Optimal. Leaf size=12 \[ -\frac {1}{\tan \left (\frac {x}{2}\right )+2} \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.75, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3114} \[ -\frac {4-5 \sin (x)}{4 (4 \cos (x)-3 \sin (x))} \]
Antiderivative was successfully verified.
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Rule 3114
Rubi steps
\begin {align*} \int \frac {1}{5+3 \cos (x)+4 \sin (x)} \, dx &=-\frac {4-5 \sin (x)}{4 (4 \cos (x)-3 \sin (x))}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 26, normalized size = 2.17 \[ \frac {\sin \left (\frac {x}{2}\right )}{2 \sin \left (\frac {x}{2}\right )+4 \cos \left (\frac {x}{2}\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 20, normalized size = 1.67 \[ -\frac {\cos \relax (x) - 2 \, \sin \relax (x) + 1}{5 \, {\left (2 \, \cos \relax (x) + \sin \relax (x) + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 10, normalized size = 0.83 \[ -\frac {1}{\tan \left (\frac {1}{2} \, x\right ) + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 11, normalized size = 0.92 \[ -\frac {1}{\tan \left (\frac {x}{2}\right )+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 15, normalized size = 1.25 \[ -\frac {1}{\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 10, normalized size = 0.83 \[ -\frac {1}{\mathrm {tan}\left (\frac {x}{2}\right )+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 8, normalized size = 0.67 \[ - \frac {1}{\tan {\left (\frac {x}{2} \right )} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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