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

Optimal. Leaf size=28 \[ 2 \sin ^{-1}\left (\frac{\sin (x)}{2}\right )+\frac{1}{2} \sin (x) \sqrt{4-\sin ^2(x)} \]

[Out]

2*ArcSin[Sin[x]/2] + (Sin[x]*Sqrt[4 - Sin[x]^2])/2

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Rubi [A]  time = 0.026376, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3190, 195, 216} \[ 2 \sin ^{-1}\left (\frac{\sin (x)}{2}\right )+\frac{1}{2} \sin (x) \sqrt{4-\sin ^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Sqrt[4 - Sin[x]^2],x]

[Out]

2*ArcSin[Sin[x]/2] + (Sin[x]*Sqrt[4 - Sin[x]^2])/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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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

Mathematica [A]  time = 0.0177558, size = 28, normalized size = 1. \[ 2 \sin ^{-1}\left (\frac{\sin (x)}{2}\right )+\frac{1}{2} \sin (x) \sqrt{4-\sin ^2(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*ArcSin[Sin[x]/2] + (Sin[x]*Sqrt[4 - Sin[x]^2])/2

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Maple [A]  time = 0.023, size = 23, normalized size = 0.8 \begin{align*} 2\,\arcsin \left ( 1/2\,\sin \left ( x \right ) \right ) +{\frac{\sin \left ( x \right ) }{2}\sqrt{4- \left ( \sin \left ( x \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arcsin(1/2*sin(x))+1/2*sin(x)*(4-sin(x)^2)^(1/2)

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Maxima [A]  time = 1.49782, size = 30, normalized size = 1.07 \begin{align*} \frac{1}{2} \, \sqrt{-\sin \left (x\right )^{2} + 4} \sin \left (x\right ) + 2 \, \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]

1/2*sqrt(-sin(x)^2 + 4)*sin(x) + 2*arcsin(1/2*sin(x))

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Fricas [B]  time = 2.22431, size = 208, normalized size = 7.43 \begin{align*} \frac{1}{2} \, \sqrt{\cos \left (x\right )^{2} + 3} \sin \left (x\right ) + \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 ) + \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*sqrt(cos(x)^2 + 3)*sin(x) + 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)) + arctan(sin(x)/cos(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- \left (\sin{\left (x \right )} - 2\right ) \left (\sin{\left (x \right )} + 2\right )} \cos{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09273, size = 30, normalized size = 1.07 \begin{align*} \frac{1}{2} \, \sqrt{-\sin \left (x\right )^{2} + 4} \sin \left (x\right ) + 2 \, \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]

1/2*sqrt(-sin(x)^2 + 4)*sin(x) + 2*arcsin(1/2*sin(x))