Integrand size = 13, antiderivative size = 5 \[ \int \cos ^2(x) \left (1-\tan ^2(x)\right ) \, dx=\cos (x) \sin (x) \]
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Time = 0.02 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3756, 391} \[ \int \cos ^2(x) \left (1-\tan ^2(x)\right ) \, dx=\sin (x) \cos (x) \]
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Rule 391
Rule 3756
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1-x^2}{\left (1+x^2\right )^2} \, dx,x,\tan (x)\right ) \\ & = \cos (x) \sin (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.60 \[ \int \cos ^2(x) \left (1-\tan ^2(x)\right ) \, dx=\frac {1}{2} \sin (2 x) \]
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Time = 3.15 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20
method | result | size |
default | \(\cos \left (x \right ) \sin \left (x \right )\) | \(6\) |
risch | \(\frac {\sin \left (2 x \right )}{2}\) | \(7\) |
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none
Time = 0.24 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \cos ^2(x) \left (1-\tan ^2(x)\right ) \, dx=\cos \left (x\right ) \sin \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (5) = 10\).
Time = 1.64 (sec) , antiderivative size = 14, normalized size of antiderivative = 2.80 \[ \int \cos ^2(x) \left (1-\tan ^2(x)\right ) \, dx=\frac {\sin {\left (x \right )} \cos {\left (x \right )}}{2} + \frac {\sin {\left (2 x \right )}}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 11 vs. \(2 (5) = 10\).
Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 2.20 \[ \int \cos ^2(x) \left (1-\tan ^2(x)\right ) \, dx=\frac {\tan \left (x\right )}{\tan \left (x\right )^{2} + 1} \]
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none
Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.80 \[ \int \cos ^2(x) \left (1-\tan ^2(x)\right ) \, dx=\frac {1}{\frac {1}{\tan \left (x\right )} + \tan \left (x\right )} \]
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Time = 25.92 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20 \[ \int \cos ^2(x) \left (1-\tan ^2(x)\right ) \, dx=\frac {\sin \left (2\,x\right )}{2} \]
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