\frac {1}{3} \arctan \left (\frac {2 \sin (x) \cos (x)}{2 \sin ^2(x)+1}\right )+\frac {x}{3} \]
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Rule 209
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1+9 x^2} \, dx,x,\tan (x)\right ) \\ & = \frac {x}{3}+\frac {1}{3} \arctan \left (\frac {2 \cos (x) \sin (x)}{1+2 \sin ^2(x)}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.33 \[ \int \frac {1}{4-3 \cos ^2(x)+5 \sin ^2(x)} \, dx=\frac {1}{3} \arctan (3 \tan (x)) \]
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Time = 0.81 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.30
method | result | size |
default | \(\frac {\arctan \left (3 \tan \left (x \right )\right )}{3}\) | \(8\) |
risch | \(\frac {i \ln \left ({\mathrm e}^{2 i x}-2\right )}{6}-\frac {i \ln \left ({\mathrm e}^{2 i x}-\frac {1}{2}\right )}{6}\) | \(24\) |
parallelrisch | \(-\frac {i \left (\ln \left (\frac {-3 i \sin \left (x \right )-\cos \left (x \right )}{\cos \left (x \right )+1}\right )-\ln \left (\frac {3 i \sin \left (x \right )-\cos \left (x \right )}{\cos \left (x \right )+1}\right )\right )}{6}\) | \(43\) |
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Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {1}{4-3 \cos ^2(x)+5 \sin ^2(x)} \, dx=-\frac {1}{6} \, \arctan \left (\frac {10 \, \cos \left (x\right )^{2} - 9}{6 \, \cos \left (x\right ) \sin \left (x\right )}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (22) = 44\).
Time = 4.41 (sec) , antiderivative size = 219, normalized size of antiderivative = 8.11 \[ \int \frac {1}{4-3 \cos ^2(x)+5 \sin ^2(x)} \, dx=\frac {4478554083 \sqrt {17 - 12 \sqrt {2}} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {17 - 12 \sqrt {2}}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{2305195203 + 1630019160 \sqrt {2}} + \frac {3166815962 \sqrt {2} \sqrt {17 - 12 \sqrt {2}} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {17 - 12 \sqrt {2}}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{2305195203 + 1630019160 \sqrt {2}} + \frac {131836323 \sqrt {12 \sqrt {2} + 17} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {12 \sqrt {2} + 17}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{2305195203 + 1630019160 \sqrt {2}} + \frac {93222358 \sqrt {2} \sqrt {12 \sqrt {2} + 17} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {12 \sqrt {2} + 17}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{2305195203 + 1630019160 \sqrt {2}} \]
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Time = 0.27 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.26 \[ \int \frac {1}{4-3 \cos ^2(x)+5 \sin ^2(x)} \, dx=\frac {1}{3} \, \arctan \left (3 \, \tan \left (x\right )\right ) \]
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Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {1}{4-3 \cos ^2(x)+5 \sin ^2(x)} \, dx=\frac {1}{3} \, x - \frac {1}{3} \, \arctan \left (\frac {\sin \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 2}\right ) \]
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Time = 0.32 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.59 \[ \int \frac {1}{4-3 \cos ^2(x)+5 \sin ^2(x)} \, dx=\frac {x}{3}-\frac {\mathrm {atan}\left (\mathrm {tan}\left (x\right )\right )}{3}+\frac {\mathrm {atan}\left (3\,\mathrm {tan}\left (x\right )\right )}{3} \]
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