Optimal. Leaf size=6 \[ \frac {1}{1+\cos (x)} \]
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Rubi [A]
time = 0.01, 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 = {2746, 32}
\begin {gather*} \frac {1}{\cos (x)+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2746
Rubi steps
\begin {align*} \int \frac {\sin (x)}{(1+\cos (x))^2} \, dx &=-\text {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 \begin {gather*} \frac {1}{2} \sec ^2\left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 7, normalized size = 1.17
method | result | size |
derivativedivides | \(\frac {1}{\cos \left (x \right )+1}\) | \(7\) |
default | \(\frac {1}{\cos \left (x \right )+1}\) | \(7\) |
norman | \(\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}\) | \(9\) |
risch | \(\frac {2 \,{\mathrm e}^{i x}}{\left ({\mathrm e}^{i x}+1\right )^{2}}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 6, normalized size = 1.00 \begin {gather*} \frac {1}{\cos \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 6, normalized size = 1.00 \begin {gather*} \frac {1}{\cos \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 5, normalized size = 0.83 \begin {gather*} \frac {1}{\cos {\left (x \right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 6, normalized size = 1.00 \begin {gather*} \frac {1}{\cos \left (x\right ) + 1} \end {gather*}
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 \begin {gather*} \frac {1}{\cos \left (x\right )+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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