3.668 \(\int \frac {\cos (x)}{\sqrt {4-\sin ^2(x)}} \, dx\)

Optimal. Leaf size=7 \[ \sin ^{-1}\left (\frac {\sin (x)}{2}\right ) \]

[Out]

arcsin(1/2*sin(x))

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Rubi [A]  time = 0.02, antiderivative size = 7, normalized size of antiderivative = 1.00, 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, 216} \[ \sin ^{-1}\left (\frac {\sin (x)}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/Sqrt[4 - Sin[x]^2],x]

[Out]

ArcSin[Sin[x]/2]

Rule 216

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 7, normalized size = 1.00 \[ \sin ^{-1}\left (\frac {\sin (x)}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]/Sqrt[4 - Sin[x]^2],x]

[Out]

ArcSin[Sin[x]/2]

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fricas [B]  time = 3.02, size = 53, normalized size = 7.57 \[ \frac {1}{2} \, \arctan \left (\frac {\sqrt {\cos \relax (x)^{2} + 3} {\left (\cos \relax (x)^{2} + 1\right )} \sin \relax (x) - 4 \, \cos \relax (x) \sin \relax (x)}{\cos \relax (x)^{4} + 6 \, \cos \relax (x)^{2} - 3}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan((sqrt(cos(x)^2 + 3)*(cos(x)^2 + 1)*sin(x) - 4*cos(x)*sin(x))/(cos(x)^4 + 6*cos(x)^2 - 3)) + 1/2*arc
tan(sin(x)/cos(x))

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giac [A]  time = 0.15, size = 5, normalized size = 0.71 \[ \arcsin \left (\frac {1}{2} \, \sin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(1/2*sin(x))

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maple [A]  time = 0.08, size = 6, normalized size = 0.86 \[ \arcsin \left (\frac {\sin \relax (x )}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/(4-sin(x)^2)^(1/2),x)

[Out]

arcsin(1/2*sin(x))

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maxima [A]  time = 0.42, size = 5, normalized size = 0.71 \[ \arcsin \left (\frac {1}{2} \, \sin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(1/2*sin(x))

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mupad [B]  time = 2.98, size = 5, normalized size = 0.71 \[ \mathrm {asin}\left (\frac {\sin \relax (x)}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/(4 - sin(x)^2)^(1/2),x)

[Out]

asin(sin(x)/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\relax (x )}}{\sqrt {- \left (\sin {\relax (x )} - 2\right ) \left (\sin {\relax (x )} + 2\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(x)/sqrt(-(sin(x) - 2)*(sin(x) + 2)), x)

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