3.8 \(\int \frac{2}{1+\cos ^2(x)} \, dx\)

Optimal. Leaf size=34 \[ \sqrt{2} x-\sqrt{2} \tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0149801, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {12, 3181, 203} \[ \sqrt{2} x-\sqrt{2} \tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[2/(1 + Cos[x]^2),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

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

Mathematica [A]  time = 0.0190037, size = 15, normalized size = 0.44 \[ \sqrt{2} \tan ^{-1}\left (\frac{\tan (x)}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[2/(1 + Cos[x]^2),x]

[Out]

Sqrt[2]*ArcTan[Tan[x]/Sqrt[2]]

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Maple [A]  time = 0.013, size = 13, normalized size = 0.4 \begin{align*} \sqrt{2}\arctan \left ({\frac{\tan \left ( x \right ) \sqrt{2}}{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/(1+cos(x)^2),x)

[Out]

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

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Maxima [A]  time = 1.41877, size = 16, normalized size = 0.47 \begin{align*} \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \tan \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.73621, size = 99, normalized size = 2.91 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{3 \, \sqrt{2} \cos \left (x\right )^{2} - \sqrt{2}}{4 \, \cos \left (x\right ) \sin \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.897785, size = 60, normalized size = 1.76 \begin{align*} \sqrt{2} \left (\operatorname{atan}{\left (\sqrt{2} \tan{\left (\frac{x}{2} \right )} - 1 \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right ) + \sqrt{2} \left (\operatorname{atan}{\left (\sqrt{2} \tan{\left (\frac{x}{2} \right )} + 1 \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(1+cos(x)**2),x)

[Out]

sqrt(2)*(atan(sqrt(2)*tan(x/2) - 1) + pi*floor((x/2 - pi/2)/pi)) + sqrt(2)*(atan(sqrt(2)*tan(x/2) + 1) + pi*fl
oor((x/2 - pi/2)/pi))

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Giac [A]  time = 1.12137, size = 61, normalized size = 1.79 \begin{align*} \sqrt{2}{\left (x + \arctan \left (-\frac{\sqrt{2} \sin \left (2 \, x\right ) - \sin \left (2 \, x\right )}{\sqrt{2} \cos \left (2 \, x\right ) + \sqrt{2} - \cos \left (2 \, x\right ) + 1}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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