Optimal. Leaf size=16 \[ x+2 \tan ^{-1}\left (\frac{\sin (x)}{2-\cos (x)}\right ) \]
[Out]
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Rubi [A] time = 0.0234557, antiderivative size = 16, 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 \[ x+2 \tan ^{-1}\left (\frac{\sin (x)}{2-\cos (x)}\right ) \]
Antiderivative was successfully verified.
[In] Int[3/(5 - 4*Cos[x]),x]
[Out]
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Rubi in Sympy [A] time = 1.00328, size = 8, normalized size = 0.5 \[ 2 \operatorname{atan}{\left (3 \tan{\left (\frac{x}{2} \right )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(3/(5-4*cos(x)),x)
[Out]
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Mathematica [A] time = 0.0137327, size = 11, normalized size = 0.69 \[ 2 \tan ^{-1}\left (3 \tan \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[3/(5 - 4*Cos[x]),x]
[Out]
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Maple [A] time = 0.016, size = 10, normalized size = 0.6 \[ 2\,\arctan \left ( 3\,\tan \left ( x/2 \right ) \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(3/(5-4*cos(x)),x)
[Out]
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Maxima [A] time = 1.52072, size = 18, normalized size = 1.12 \[ 2 \, \arctan \left (\frac{3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-3/(4*cos(x) - 5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240974, size = 20, normalized size = 1.25 \[ -\arctan \left (\frac{5 \, \cos \left (x\right ) - 4}{3 \, \sin \left (x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-3/(4*cos(x) - 5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.327733, size = 22, normalized size = 1.38 \[ 2 \operatorname{atan}{\left (3 \tan{\left (\frac{x}{2} \right )} \right )} + 2 \pi \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\rfloor \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(3/(5-4*cos(x)),x)
[Out]
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GIAC/XCAS [A] time = 0.203195, size = 19, normalized size = 1.19 \[ x - 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) - 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-3/(4*cos(x) - 5),x, algorithm="giac")
[Out]