Integrand size = 13, antiderivative size = 36 \[ \int \frac {\sin ^2(x)}{1+\sin ^2(x)} \, dx=x-\frac {x}{\sqrt {2}}-\frac {\arctan \left (\frac {\cos (x) \sin (x)}{1+\sqrt {2}+\sin ^2(x)}\right )}{\sqrt {2}} \]
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Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3250, 3260, 209} \[ \int \frac {\sin ^2(x)}{1+\sin ^2(x)} \, dx=-\frac {\arctan \left (\frac {\sin (x) \cos (x)}{\sin ^2(x)+\sqrt {2}+1}\right )}{\sqrt {2}}-\frac {x}{\sqrt {2}}+x \]
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Rule 209
Rule 3250
Rule 3260
Rubi steps \begin{align*} \text {integral}& = x-\int \frac {1}{1+\sin ^2(x)} \, dx \\ & = x-\text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\tan (x)\right ) \\ & = x-\frac {x}{\sqrt {2}}-\frac {\arctan \left (\frac {\cos (x) \sin (x)}{1+\sqrt {2}+\sin ^2(x)}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.50 \[ \int \frac {\sin ^2(x)}{1+\sin ^2(x)} \, dx=x-\frac {\arctan \left (\sqrt {2} \tan (x)\right )}{\sqrt {2}} \]
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Time = 0.14 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.47
method | result | size |
default | \(-\frac {\sqrt {2}\, \arctan \left (\tan \left (x \right ) \sqrt {2}\right )}{2}+\arctan \left (\tan \left (x \right )\right )\) | \(17\) |
risch | \(x -\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-2 \sqrt {2}-3\right )}{4}+\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+2 \sqrt {2}-3\right )}{4}\) | \(41\) |
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Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \frac {\sin ^2(x)}{1+\sin ^2(x)} \, dx=\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (x\right )^{2} - 2 \, \sqrt {2}}{4 \, \cos \left (x\right ) \sin \left (x\right )}\right ) + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (36) = 72\).
Time = 24.44 (sec) , antiderivative size = 248, normalized size of antiderivative = 6.89 \[ \int \frac {\sin ^2(x)}{1+\sin ^2(x)} \, dx=\frac {31988856 \sqrt {2} x}{31988856 \sqrt {2} + 45239074} + \frac {45239074 x}{31988856 \sqrt {2} + 45239074} - \frac {77227930 \sqrt {3 - 2 \sqrt {2}} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {3 - 2 \sqrt {2}}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{31988856 \sqrt {2} + 45239074} - \frac {54608393 \sqrt {2} \sqrt {3 - 2 \sqrt {2}} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {3 - 2 \sqrt {2}}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{31988856 \sqrt {2} + 45239074} - \frac {13250218 \sqrt {2 \sqrt {2} + 3} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {2 \sqrt {2} + 3}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{31988856 \sqrt {2} + 45239074} - \frac {9369319 \sqrt {2} \sqrt {2 \sqrt {2} + 3} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {2 \sqrt {2} + 3}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{31988856 \sqrt {2} + 45239074} \]
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Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.39 \[ \int \frac {\sin ^2(x)}{1+\sin ^2(x)} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} \tan \left (x\right )\right ) + x \]
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Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \frac {\sin ^2(x)}{1+\sin ^2(x)} \, dx=-\frac {1}{2} \, \sqrt {2} {\left (x + \arctan \left (-\frac {\sqrt {2} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )} + x \]
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Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {\sin ^2(x)}{1+\sin ^2(x)} \, dx=x-\frac {\sqrt {2}\,\left (x-\mathrm {atan}\left (\mathrm {tan}\left (x\right )\right )\right )}{2}-\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\mathrm {tan}\left (x\right )\right )}{2} \]
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