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

Optimal. Leaf size=9 \[ \sinh ^{-1}\left (\frac{\sin (x)}{\sqrt{3}}\right ) \]

[Out]

ArcSinh[Sin[x]/Sqrt[3]]

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Rubi [A]  time = 0.0257469, antiderivative size = 9, 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 = {3186, 215} \[ \sinh ^{-1}\left (\frac{\sin (x)}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/Sqrt[4 - Cos[x]^2],x]

[Out]

ArcSinh[Sin[x]/Sqrt[3]]

Rule 3186

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 215

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

Rubi steps

\begin{align*} \int \frac{\cos (x)}{\sqrt{4-\cos ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{3+x^2}} \, dx,x,\sin (x)\right )\\ &=\sinh ^{-1}\left (\frac{\sin (x)}{\sqrt{3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0093868, size = 9, normalized size = 1. \[ \sinh ^{-1}\left (\frac{\sin (x)}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSinh[Sin[x]/Sqrt[3]]

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Maple [B]  time = 0.897, size = 53, normalized size = 5.9 \begin{align*} -{\frac{1}{2\,\sin \left ( x \right ) }\sqrt{- \left ( -4+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \left ( \sin \left ( x \right ) \right ) ^{2}}\ln \left ( - \left ( \sin \left ( x \right ) \right ) ^{2}+\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{4}+3\, \left ( \sin \left ( x \right ) \right ) ^{2}}-{\frac{3}{2}} \right ){\frac{1}{\sqrt{4- \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-(-4+cos(x)^2)*sin(x)^2)^(1/2)*ln(-sin(x)^2+(sin(x)^4+3*sin(x)^2)^(1/2)-3/2)/sin(x)/(4-cos(x)^2)^(1/2)

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Maxima [A]  time = 1.52196, size = 11, normalized size = 1.22 \begin{align*} \operatorname{arsinh}\left (\frac{1}{3} \, \sqrt{3} \sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(1/3*sqrt(3)*sin(x))

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Fricas [B]  time = 1.66927, size = 119, normalized size = 13.22 \begin{align*} \frac{1}{4} \, \log \left (8 \, \cos \left (x\right )^{4} - 4 \,{\left (2 \, \cos \left (x\right )^{2} - 5\right )} \sqrt{-\cos \left (x\right )^{2} + 4} \sin \left (x\right ) - 40 \, \cos \left (x\right )^{2} + 41\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(8*cos(x)^4 - 4*(2*cos(x)^2 - 5)*sqrt(-cos(x)^2 + 4)*sin(x) - 40*cos(x)^2 + 41)

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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-cos(x)**2)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.23394, size = 22, normalized size = 2.44 \begin{align*} -\log \left (\sqrt{\sin \left (x\right )^{2} + 3} - \sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sqrt(sin(x)^2 + 3) - sin(x))