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

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Rubi [A]  time = 0.03, antiderivative size = 28, 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 = {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 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]

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 \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*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 1.00 \[ 2 \sin ^{-1}\left (\frac {\sin (x)}{2}\right )+\frac {1}{2} \sin (x) \sqrt {4-\sin ^2(x)} \]

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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fricas [B]  time = 1.07, size = 61, normalized size = 2.18 \[ \frac {1}{2} \, \sqrt {\cos \relax (x)^{2} + 3} \sin \relax (x) + \arctan \left (\frac {\sqrt {\cos \relax (x)^{2} + 3} {\left (\cos \relax (x)^{2} + 1\right )} \sin \relax (x) - 4 \, \cos \relax (x) \sin \relax (x)}{\cos \relax (x)^{4} + 6 \, \cos \relax (x)^{2} - 3}\right ) + \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right ) \]

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*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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giac [A]  time = 0.14, size = 22, normalized size = 0.79 \[ \frac {1}{2} \, \sqrt {-\sin \relax (x)^{2} + 4} \sin \relax (x) + 2 \, \arcsin \left (\frac {1}{2} \, \sin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 23, normalized size = 0.82 \[ 2 \arcsin \left (\frac {\sin \relax (x )}{2}\right )+\frac {\sin \relax (x ) \sqrt {4-\left (\sin ^{2}\relax (x )\right )}}{2} \]

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 = 0.46, size = 22, normalized size = 0.79 \[ \frac {1}{2} \, \sqrt {-\sin \relax (x)^{2} + 4} \sin \relax (x) + 2 \, \arcsin \left (\frac {1}{2} \, \sin \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.97, size = 20, normalized size = 0.71 \[ 2\,\mathrm {asin}\left (\frac {\sin \relax (x)}{2}\right )+\frac {\sin \relax (x)\,\sqrt {{\cos \relax (x)}^2+3}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*(4 - sin(x)^2)^(1/2),x)

[Out]

2*asin(sin(x)/2) + (sin(x)*(cos(x)^2 + 3)^(1/2))/2

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

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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