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[Sin[x]/2]

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Rubi [A]  time = 0.0249788, antiderivative size = 7, 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, 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 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 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]

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*}

Mathematica [A]  time = 0.0089002, size = 7, normalized size = 1. \[ \sin ^{-1}\left (\frac{\sin (x)}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[Sin[x]/2]

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Maple [A]  time = 0.026, size = 6, normalized size = 0.9 \begin{align*} \arcsin \left ({\frac{\sin \left ( x \right ) }{2}} \right ) \end{align*}

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 = 1.44222, size = 7, normalized size = 1. \begin{align*} \arcsin \left (\frac{1}{2} \, \sin \left (x\right )\right ) \end{align*}

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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Fricas [B]  time = 2.21549, size = 176, normalized size = 25.14 \begin{align*} \frac{1}{2} \, \arctan \left (\frac{\sqrt{\cos \left (x\right )^{2} + 3}{\left (\cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 4 \, \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{4} + 6 \, \cos \left (x\right )^{2} - 3}\right ) + \frac{1}{2} \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right )}\right ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.10703, size = 7, normalized size = 1. \begin{align*} \arcsin \left (\frac{1}{2} \, \sin \left (x\right )\right ) \end{align*}

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))