3.1.68 \(\int \frac {\cos (5+3 x)}{\sqrt {3+\cos ^2(5+3 x)}} \, dx\) [68]

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

[Out]

1/3*arcsin(1/2*sin(5+3*x))

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Rubi [A]
time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = {3265, 222} \begin {gather*} \frac {1}{3} \text {ArcSin}\left (\frac {1}{2} \sin (3 x+5)\right ) \end {gather*}

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 222

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 3265

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]

Rubi steps

\begin {align*} \int \frac {\cos (5+3 x)}{\sqrt {3+\cos ^2(5+3 x)}} \, dx &=\frac {1}{3} \text {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*}

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Mathematica [A]
time = 0.03, size = 15, normalized size = 1.00 \begin {gather*} \frac {1}{3} \text {ArcSin}\left (\frac {1}{2} \sin (5+3 x)\right ) \end {gather*}

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(11)=22\).
time = 0.32, size = 57, normalized size = 3.80

method	result	size
default	\(\frac {\sqrt {\left (3+\cos ^{2}\left (5+3 x \right )\right ) \left (\sin ^{2}\left (5+3 x \right )\right )}\, \arcsin \left (-1+\frac {\left (\sin ^{2}\left (5+3 x \right )\right )}{2}\right )}{6 \sin \left (5+3 x \right ) \sqrt {3+\cos ^{2}\left (5+3 x \right )}}\)	\(57\)

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,method=_RETURNVERBOSE)

[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)

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Maxima [A]
time = 0.47, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{3} \, \arcsin \left (\frac {1}{2} \, \sin \left (3 \, x + 5\right )\right ) \end {gather*}

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))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (11) = 22\).
time = 0.39, size = 89, normalized size = 5.93 \begin {gather*} \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 {gather*}

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))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (3 x + 5 \right )}}{\sqrt {\cos ^{2}{\left (3 x + 5 \right )} + 3}}\, dx \end {gather*}

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]

Integral(cos(3*x + 5)/sqrt(cos(3*x + 5)**2 + 3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

integrate(cos(3*x + 5)/sqrt(cos(3*x + 5)^2 + 3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {\cos \left (3\,x+5\right )}{\sqrt {{\cos \left (3\,x+5\right )}^2+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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