Optimal. Leaf size=36 \[ \sqrt {2} x-x+\sqrt {2} \tan ^{-1}\left (\frac {\sin (x) \cos (x)}{\sin ^2(x)+\sqrt {2}+1}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3171, 3181, 203} \[ \sqrt {2} x-x+\sqrt {2} \tan ^{-1}\left (\frac {\sin (x) \cos (x)}{\sin ^2(x)+\sqrt {2}+1}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 3171
Rule 3181
Rubi steps
\begin {align*} \int \frac {1-\sin ^2(x)}{1+\sin ^2(x)} \, dx &=-x+2 \int \frac {1}{1+\sin ^2(x)} \, dx\\ &=-x+2 \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\tan (x)\right )\\ &=-x+\sqrt {2} x+\sqrt {2} \tan ^{-1}\left (\frac {\cos (x) \sin (x)}{1+\sqrt {2}+\sin ^2(x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 24, normalized size = 0.67 \[ -2 \left (\frac {x}{2}-\frac {\tan ^{-1}\left (\sqrt {2} \tan (x)\right )}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 35, normalized size = 0.97 \[ -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \relax (x)^{2} - 2 \, \sqrt {2}}{4 \, \cos \relax (x) \sin \relax (x)}\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 49, normalized size = 1.36 \[ \sqrt {2} {\left (x + \arctan \left (-\frac {\sqrt {2} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )} - x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 16, normalized size = 0.44 \[ \sqrt {2}\, \arctan \left (\sqrt {2}\, \tan \relax (x )\right )-x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 15, normalized size = 0.42 \[ \sqrt {2} \arctan \left (\sqrt {2} \tan \relax (x)\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.32, size = 26, normalized size = 0.72 \[ \sqrt {2}\,\left (x-\mathrm {atan}\left (\mathrm {tan}\relax (x)\right )\right )-x+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\mathrm {tan}\relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 48.46, size = 248, normalized size = 6.89 \[ - \frac {22619537 x}{15994428 \sqrt {2} + 22619537} - \frac {15994428 \sqrt {2} x}{15994428 \sqrt {2} + 22619537} + \frac {54608393 \sqrt {2} \sqrt {3 - 2 \sqrt {2}} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {3 - 2 \sqrt {2}}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{15994428 \sqrt {2} + 22619537} + \frac {77227930 \sqrt {3 - 2 \sqrt {2}} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {3 - 2 \sqrt {2}}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{15994428 \sqrt {2} + 22619537} + \frac {9369319 \sqrt {2} \sqrt {2 \sqrt {2} + 3} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {2 \sqrt {2} + 3}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{15994428 \sqrt {2} + 22619537} + \frac {13250218 \sqrt {2 \sqrt {2} + 3} \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\sqrt {2 \sqrt {2} + 3}} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{15994428 \sqrt {2} + 22619537} \]
Verification of antiderivative is not currently implemented for this CAS.
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