Optimal. Leaf size=14 \[ \frac {2 \sin (x)}{\cos (x)+1}-x \]
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Rubi [A] time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2680, 8} \[ \frac {2 \sin (x)}{\cos (x)+1}-x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2680
Rubi steps
\begin {align*} \int \frac {\sin ^2(x)}{(1+\cos (x))^2} \, dx &=\frac {2 \sin (x)}{1+\cos (x)}-\int 1 \, dx\\ &=-x+\frac {2 \sin (x)}{1+\cos (x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.29 \[ 2 \tan \left (\frac {x}{2}\right )-2 \tan ^{-1}\left (\tan \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 18, normalized size = 1.29 \[ -\frac {x \cos \relax (x) + x - 2 \, \sin \relax (x)}{\cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 10, normalized size = 0.71 \[ -x + 2 \, \tan \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 11, normalized size = 0.79 \[ 2 \tan \left (\frac {x}{2}\right )-x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 23, normalized size = 1.64 \[ \frac {2 \, \sin \relax (x)}{\cos \relax (x) + 1} - 2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 10, normalized size = 0.71 \[ 2\,\mathrm {tan}\left (\frac {x}{2}\right )-x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 7, normalized size = 0.50 \[ - x + 2 \tan {\left (\frac {x}{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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