Optimal. Leaf size=6 \[ x-\cos (x) \]
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Rubi [A] time = 0.06, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4391, 2682, 8} \[ x-\cos (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2682
Rule 4391
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*}
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Mathematica [A] time = 0.02, size = 6, normalized size = 1.00 \[ x-\cos (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 6, normalized size = 1.00 \[ x - \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.03, size = 14, normalized size = 2.33 \[ x - \frac {2}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 15, normalized size = 2.50 \[ -\frac {2}{\tan ^{2}\left (\frac {x}{2}\right )+1}+x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 30, normalized size = 5.00 \[ -\frac {2}{\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1} + 2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 6, normalized size = 1.00 \[ x-\cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\relax (x )}}{- \tan {\relax (x )} + \sec {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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