Optimal. Leaf size=14 \[ x+2 \tan ^{-1}\left (\frac{\cos (x)}{\sin (x)+2}\right ) \]
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Rubi [A] time = 0.0107162, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {12, 2657} \[ x+2 \tan ^{-1}\left (\frac{\cos (x)}{\sin (x)+2}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2657
Rubi steps
\begin{align*} \int \frac{3}{5+4 \sin (x)} \, dx &=3 \int \frac{1}{5+4 \sin (x)} \, dx\\ &=x+2 \tan ^{-1}\left (\frac{\cos (x)}{2+\sin (x)}\right )\\ \end{align*}
Mathematica [B] time = 0.0155187, size = 79, normalized size = 5.64 \[ 3 \left (\frac{1}{3} \tan ^{-1}\left (\frac{2 \sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+2 \cos \left (\frac{x}{2}\right )}\right )-\frac{1}{3} \tan ^{-1}\left (\frac{\sin \left (\frac{x}{2}\right )+2 \cos \left (\frac{x}{2}\right )}{2 \sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 12, normalized size = 0.9 \begin{align*} 2\,\arctan \left ( 5/3\,\tan \left ( x/2 \right ) +4/3 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41181, size = 20, normalized size = 1.43 \begin{align*} 2 \, \arctan \left (\frac{5 \, \sin \left (x\right )}{3 \,{\left (\cos \left (x\right ) + 1\right )}} + \frac{4}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6957, size = 47, normalized size = 3.36 \begin{align*} \arctan \left (\frac{5 \, \sin \left (x\right ) + 4}{3 \, \cos \left (x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.268381, size = 27, normalized size = 1.93 \begin{align*} 2 \operatorname{atan}{\left (\frac{5 \tan{\left (\frac{x}{2} \right )}}{3} + \frac{4}{3} \right )} + 2 \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0905, size = 34, normalized size = 2.43 \begin{align*} x + 2 \, \arctan \left (-\frac{2 \, \cos \left (x\right ) + \sin \left (x\right ) + 2}{\cos \left (x\right ) - 2 \, \sin \left (x\right ) - 4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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