3.7 \(\int \frac{3}{5+4 \sin (x)} \, dx\)

Optimal. Leaf size=14 \[ x+2 \tan ^{-1}\left (\frac{\cos (x)}{\sin (x)+2}\right ) \]

[Out]

x + 2*ArcTan[Cos[x]/(2 + Sin[x])]

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Rubi [A]  time = 0.0180854, 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 \[ x+2 \tan ^{-1}\left (\frac{\cos (x)}{\sin (x)+2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[3/(5 + 4*Sin[x]),x]

[Out]

x + 2*ArcTan[Cos[x]/(2 + Sin[x])]

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Rubi in Sympy [A]  time = 1.16049, size = 14, normalized size = 1. \[ 2 \operatorname{atan}{\left (\frac{5 \tan{\left (\frac{x}{2} \right )}}{3} + \frac{4}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(3/(5+4*sin(x)),x)

[Out]

2*atan(5*tan(x/2)/3 + 4/3)

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Mathematica [B]  time = 0.0155653, 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.

[In]  Integrate[3/(5 + 4*Sin[x]),x]

[Out]

3*(-ArcTan[(2*Cos[x/2] + Sin[x/2])/(Cos[x/2] + 2*Sin[x/2])]/3 + ArcTan[(Cos[x/2]
+ 2*Sin[x/2])/(2*Cos[x/2] + Sin[x/2])]/3)

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Maple [A]  time = 0.019, size = 12, normalized size = 0.9 \[ 2\,\arctan \left ( 5/3\,\tan \left ( x/2 \right ) +4/3 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(3/(5+4*sin(x)),x)

[Out]

2*arctan(5/3*tan(1/2*x)+4/3)

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Maxima [A]  time = 1.47763, size = 20, normalized size = 1.43 \[ 2 \, \arctan \left (\frac{5 \, \sin \left (x\right )}{3 \,{\left (\cos \left (x\right ) + 1\right )}} + \frac{4}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(3/(4*sin(x) + 5),x, algorithm="maxima")

[Out]

2*arctan(5/3*sin(x)/(cos(x) + 1) + 4/3)

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Fricas [A]  time = 0.23477, size = 18, normalized size = 1.29 \[ \arctan \left (\frac{5 \, \sin \left (x\right ) + 4}{3 \, \cos \left (x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(3/(4*sin(x) + 5),x, algorithm="fricas")

[Out]

arctan(1/3*(5*sin(x) + 4)/cos(x))

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Sympy [A]  time = 0.427045, size = 27, normalized size = 1.93 \[ 2 \operatorname{atan}{\left (\frac{5 \tan{\left (\frac{x}{2} \right )}}{3} + \frac{4}{3} \right )} + 2 \pi \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\rfloor \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(3/(5+4*sin(x)),x)

[Out]

2*atan(5*tan(x/2)/3 + 4/3) + 2*pi*floor((x/2 - pi/2)/pi)

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GIAC/XCAS [A]  time = 0.202492, size = 34, normalized size = 2.43 \[ x + 2 \, \arctan \left (-\frac{2 \, \cos \left (x\right ) + \sin \left (x\right ) + 2}{\cos \left (x\right ) - 2 \, \sin \left (x\right ) - 4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(3/(4*sin(x) + 5),x, algorithm="giac")

[Out]

x + 2*arctan(-(2*cos(x) + sin(x) + 2)/(cos(x) - 2*sin(x) - 4))