Optimal. Leaf size=55 \[ \frac {3 x}{4 \sqrt {2}}-\frac {3 \text {ArcTan}\left (\frac {\cos (x) \sin (x)}{1+\sqrt {2}+\cos ^2(x)}\right )}{4 \sqrt {2}}-\frac {\cos (x) \sin (x)}{4 \left (1+\cos ^2(x)\right )} \]
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Rubi [A]
time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3263, 12, 3260,
209} \begin {gather*} -\frac {3 \text {ArcTan}\left (\frac {\sin (x) \cos (x)}{\cos ^2(x)+\sqrt {2}+1}\right )}{4 \sqrt {2}}+\frac {3 x}{4 \sqrt {2}}-\frac {\sin (x) \cos (x)}{4 \left (\cos ^2(x)+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 3260
Rule 3263
Rubi steps
\begin {align*} \int \frac {1}{\left (1+\cos ^2(x)\right )^2} \, dx &=-\frac {\cos (x) \sin (x)}{4 \left (1+\cos ^2(x)\right )}-\frac {1}{4} \int -\frac {3}{1+\cos ^2(x)} \, dx\\ &=-\frac {\cos (x) \sin (x)}{4 \left (1+\cos ^2(x)\right )}+\frac {3}{4} \int \frac {1}{1+\cos ^2(x)} \, dx\\ &=-\frac {\cos (x) \sin (x)}{4 \left (1+\cos ^2(x)\right )}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\cot (x)\right )\\ &=\frac {3 x}{4 \sqrt {2}}-\frac {3 \tan ^{-1}\left (\frac {\cos (x) \sin (x)}{1+\sqrt {2}+\cos ^2(x)}\right )}{4 \sqrt {2}}-\frac {\cos (x) \sin (x)}{4 \left (1+\cos ^2(x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 35, normalized size = 0.64 \begin {gather*} \frac {3 \text {ArcTan}\left (\frac {\tan (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {\sin (2 x)}{4 (3+\cos (2 x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 27, normalized size = 0.49
method | result | size |
default | \(-\frac {\tan \left (x \right )}{4 \left (\tan ^{2}\left (x \right )+2\right )}+\frac {3 \arctan \left (\frac {\tan \left (x \right ) \sqrt {2}}{2}\right ) \sqrt {2}}{8}\) | \(27\) |
risch | \(-\frac {i \left (3 \,{\mathrm e}^{2 i x}+1\right )}{2 \left ({\mathrm e}^{4 i x}+6 \,{\mathrm e}^{2 i x}+1\right )}+\frac {3 i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+2 \sqrt {2}+3\right )}{16}-\frac {3 i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-2 \sqrt {2}+3\right )}{16}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 26, normalized size = 0.47 \begin {gather*} \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \tan \left (x\right )\right ) - \frac {\tan \left (x\right )}{4 \, {\left (\tan \left (x\right )^{2} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 57, normalized size = 1.04 \begin {gather*} -\frac {3 \, {\left (\sqrt {2} \cos \left (x\right )^{2} + \sqrt {2}\right )} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (x\right )^{2} - \sqrt {2}}{4 \, \cos \left (x\right ) \sin \left (x\right )}\right ) + 4 \, \cos \left (x\right ) \sin \left (x\right )}{16 \, {\left (\cos \left (x\right )^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs.
\(2 (53) = 106\).
time = 1.09, size = 218, normalized size = 3.96 \begin {gather*} \frac {3 \sqrt {2} \left (\operatorname {atan}{\left (\sqrt {2} \tan {\left (\frac {x}{2} \right )} - 1 \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{4}{\left (\frac {x}{2} \right )}}{8 \tan ^{4}{\left (\frac {x}{2} \right )} + 8} + \frac {3 \sqrt {2} \left (\operatorname {atan}{\left (\sqrt {2} \tan {\left (\frac {x}{2} \right )} - 1 \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{8 \tan ^{4}{\left (\frac {x}{2} \right )} + 8} + \frac {3 \sqrt {2} \left (\operatorname {atan}{\left (\sqrt {2} \tan {\left (\frac {x}{2} \right )} + 1 \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{4}{\left (\frac {x}{2} \right )}}{8 \tan ^{4}{\left (\frac {x}{2} \right )} + 8} + \frac {3 \sqrt {2} \left (\operatorname {atan}{\left (\sqrt {2} \tan {\left (\frac {x}{2} \right )} + 1 \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{8 \tan ^{4}{\left (\frac {x}{2} \right )} + 8} + \frac {2 \tan ^{3}{\left (\frac {x}{2} \right )}}{8 \tan ^{4}{\left (\frac {x}{2} \right )} + 8} - \frac {2 \tan {\left (\frac {x}{2} \right )}}{8 \tan ^{4}{\left (\frac {x}{2} \right )} + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 59, normalized size = 1.07 \begin {gather*} \frac {3}{8} \, \sqrt {2} {\left (x + \arctan \left (-\frac {\sqrt {2} \sin \left (2 \, x\right ) - \sin \left (2 \, x\right )}{\sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2} - \cos \left (2 \, x\right ) + 1}\right )\right )} - \frac {\tan \left (x\right )}{4 \, {\left (\tan \left (x\right )^{2} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.17, size = 40, normalized size = 0.73 \begin {gather*} \frac {3\,\sqrt {2}\,\left (x-\mathrm {atan}\left (\mathrm {tan}\left (x\right )\right )\right )}{8}-\frac {\mathrm {tan}\left (x\right )}{4\,\left ({\mathrm {tan}\left (x\right )}^2+2\right )}+\frac {3\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {tan}\left (x\right )}{2}\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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