3.11 \(\int \frac {\sin (x)}{3-2 \cos (x)+\cos ^2(x)} \, dx\)

Optimal. Leaf size=19 \[ \frac {\tan ^{-1}\left (\frac {1-\cos (x)}{\sqrt {2}}\right )}{\sqrt {2}} \]

[Out]

1/2*arctan(1/2*(1-cos(x))*2^(1/2))*2^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3259, 618, 204} \[ \frac {\tan ^{-1}\left (\frac {1-\cos (x)}{\sqrt {2}}\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(3 - 2*Cos[x] + Cos[x]^2),x]

[Out]

ArcTan[(1 - Cos[x])/Sqrt[2]]/Sqrt[2]

Rule 204

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 3259

Int[((a_.) + (b_.)*(cos[(d_.) + (e_.)*(x_)]*(f_.))^(n_.) + (c_.)*(cos[(d_.) + (e_.)*(x_)]*(f_.))^(n2_.))^(p_.)
*sin[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :> Module[{g = FreeFactors[Cos[d + e*x], x]}, -Dist[g/e, Subst[Int[(
1 - g^2*x^2)^((m - 1)/2)*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p, x], x, Cos[d + e*x]/g], x]] /; FreeQ[{a, b, c,
d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\sin (x)}{3-2 \cos (x)+\cos ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{3-2 x+x^2} \, dx,x,\cos (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,-2+2 \cos (x)\right )\\ &=\frac {\tan ^{-1}\left (\frac {1-\cos (x)}{\sqrt {2}}\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 18, normalized size = 0.95 \[ -\frac {\tan ^{-1}\left (\frac {\cos (x)-1}{\sqrt {2}}\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(3 - 2*Cos[x] + Cos[x]^2),x]

[Out]

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

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fricas [A]  time = 0.76, size = 19, normalized size = 1.00 \[ -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \cos \relax (x) - \frac {1}{2} \, \sqrt {2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(3-2*cos(x)+cos(x)^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.60, size = 15, normalized size = 0.79 \[ -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\cos \relax (x) - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(3-2*cos(x)+cos(x)^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.07, size = 18, normalized size = 0.95 \[ -\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \cos \relax (x )\right ) \sqrt {2}}{4}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(3-2*cos(x)+cos(x)^2),x)

[Out]

-1/2*2^(1/2)*arctan(1/4*(-2+2*cos(x))*2^(1/2))

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maxima [A]  time = 0.85, size = 15, normalized size = 0.79 \[ -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\cos \relax (x) - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(3-2*cos(x)+cos(x)^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 15, normalized size = 0.79 \[ -\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\left (\cos \relax (x)-1\right )}{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(cos(x)^2 - 2*cos(x) + 3),x)

[Out]

-(2^(1/2)*atan((2^(1/2)*(cos(x) - 1))/2))/2

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sympy [A]  time = 0.27, size = 26, normalized size = 1.37 \[ - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \cos {\relax (x )}}{2} - \frac {\sqrt {2}}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(3-2*cos(x)+cos(x)**2),x)

[Out]

-sqrt(2)*atan(sqrt(2)*cos(x)/2 - sqrt(2)/2)/2

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