Optimal. Leaf size=6 \[ \frac {1}{\cos (x)+1} \]
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Rubi [A] time = 0.02, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2667, 32} \[ \frac {1}{\cos (x)+1} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sin (x)}{(1+\cos (x))^2} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,\cos (x)\right )\\ &=\frac {1}{1+\cos (x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 2.00 \[ \frac {1}{2} \sec ^2\left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 6, normalized size = 1.00 \[ \frac {1}{\cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 6, normalized size = 1.00 \[ \frac {1}{\cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 7, normalized size = 1.17 \[ \frac {1}{\cos \relax (x )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 6, normalized size = 1.00 \[ \frac {1}{\cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 6, normalized size = 1.00 \[ \frac {1}{\cos \relax (x)+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 5, normalized size = 0.83 \[ \frac {1}{\cos {\relax (x )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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