3.18 \(\int \frac {1}{4-\cos ^2(x)} \, dx\)

Optimal. Leaf size=41 \[ \frac {x}{2 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sin (x) \cos (x)}{\sin ^2(x)+2 \sqrt {3}+3}\right )}{2 \sqrt {3}} \]

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Rubi [A]  time = 0.02, antiderivative size = 41, 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 = {3181, 203} \[ \frac {x}{2 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sin (x) \cos (x)}{\sin ^2(x)+2 \sqrt {3}+3}\right )}{2 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[(4 - Cos[x]^2)^(-1),x]

[Out]

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 3181

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

Rubi steps

\begin {align*} \int \frac {1}{4-\cos ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{4+3 x^2} \, dx,x,\cot (x)\right )\\ &=\frac {x}{2 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\cos (x) \sin (x)}{3+2 \sqrt {3}+\sin ^2(x)}\right )}{2 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 19, normalized size = 0.46 \[ \frac {\tan ^{-1}\left (\frac {2 \tan (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(4 - Cos[x]^2)^(-1),x]

[Out]

ArcTan[(2*Tan[x])/Sqrt[3]]/(2*Sqrt[3])

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{4-\cos ^2(x)} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[(4 - Cos[x]^2)^(-1),x]

[Out]

Could not integrate

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fricas [A]  time = 1.20, size = 31, normalized size = 0.76 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {7 \, \sqrt {3} \cos \relax (x)^{2} - 4 \, \sqrt {3}}{12 \, \cos \relax (x) \sin \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-cos(x)^2),x, algorithm="fricas")

[Out]

-1/12*sqrt(3)*arctan(1/12*(7*sqrt(3)*cos(x)^2 - 4*sqrt(3))/(cos(x)*sin(x)))

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giac [A]  time = 0.83, size = 46, normalized size = 1.12 \[ \frac {1}{6} \, \sqrt {3} {\left (x + \arctan \left (-\frac {\sqrt {3} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt {3} \cos \left (2 \, x\right ) + \sqrt {3} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-cos(x)^2),x, algorithm="giac")

[Out]

1/6*sqrt(3)*(x + arctan(-(sqrt(3)*sin(2*x) - 2*sin(2*x))/(sqrt(3)*cos(2*x) + sqrt(3) - 2*cos(2*x) + 2)))

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maple [A]  time = 0.07, size = 14, normalized size = 0.34

method	result	size
default	\(\frac {\sqrt {3}\, \arctan \left (\frac {2 \tan \relax (x ) \sqrt {3}}{3}\right )}{6}\)	\(14\)
risch	\(\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}-4 \sqrt {3}-7\right )}{12}-\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}+4 \sqrt {3}-7\right )}{12}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4-cos(x)^2),x,method=_RETURNVERBOSE)

[Out]

1/6*3^(1/2)*arctan(2/3*tan(x)*3^(1/2))

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maxima [A]  time = 0.95, size = 13, normalized size = 0.32 \[ \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \tan \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-cos(x)^2),x, algorithm="maxima")

[Out]

1/6*sqrt(3)*arctan(2/3*sqrt(3)*tan(x))

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mupad [B]  time = 0.23, size = 26, normalized size = 0.63 \[ \frac {\sqrt {3}\,\left (x-\mathrm {atan}\left (\mathrm {tan}\relax (x)\right )\right )}{6}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,\mathrm {tan}\relax (x)}{3}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(cos(x)^2 - 4),x)

[Out]

(3^(1/2)*(x - atan(tan(x))))/6 + (3^(1/2)*atan((2*3^(1/2)*tan(x))/3))/6

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sympy [A]  time = 0.68, size = 61, normalized size = 1.49 \[ \frac {\sqrt {3} \left (\operatorname {atan}{\left (\frac {\sqrt {3} \tan {\left (\frac {x}{2} \right )}}{3} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{6} + \frac {\sqrt {3} \left (\operatorname {atan}{\left (\sqrt {3} \tan {\left (\frac {x}{2} \right )} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-cos(x)**2),x)

[Out]

sqrt(3)*(atan(sqrt(3)*tan(x/2)/3) + pi*floor((x/2 - pi/2)/pi))/6 + sqrt(3)*(atan(sqrt(3)*tan(x/2)) + pi*floor(
(x/2 - pi/2)/pi))/6

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