Optimal. Leaf size=15 \[ \frac{1}{4} \log \left (4 \tan \left (\frac{x}{2}\right )+3\right ) \]
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Rubi [A] time = 0.0131235, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3124, 31} \[ \frac{1}{4} \log \left (4 \tan \left (\frac{x}{2}\right )+3\right ) \]
Antiderivative was successfully verified.
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Rule 3124
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{3+3 \cos (x)+4 \sin (x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{6+8 x} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{1}{4} \log \left (3+4 \tan \left (\frac{x}{2}\right )\right )\\ \end{align*}
Mathematica [B] time = 0.017616, size = 34, normalized size = 2.27 \[ \frac{1}{4} \log \left (4 \sin \left (\frac{x}{2}\right )+3 \cos \left (\frac{x}{2}\right )\right )-\frac{1}{4} \log \left (\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 12, normalized size = 0.8 \begin{align*}{\frac{1}{4}\ln \left ( 3+4\,\tan \left ( x/2 \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.999468, size = 20, normalized size = 1.33 \begin{align*} \frac{1}{4} \, \log \left (\frac{4 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06574, size = 95, normalized size = 6.33 \begin{align*} -\frac{1}{8} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{8} \, \log \left (-\frac{7}{2} \, \cos \left (x\right ) + 12 \, \sin \left (x\right ) + \frac{25}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.250092, size = 10, normalized size = 0.67 \begin{align*} \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} + \frac{3}{4} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10376, size = 16, normalized size = 1.07 \begin{align*} \frac{1}{4} \, \log \left ({\left | 4 \, \tan \left (\frac{1}{2} \, x\right ) + 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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