3.763 \(\int \frac{\cos (2 x)}{8+\sin ^2(2 x)} \, dx\)

Optimal. Leaf size=23 \[ \frac{\tan ^{-1}\left (\frac{\sin (2 x)}{2 \sqrt{2}}\right )}{4 \sqrt{2}} \]

[Out]

ArcTan[Sin[2*x]/(2*Sqrt[2])]/(4*Sqrt[2])

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Rubi [A]  time = 0.0218966, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3190, 203} \[ \frac{\tan ^{-1}\left (\frac{\sin (2 x)}{2 \sqrt{2}}\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[2*x]/(8 + Sin[2*x]^2),x]

[Out]

ArcTan[Sin[2*x]/(2*Sqrt[2])]/(4*Sqrt[2])

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

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])

Rubi steps

\begin{align*} \int \frac{\cos (2 x)}{8+\sin ^2(2 x)} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{8+x^2} \, dx,x,\sin (2 x)\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sin (2 x)}{2 \sqrt{2}}\right )}{4 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0127909, size = 20, normalized size = 0.87 \[ \frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[2*x]/(8 + Sin[2*x]^2),x]

[Out]

ArcTan[(Cos[x]*Sin[x])/Sqrt[2]]/(4*Sqrt[2])

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Maple [A]  time = 0.013, size = 16, normalized size = 0.7 \begin{align*}{\frac{\sqrt{2}}{8}\arctan \left ({\frac{\sin \left ( 2\,x \right ) \sqrt{2}}{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(2*x)/(8+sin(2*x)^2),x)

[Out]

1/8*arctan(1/4*sin(2*x)*2^(1/2))*2^(1/2)

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Maxima [A]  time = 1.45835, size = 20, normalized size = 0.87 \begin{align*} \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{4} \, \sqrt{2} \sin \left (2 \, x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)/(8+sin(2*x)^2),x, algorithm="maxima")

[Out]

1/8*sqrt(2)*arctan(1/4*sqrt(2)*sin(2*x))

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Fricas [A]  time = 2.00101, size = 57, normalized size = 2.48 \begin{align*} \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{4} \, \sqrt{2} \sin \left (2 \, x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)/(8+sin(2*x)^2),x, algorithm="fricas")

[Out]

1/8*sqrt(2)*arctan(1/4*sqrt(2)*sin(2*x))

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Sympy [A]  time = 0.318515, size = 19, normalized size = 0.83 \begin{align*} \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sin{\left (2 x \right )}}{4} \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)/(8+sin(2*x)**2),x)

[Out]

sqrt(2)*atan(sqrt(2)*sin(2*x)/4)/8

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Giac [A]  time = 1.08792, size = 20, normalized size = 0.87 \begin{align*} \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{4} \, \sqrt{2} \sin \left (2 \, x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)/(8+sin(2*x)^2),x, algorithm="giac")

[Out]

1/8*sqrt(2)*arctan(1/4*sqrt(2)*sin(2*x))