3.378 \(\int \frac{1}{4-3 \cos ^2(x)+5 \sin ^2(x)} \, dx\)

Optimal. Leaf size=27 \[ \frac{x}{3}+\frac{1}{3} \tan ^{-1}\left (\frac{2 \sin (x) \cos (x)}{2 \sin ^2(x)+1}\right ) \]

[Out]

x/3 + ArcTan[(2*Cos[x]*Sin[x])/(1 + 2*Sin[x]^2)]/3

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Rubi [A]  time = 0.0364624, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{x}{3}+\frac{1}{3} \tan ^{-1}\left (\frac{2 \sin (x) \cos (x)}{2 \sin ^2(x)+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(4 - 3*Cos[x]^2 + 5*Sin[x]^2)^(-1),x]

[Out]

x/3 + ArcTan[(2*Cos[x]*Sin[x])/(1 + 2*Sin[x]^2)]/3

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Rubi in Sympy [A]  time = 9.27266, size = 7, normalized size = 0.26 \[ \frac{\operatorname{atan}{\left (3 \tan{\left (x \right )} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(4-3*cos(x)**2+5*sin(x)**2),x)

[Out]

atan(3*tan(x))/3

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Mathematica [A]  time = 0.0159438, size = 9, normalized size = 0.33 \[ \frac{1}{3} \tan ^{-1}(3 \tan (x)) \]

Antiderivative was successfully verified.

[In]  Integrate[(4 - 3*Cos[x]^2 + 5*Sin[x]^2)^(-1),x]

[Out]

ArcTan[3*Tan[x]]/3

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Maple [A]  time = 0.039, size = 8, normalized size = 0.3 \[{\frac{\arctan \left ( 3\,\tan \left ( x \right ) \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(4-3*cos(x)^2+5*sin(x)^2),x)

[Out]

1/3*arctan(3*tan(x))

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Maxima [A]  time = 1.53778, size = 9, normalized size = 0.33 \[ \frac{1}{3} \, \arctan \left (3 \, \tan \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(3*cos(x)^2 - 5*sin(x)^2 - 4),x, algorithm="maxima")

[Out]

1/3*arctan(3*tan(x))

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Fricas [A]  time = 0.222552, size = 28, normalized size = 1.04 \[ -\frac{1}{6} \, \arctan \left (\frac{10 \, \cos \left (x\right )^{2} - 9}{6 \, \cos \left (x\right ) \sin \left (x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(3*cos(x)^2 - 5*sin(x)^2 - 4),x, algorithm="fricas")

[Out]

-1/6*arctan(1/6*(10*cos(x)^2 - 9)/(cos(x)*sin(x)))

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Sympy [A]  time = 30.8507, size = 148, normalized size = 5.48 \[ \frac{\operatorname{atan}{\left (\frac{\tan{\left (\frac{x}{2} \right )}}{\sqrt{- 12 \sqrt{2} + 17}} \right )} + \pi \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\rfloor }{6 \sqrt{2} \sqrt{- 12 \sqrt{2} + 17} + 9 \sqrt{- 12 \sqrt{2} + 17}} + \frac{\sqrt{- 12 \sqrt{2} + 17} \sqrt{12 \sqrt{2} + 17} \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{x}{2} \right )}}{\sqrt{12 \sqrt{2} + 17}} \right )} + \pi \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\rfloor \right )}{6 \sqrt{2} \sqrt{- 12 \sqrt{2} + 17} + 9 \sqrt{- 12 \sqrt{2} + 17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4-3*cos(x)**2+5*sin(x)**2),x)

[Out]

(atan(tan(x/2)/sqrt(-12*sqrt(2) + 17)) + pi*floor((x/2 - pi/2)/pi))/(6*sqrt(2)*s
qrt(-12*sqrt(2) + 17) + 9*sqrt(-12*sqrt(2) + 17)) + sqrt(-12*sqrt(2) + 17)*sqrt(
12*sqrt(2) + 17)*(atan(tan(x/2)/sqrt(12*sqrt(2) + 17)) + pi*floor((x/2 - pi/2)/p
i))/(6*sqrt(2)*sqrt(-12*sqrt(2) + 17) + 9*sqrt(-12*sqrt(2) + 17))

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GIAC/XCAS [A]  time = 0.208632, size = 27, normalized size = 1. \[ \frac{1}{3} \, x - \frac{1}{3} \, \arctan \left (\frac{\sin \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(3*cos(x)^2 - 5*sin(x)^2 - 4),x, algorithm="giac")

[Out]

1/3*x - 1/3*arctan(sin(2*x)/(cos(2*x) - 2))