Optimal. Leaf size=16 \[ x+2 \tan ^{-1}\left (\frac {\sin (x)}{2-\cos (x)}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 2657} \[ x+2 \tan ^{-1}\left (\frac {\sin (x)}{2-\cos (x)}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2657
Rubi steps
\begin {align*} \int \frac {3}{5-4 \cos (x)} \, dx &=3 \int \frac {1}{5-4 \cos (x)} \, dx\\ &=x+2 \tan ^{-1}\left (\frac {\sin (x)}{2-\cos (x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 11, normalized size = 0.69 \[ 2 \tan ^{-1}\left (3 \tan \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 15, normalized size = 0.94 \[ -\arctan \left (\frac {5 \, \cos \relax (x) - 4}{3 \, \sin \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 14, normalized size = 0.88 \[ x - 2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) - 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 10, normalized size = 0.62 \[ 2 \arctan \left (3 \tan \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 13, normalized size = 0.81 \[ 2 \, \arctan \left (\frac {3 \, \sin \relax (x)}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 18, normalized size = 1.12 \[ x+2\,\mathrm {atan}\left (3\,\mathrm {tan}\left (\frac {x}{2}\right )\right )-2\,\mathrm {atan}\left (\mathrm {tan}\left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 22, normalized size = 1.38 \[ 2 \operatorname {atan}{\left (3 \tan {\left (\frac {x}{2} \right )} \right )} + 2 \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \]
Verification of antiderivative is not currently implemented for this CAS.
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