3.68 \(\int \frac{\cos (5+3 x)}{\sqrt{3+\cos ^2(5+3 x)}} \, dx\)

Optimal. Leaf size=15 \[ \frac{1}{3} \sin ^{-1}\left (\frac{1}{2} \sin (3 x+5)\right ) \]

[Out]

ArcSin[Sin[5 + 3*x]/2]/3

________________________________________________________________________________________

Rubi [A]  time = 0.0272563, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3186, 216} \[ \frac{1}{3} \sin ^{-1}\left (\frac{1}{2} \sin (3 x+5)\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[5 + 3*x]/Sqrt[3 + Cos[5 + 3*x]^2],x]

[Out]

ArcSin[Sin[5 + 3*x]/2]/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 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 (5+3 x)}{\sqrt{3+\cos ^2(5+3 x)}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{4-x^2}} \, dx,x,\sin (5+3 x)\right )\\ &=\frac{1}{3} \sin ^{-1}\left (\frac{1}{2} \sin (5+3 x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0239926, size = 15, normalized size = 1. \[ \frac{1}{3} \sin ^{-1}\left (\frac{1}{2} \sin (3 x+5)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[5 + 3*x]/Sqrt[3 + Cos[5 + 3*x]^2],x]

[Out]

ArcSin[Sin[5 + 3*x]/2]/3

________________________________________________________________________________________

Maple [B]  time = 0.484, size = 57, normalized size = 3.8 \begin{align*}{\frac{1}{6\,\sin \left ( 5+3\,x \right ) }\sqrt{ \left ( 3+ \left ( \cos \left ( 5+3\,x \right ) \right ) ^{2} \right ) \left ( \sin \left ( 5+3\,x \right ) \right ) ^{2}}\arcsin \left ( -1+{\frac{ \left ( \sin \left ( 5+3\,x \right ) \right ) ^{2}}{2}} \right ){\frac{1}{\sqrt{3+ \left ( \cos \left ( 5+3\,x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(5+3*x)/(3+cos(5+3*x)^2)^(1/2),x)

[Out]

1/6*((3+cos(5+3*x)^2)*sin(5+3*x)^2)^(1/2)*arcsin(-1+1/2*sin(5+3*x)^2)/sin(5+3*x)/(3+cos(5+3*x)^2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.45339, size = 15, normalized size = 1. \begin{align*} \frac{1}{3} \, \arcsin \left (\frac{1}{2} \, \sin \left (3 \, x + 5\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5+3*x)/(3+cos(5+3*x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/3*arcsin(1/2*sin(3*x + 5))

________________________________________________________________________________________

Fricas [B]  time = 1.72898, size = 248, normalized size = 16.53 \begin{align*} \frac{1}{6} \, \arctan \left (\frac{\sqrt{\cos \left (3 \, x + 5\right )^{2} + 3}{\left (\cos \left (3 \, x + 5\right )^{2} + 1\right )} \sin \left (3 \, x + 5\right ) - 4 \, \cos \left (3 \, x + 5\right ) \sin \left (3 \, x + 5\right )}{\cos \left (3 \, x + 5\right )^{4} + 6 \, \cos \left (3 \, x + 5\right )^{2} - 3}\right ) + \frac{1}{6} \, \arctan \left (\frac{\sin \left (3 \, x + 5\right )}{\cos \left (3 \, x + 5\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5+3*x)/(3+cos(5+3*x)^2)^(1/2),x, algorithm="fricas")

[Out]

1/6*arctan((sqrt(cos(3*x + 5)^2 + 3)*(cos(3*x + 5)^2 + 1)*sin(3*x + 5) - 4*cos(3*x + 5)*sin(3*x + 5))/(cos(3*x
+ 5)^4 + 6*cos(3*x + 5)^2 - 3)) + 1/6*arctan(sin(3*x + 5)/cos(3*x + 5))

________________________________________________________________________________________

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(5+3*x)/(3+cos(5+3*x)**2)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.41556, size = 15, normalized size = 1. \begin{align*} \frac{1}{3} \, \arcsin \left (\frac{1}{2} \, \sin \left (3 \, x + 5\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5+3*x)/(3+cos(5+3*x)^2)^(1/2),x, algorithm="giac")

[Out]

1/3*arcsin(1/2*sin(3*x + 5))