Optimal. Leaf size=25 \[ \frac {\sin (x)}{3 (\cos (x)+1)}+\frac {\sin (x)}{3 (\cos (x)+1)^2} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2650, 2648} \[ \frac {\sin (x)}{3 (\cos (x)+1)}+\frac {\sin (x)}{3 (\cos (x)+1)^2} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2650
Rubi steps
\begin {align*} \int \frac {1}{(1+\cos (x))^2} \, dx &=\frac {\sin (x)}{3 (1+\cos (x))^2}+\frac {1}{3} \int \frac {1}{1+\cos (x)} \, dx\\ &=\frac {\sin (x)}{3 (1+\cos (x))^2}+\frac {\sin (x)}{3 (1+\cos (x))}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 0.64 \[ \frac {\sin (x) (\cos (x)+2)}{3 (\cos (x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 20, normalized size = 0.80 \[ \frac {{\left (\cos \relax (x) + 2\right )} \sin \relax (x)}{3 \, {\left (\cos \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 15, normalized size = 0.60 \[ \frac {1}{6} \, \tan \left (\frac {1}{2} \, x\right )^{3} + \frac {1}{2} \, \tan \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 16, normalized size = 0.64 \[ \frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{6}+\frac {\tan \left (\frac {x}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 23, normalized size = 0.92 \[ \frac {\sin \relax (x)}{2 \, {\left (\cos \relax (x) + 1\right )}} + \frac {\sin \relax (x)^{3}}{6 \, {\left (\cos \relax (x) + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 14, normalized size = 0.56 \[ \frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+3\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 14, normalized size = 0.56 \[ \frac {\tan ^{3}{\left (\frac {x}{2} \right )}}{6} + \frac {\tan {\left (\frac {x}{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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