Optimal. Leaf size=21 \[ -\frac{1}{3} \log \left (3 \cot \left (\frac{x}{2}+\frac{\pi }{4}\right )+4\right ) \]
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Rubi [A] time = 0.0119244, antiderivative size = 21, 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 = {3123, 31} \[ -\frac{1}{3} \log \left (3 \cot \left (\frac{x}{2}+\frac{\pi }{4}\right )+4\right ) \]
Antiderivative was successfully verified.
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Rule 3123
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{4+3 \cos (x)+4 \sin (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{4+3 x} \, dx,x,\cot \left (\frac{\pi }{4}+\frac{x}{2}\right )\right )\\ &=-\frac{1}{3} \log \left (4+3 \cot \left (\frac{\pi }{4}+\frac{x}{2}\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0162603, size = 39, normalized size = 1.86 \[ \frac{1}{3} \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\frac{1}{3} \log \left (\sin \left (\frac{x}{2}\right )+7 \cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 20, normalized size = 1. \begin{align*}{\frac{1}{3}\ln \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) }-{\frac{1}{3}\ln \left ( \tan \left ({\frac{x}{2}} \right ) +7 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.932742, size = 39, normalized size = 1.86 \begin{align*} -\frac{1}{3} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 7\right ) + \frac{1}{3} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11437, size = 80, normalized size = 3.81 \begin{align*} -\frac{1}{6} \, \log \left (24 \, \cos \left (x\right ) + 7 \, \sin \left (x\right ) + 25\right ) + \frac{1}{6} \, \log \left (\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.257689, size = 19, normalized size = 0.9 \begin{align*} \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )}}{3} - \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} + 7 \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10748, size = 28, normalized size = 1.33 \begin{align*} -\frac{1}{3} \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 7 \right |}\right ) + \frac{1}{3} \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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