Optimal. Leaf size=12 \[ -\frac{1}{\tan \left (\frac{x}{2}\right )+2} \]
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Rubi [A] time = 0.0113994, 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*}
Mathematica [B] time = 0.0312558, 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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Maple [A] time = 0.03, size = 11, normalized size = 0.9 \begin{align*} - \left ( 2+\tan \left ({\frac{x}{2}} \right ) \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.932105, size = 20, normalized size = 1.67 \begin{align*} -\frac{1}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74958, size = 73, normalized size = 6.08 \begin{align*} -\frac{\cos \left (x\right ) - 2 \, \sin \left (x\right ) + 1}{5 \,{\left (2 \, \cos \left (x\right ) + \sin \left (x\right ) + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.442646, size = 8, normalized size = 0.67 \begin{align*} - \frac{1}{\tan{\left (\frac{x}{2} \right )} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09093, size = 14, normalized size = 1.17 \begin{align*} -\frac{1}{\tan \left (\frac{1}{2} \, x\right ) + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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