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 = 0.0410113, antiderivative size = 36, normalized size of antiderivative = 1., 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 3171
Rule 3181
Rule 203
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*}
Mathematica [A] time = 0.0286231, 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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Maple [A] time = 0.022, size = 16, normalized size = 0.4 \begin{align*} \sqrt{2}\arctan \left ( \tan \left ( x \right ) \sqrt{2} \right ) -x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48226, size = 20, normalized size = 0.56 \begin{align*} \sqrt{2} \arctan \left (\sqrt{2} \tan \left (x\right )\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.39607, size = 107, normalized size = 2.97 \begin{align*} -\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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 77.8847, size = 416, normalized size = 11.56 \begin{align*} - \frac{58 x \sqrt{3 - 2 \sqrt{2}}}{41 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 58 \sqrt{3 - 2 \sqrt{2}}} - \frac{41 \sqrt{2} x \sqrt{3 - 2 \sqrt{2}}}{41 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 58 \sqrt{3 - 2 \sqrt{2}}} + \frac{24 \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 )}{41 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 58 \sqrt{3 - 2 \sqrt{2}}} + \frac{34 \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 )}{41 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 58 \sqrt{3 - 2 \sqrt{2}}} + \frac{24 \sqrt{2} \sqrt{3 - 2 \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 )}{41 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 58 \sqrt{3 - 2 \sqrt{2}}} + \frac{34 \sqrt{3 - 2 \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 )}{41 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 58 \sqrt{3 - 2 \sqrt{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11781, size = 66, normalized size = 1.83 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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