Optimal. Leaf size=6 \[ x-\cos (x) \]
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Rubi [A] time = 0.0600477, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4391, 2682, 8} \[ x-\cos (x) \]
Antiderivative was successfully verified.
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Rule 4391
Rule 2682
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos (x)}{\sec (x)-\tan (x)} \, dx &=\int \frac{\cos ^2(x)}{1-\sin (x)} \, dx\\ &=-\cos (x)+\int 1 \, dx\\ &=x-\cos (x)\\ \end{align*}
Mathematica [A] time = 0.0214677, size = 6, normalized size = 1. \[ x-\cos (x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.099, size = 15, normalized size = 2.5 \begin{align*} -2\, \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{-1}+x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.48269, size = 41, normalized size = 6.83 \begin{align*} -\frac{2}{\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1} + 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.470495, size = 16, normalized size = 2.67 \begin{align*} x - \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (x \right )}}{- \tan{\left (x \right )} + \sec{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12075, size = 19, normalized size = 3.17 \begin{align*} x - \frac{2}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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