Optimal. Leaf size=5 \[ \sin (x) \cos (x) \]
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Rubi [A] time = 0.0220327, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3675, 383} \[ \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 3675
Rule 383
Rubi steps
\begin{align*} \int \cos ^2(x) \left (1-\tan ^2(x)\right ) \, dx &=\operatorname{Subst}\left (\int \frac{1-x^2}{\left (1+x^2\right )^2} \, dx,x,\tan (x)\right )\\ &=\cos (x) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0016832, size = 8, normalized size = 1.6 \[ \frac{1}{2} \sin (2 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 6, normalized size = 1.2 \begin{align*} \cos \left ( x \right ) \sin \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.930044, size = 15, normalized size = 3. \begin{align*} \frac{\tan \left (x\right )}{\tan \left (x\right )^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12409, size = 20, normalized size = 4. \begin{align*} \cos \left (x\right ) \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.414073, size = 7, normalized size = 1.4 \begin{align*} \frac{\tan{\left (x \right )}}{\sec ^{2}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06252, size = 12, normalized size = 2.4 \begin{align*} \frac{1}{\frac{1}{\tan \left (x\right )} + \tan \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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