3.18 \(\int \frac {\cos (x)}{10-6 \sin (x)+\sin ^2(x)} \, dx\)

Optimal. Leaf size=9 \[ -\tan ^{-1}(3-\sin (x)) \]

[Out]

arctan(-3+sin(x))

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Rubi [A]  time = 0.03, antiderivative size = 9, 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 = {3258, 618, 204} \[ -\tan ^{-1}(3-\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/(10 - 6*Sin[x] + Sin[x]^2),x]

[Out]

-ArcTan[3 - Sin[x]]

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 3258

Int[cos[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*sin[(d_.) + (e_.)*(x_)])^(n_.) + (c_.)*((f_.)*sin[(d_.
) + (e_.)*(x_)])^(n2_.))^(p_.), x_Symbol] :> Module[{g = FreeFactors[Sin[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, Sin[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 {\cos (x)}{10-6 \sin (x)+\sin ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{10-6 x+x^2} \, dx,x,\sin (x)\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,-6+2 \sin (x)\right )\right )\\ &=-\tan ^{-1}(3-\sin (x))\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 9, normalized size = 1.00 \[ -\tan ^{-1}(3-\sin (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]/(10 - 6*Sin[x] + Sin[x]^2),x]

[Out]

-ArcTan[3 - Sin[x]]

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fricas [A]  time = 0.44, size = 5, normalized size = 0.56 \[ \arctan \left (\sin \relax (x) - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/(10-6*sin(x)+sin(x)^2),x, algorithm="fricas")

[Out]

arctan(sin(x) - 3)

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giac [A]  time = 0.12, size = 5, normalized size = 0.56 \[ \arctan \left (\sin \relax (x) - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/(10-6*sin(x)+sin(x)^2),x, algorithm="giac")

[Out]

arctan(sin(x) - 3)

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maple [A]  time = 0.22, size = 6, normalized size = 0.67 \[ \arctan \left (-3+\sin \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/(10-6*sin(x)+sin(x)^2),x)

[Out]

arctan(-3+sin(x))

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maxima [A]  time = 0.46, size = 5, normalized size = 0.56 \[ \arctan \left (\sin \relax (x) - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/(10-6*sin(x)+sin(x)^2),x, algorithm="maxima")

[Out]

arctan(sin(x) - 3)

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mupad [B]  time = 0.12, size = 5, normalized size = 0.56 \[ \mathrm {atan}\left (\sin \relax (x)-3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/(sin(x)^2 - 6*sin(x) + 10),x)

[Out]

atan(sin(x) - 3)

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sympy [A]  time = 0.50, size = 5, normalized size = 0.56 \[ \operatorname {atan}{\left (\sin {\relax (x )} - 3 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/(10-6*sin(x)+sin(x)**2),x)

[Out]

atan(sin(x) - 3)

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