3.115 \(\int \frac{\sin (7+3 x)}{\sqrt{3+\sin ^2(7+3 x)}} \, dx\)

Optimal. Leaf size=15 \[ -\frac{1}{3} \sin ^{-1}\left (\frac{1}{2} \cos (3 x+7)\right ) \]

[Out]

-ArcSin[Cos[7 + 3*x]/2]/3

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Rubi [A]  time = 0.029909, 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} \[ -\frac{1}{3} \sin ^{-1}\left (\frac{1}{2} \cos (3 x+7)\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sin[7 + 3*x]/Sqrt[3 + Sin[7 + 3*x]^2],x]

[Out]

-ArcSin[Cos[7 + 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{\sin (7+3 x)}{\sqrt{3+\sin ^2(7+3 x)}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{4-x^2}} \, dx,x,\cos (7+3 x)\right )\right )\\ &=-\frac{1}{3} \sin ^{-1}\left (\frac{1}{2} \cos (7+3 x)\right )\\ \end{align*}

Mathematica [C]  time = 0.0933488, size = 39, normalized size = 2.6 \[ \frac{1}{3} i \log \left (\sqrt{7-\cos (2 (3 x+7))}+i \sqrt{2} \cos (3 x+7)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[7 + 3*x]/Sqrt[3 + Sin[7 + 3*x]^2],x]

[Out]

(I/3)*Log[I*Sqrt[2]*Cos[7 + 3*x] + Sqrt[7 - Cos[2*(7 + 3*x)]]]

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Maple [B]  time = 0.797, size = 57, normalized size = 3.8 \begin{align*} -{\frac{1}{6\,\cos \left ( 7+3\,x \right ) }\sqrt{ \left ( 3+ \left ( \sin \left ( 7+3\,x \right ) \right ) ^{2} \right ) \left ( \cos \left ( 7+3\,x \right ) \right ) ^{2}}\arcsin \left ( -1+{\frac{ \left ( \cos \left ( 7+3\,x \right ) \right ) ^{2}}{2}} \right ){\frac{1}{\sqrt{3+ \left ( \sin \left ( 7+3\,x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(7+3*x)/(3+sin(7+3*x)^2)^(1/2),x)

[Out]

-1/6*((3+sin(7+3*x)^2)*cos(7+3*x)^2)^(1/2)*arcsin(-1+1/2*cos(7+3*x)^2)/cos(7+3*x)/(3+sin(7+3*x)^2)^(1/2)

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Maxima [A]  time = 1.41911, size = 15, normalized size = 1. \begin{align*} -\frac{1}{3} \, \arcsin \left (\frac{1}{2} \, \cos \left (3 \, x + 7\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(7+3*x)/(3+sin(7+3*x)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/3*arcsin(1/2*cos(3*x + 7))

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Fricas [B]  time = 1.81811, size = 251, normalized size = 16.73 \begin{align*} \frac{1}{6} \, \arctan \left (-\frac{4 \, \cos \left (3 \, x + 7\right ) \sin \left (3 \, x + 7\right ) -{\left (\cos \left (3 \, x + 7\right )^{3} - 2 \, \cos \left (3 \, x + 7\right )\right )} \sqrt{-\cos \left (3 \, x + 7\right )^{2} + 4}}{\cos \left (3 \, x + 7\right )^{4} - 8 \, \cos \left (3 \, x + 7\right )^{2} + 4}\right ) - \frac{1}{6} \, \arctan \left (\frac{\sin \left (3 \, x + 7\right )}{\cos \left (3 \, x + 7\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(7+3*x)/(3+sin(7+3*x)^2)^(1/2),x, algorithm="fricas")

[Out]

1/6*arctan(-(4*cos(3*x + 7)*sin(3*x + 7) - (cos(3*x + 7)^3 - 2*cos(3*x + 7))*sqrt(-cos(3*x + 7)^2 + 4))/(cos(3
*x + 7)^4 - 8*cos(3*x + 7)^2 + 4)) - 1/6*arctan(sin(3*x + 7)/cos(3*x + 7))

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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(sin(7+3*x)/(3+sin(7+3*x)**2)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.30528, size = 15, normalized size = 1. \begin{align*} -\frac{1}{3} \, \arcsin \left (\frac{1}{2} \, \cos \left (3 \, x + 7\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(7+3*x)/(3+sin(7+3*x)^2)^(1/2),x, algorithm="giac")

[Out]

-1/3*arcsin(1/2*cos(3*x + 7))