Integrand size = 17, antiderivative size = 36 \[ \int \frac {1-\sin ^2(x)}{1+\sin ^2(x)} \, dx=-x+\sqrt {2} x+\sqrt {2} \arctan \left (\frac {\cos (x) \sin (x)}{1+\sqrt {2}+\sin ^2(x)}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3250, 3260, 209} \[ \int \frac {1-\sin ^2(x)}{1+\sin ^2(x)} \, dx=\sqrt {2} \arctan \left (\frac {\sin (x) \cos (x)}{\sin ^2(x)+\sqrt {2}+1}\right )+\sqrt {2} x-x \]
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Rule 209
Rule 3250
Rule 3260
Rubi steps \begin{align*} \text {integral}& = -x+2 \int \frac {1}{1+\sin ^2(x)} \, dx \\ & = -x+2 \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\tan (x)\right ) \\ & = -x+\sqrt {2} x+\sqrt {2} \arctan \left (\frac {\cos (x) \sin (x)}{1+\sqrt {2}+\sin ^2(x)}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int \frac {1-\sin ^2(x)}{1+\sin ^2(x)} \, dx=-2 \left (\frac {x}{2}-\frac {\arctan \left (\sqrt {2} \tan (x)\right )}{\sqrt {2}}\right ) \]
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Time = 0.68 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.50
method | result | size |
default | \(-\arctan \left (\tan \left (x \right )\right )+\sqrt {2}\, \arctan \left (\sqrt {2}\, \tan \left (x \right )\right )\) | \(18\) |
risch | \(-x +\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-2 \sqrt {2}-3\right )}{2}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+2 \sqrt {2}-3\right )}{2}\) | \(43\) |
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {1-\sin ^2(x)}{1+\sin ^2(x)} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (x\right )^{2} - 2 \, \sqrt {2}}{4 \, \cos \left (x\right ) \sin \left (x\right )}\right ) - x \]
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Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (32) = 64\).
Time = 21.85 (sec) , antiderivative size = 248, normalized size of antiderivative = 6.89 \[ \int \frac {1-\sin ^2(x)}{1+\sin ^2(x)} \, dx=- \frac {22619537 x}{15994428 \sqrt {2} + 22619537} - \frac {15994428 \sqrt {2} x}{15994428 \sqrt {2} + 22619537} + \frac {54608393 \sqrt {2} \sqrt {3 - 2 \sqrt {2}} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {3 - 2 \sqrt {2}}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{15994428 \sqrt {2} + 22619537} + \frac {77227930 \sqrt {3 - 2 \sqrt {2}} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {3 - 2 \sqrt {2}}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{15994428 \sqrt {2} + 22619537} + \frac {9369319 \sqrt {2} \sqrt {2 \sqrt {2} + 3} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {2 \sqrt {2} + 3}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{15994428 \sqrt {2} + 22619537} + \frac {13250218 \sqrt {2 \sqrt {2} + 3} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {2 \sqrt {2} + 3}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{15994428 \sqrt {2} + 22619537} \]
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Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.42 \[ \int \frac {1-\sin ^2(x)}{1+\sin ^2(x)} \, dx=\sqrt {2} \arctan \left (\sqrt {2} \tan \left (x\right )\right ) - x \]
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Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.36 \[ \int \frac {1-\sin ^2(x)}{1+\sin ^2(x)} \, dx=\sqrt {2} {\left (x + \arctan \left (-\frac {\sqrt {2} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )} - x \]
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Time = 26.93 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {1-\sin ^2(x)}{1+\sin ^2(x)} \, dx=\sqrt {2}\,\left (x-\mathrm {atan}\left (\mathrm {tan}\left (x\right )\right )\right )-x+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\mathrm {tan}\left (x\right )\right ) \]
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