Optimal. Leaf size=6 \[ \frac {1}{3+\cos (x)} \]
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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 = {2747, 32}
\begin {gather*} \frac {1}{\cos (x)+3} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2747
Rubi steps
\begin {align*} \int \frac {\sin (x)}{(3+\cos (x))^2} \, dx &=-\text {Subst}\left (\int \frac {1}{(3+x)^2} \, dx,x,\cos (x)\right )\\ &=\frac {1}{3+\cos (x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 6, normalized size = 1.00 \begin {gather*} \frac {1}{3+\cos (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 7, normalized size = 1.17
method | result | size |
derivativedivides | \(\frac {1}{3+\cos \left (x \right )}\) | \(7\) |
default | \(\frac {1}{3+\cos \left (x \right )}\) | \(7\) |
risch | \(\frac {2 \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}+6 \,{\mathrm e}^{i x}+1}\) | \(24\) |
norman | \(\frac {-\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-\frac {1}{2}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )+2\right )}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.61, size = 6, normalized size = 1.00 \begin {gather*} \frac {1}{\cos \left (x\right ) + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.54, size = 6, normalized size = 1.00 \begin {gather*} \frac {1}{\cos \left (x\right ) + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 5, normalized size = 0.83 \begin {gather*} \frac {1}{\cos {\left (x \right )} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.81, size = 6, normalized size = 1.00 \begin {gather*} \frac {1}{\cos \left (x\right ) + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 6, normalized size = 1.00 \begin {gather*} \frac {1}{\cos \left (x\right )+3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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