3.1.37 \(\int \frac {1}{9 \cos ^2(x)+4 \sin ^2(x)} \, dx\) [37]

Optimal. Leaf size=24 \[ \frac {x}{6}-\frac {1}{6} \arctan \left (\frac {\cos (x) \sin (x)}{2+\cos ^2(x)}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {209} \begin {gather*} \frac {x}{6}-\frac {1}{6} \arctan \left (\frac {\sin (x) \cos (x)}{\cos ^2(x)+2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(9*Cos[x]^2 + 4*Sin[x]^2)^(-1),x]

[Out]

x/6 - ArcTan[(Cos[x]*Sin[x])/(2 + Cos[x]^2)]/6

Rule 209

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

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\text {Subst}\left (\int \frac {1}{9+4 x^2} \, dx,x,\tan (x)\right )\\ &=\frac {x}{6}-\frac {1}{6} \arctan \left (\frac {\cos (x) \sin (x)}{2+\cos ^2(x)}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 11, normalized size = 0.46 \begin {gather*} \frac {1}{6} \arctan \left (\frac {2 \tan (x)}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9*Cos[x]^2 + 4*Sin[x]^2)^(-1),x]

[Out]

ArcTan[(2*Tan[x])/3]/6

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Maple [A]
time = 0.22, size = 8, normalized size = 0.33

method	result	size
default	\(\frac {\arctan \left (\frac {2 \tan \left (x \right )}{3}\right )}{6}\)	\(8\)
risch	\(-\frac {i \ln \left ({\mathrm e}^{2 i x}+\frac {1}{5}\right )}{12}+\frac {i \ln \left ({\mathrm e}^{2 i x}+5\right )}{12}\)	\(24\)
parallelrisch	\(-\frac {i \left (\ln \left (\frac {-2 i \sin \left (x \right )-3 \cos \left (x \right )}{1+\cos \left (x \right )}\right )-\ln \left (\frac {2 i \sin \left (x \right )-3 \cos \left (x \right )}{1+\cos \left (x \right )}\right )\right )}{12}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(9*cos(x)^2+4*sin(x)^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.47, size = 7, normalized size = 0.29 \begin {gather*} \frac {1}{6} \, \arctan \left (\frac {2}{3} \, \tan \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*cos(x)^2+4*sin(x)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.61, size = 21, normalized size = 0.88 \begin {gather*} -\frac {1}{12} \, \arctan \left (\frac {13 \, \cos \left (x\right )^{2} - 4}{12 \, \cos \left (x\right ) \sin \left (x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*cos(x)^2+4*sin(x)^2),x, algorithm="fricas")

[Out]

-1/12*arctan(1/12*(13*cos(x)^2 - 4)/(cos(x)*sin(x)))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{4 \sin ^{2}{\left (x \right )} + 9 \cos ^{2}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*cos(x)**2+4*sin(x)**2),x)

[Out]

Integral(1/(4*sin(x)**2 + 9*cos(x)**2), x)

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Giac [A]
time = 0.47, size = 20, normalized size = 0.83 \begin {gather*} \frac {1}{6} \, x - \frac {1}{6} \, \arctan \left (\frac {\sin \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*cos(x)^2+4*sin(x)^2),x, algorithm="giac")

[Out]

1/6*x - 1/6*arctan(sin(2*x)/(cos(2*x) + 5))

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Mupad [B]
time = 0.22, size = 16, normalized size = 0.67 \begin {gather*} \frac {x}{6}-\frac {\mathrm {atan}\left (\mathrm {tan}\left (x\right )\right )}{6}+\frac {\mathrm {atan}\left (\frac {2\,\mathrm {tan}\left (x\right )}{3}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(9*cos(x)^2 + 4*sin(x)^2),x)

[Out]

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

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Chatgpt [F] Failed to verify
time = 1.00, size = 10, normalized size = 0.42 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {5}\, \sin \left (x \right )}{3}\right )}{3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(1/(9*cos(x)^2+4*sin(x)^2),x)

[Out]

1/3*arctan(1/3*5^(1/2)*sin(x))

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