3.905 \(\int \frac{\sin (2 x)}{\sqrt{9-\sin ^2(x)}} \, dx\)

Optimal. Leaf size=14 \[ -2 \sqrt{9-\sin ^2(x)} \]

[Out]

-2*Sqrt[9 - Sin[x]^2]

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Rubi [A]  time = 0.0379825, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {12, 261} \[ -2 \sqrt{9-\sin ^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[2*x]/Sqrt[9 - Sin[x]^2],x]

[Out]

-2*Sqrt[9 - Sin[x]^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\sin (2 x)}{\sqrt{9-\sin ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{2 x}{\sqrt{9-x^2}} \, dx,x,\sin (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{9-x^2}} \, dx,x,\sin (x)\right )\\ &=-2 \sqrt{9-\sin ^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.0151059, size = 14, normalized size = 1. \[ -2 \sqrt{9-\sin ^2(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[2*x]/Sqrt[9 - Sin[x]^2],x]

[Out]

-2*Sqrt[9 - Sin[x]^2]

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Maple [A]  time = 0.018, size = 13, normalized size = 0.9 \begin{align*} -2\,\sqrt{9- \left ( \sin \left ( x \right ) \right ) ^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x)/(9-sin(x)^2)^(1/2),x)

[Out]

-2*(9-sin(x)^2)^(1/2)

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Maxima [A]  time = 0.943657, size = 16, normalized size = 1.14 \begin{align*} -2 \, \sqrt{-\sin \left (x\right )^{2} + 9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/(9-sin(x)^2)^(1/2),x, algorithm="maxima")

[Out]

-2*sqrt(-sin(x)^2 + 9)

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Fricas [A]  time = 2.13119, size = 31, normalized size = 2.21 \begin{align*} -2 \, \sqrt{\cos \left (x\right )^{2} + 8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/(9-sin(x)^2)^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(cos(x)^2 + 8)

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Sympy [A]  time = 1.26045, size = 12, normalized size = 0.86 \begin{align*} - 2 \sqrt{9 - \sin ^{2}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/(9-sin(x)**2)**(1/2),x)

[Out]

-2*sqrt(9 - sin(x)**2)

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Giac [B]  time = 1.16975, size = 136, normalized size = 9.71 \begin{align*} -\frac{8 \,{\left (3 \, \tan \left (\frac{1}{2} \, x\right )^{2} - \sqrt{9 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 14 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 9} - 3\right )}}{{\left (3 \, \tan \left (\frac{1}{2} \, x\right )^{2} - \sqrt{9 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 14 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 9}\right )}^{2} + 18 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 6 \, \sqrt{9 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 14 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 9} + 5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/(9-sin(x)^2)^(1/2),x, algorithm="giac")

[Out]

-8*(3*tan(1/2*x)^2 - sqrt(9*tan(1/2*x)^4 + 14*tan(1/2*x)^2 + 9) - 3)/((3*tan(1/2*x)^2 - sqrt(9*tan(1/2*x)^4 +
14*tan(1/2*x)^2 + 9))^2 + 18*tan(1/2*x)^2 - 6*sqrt(9*tan(1/2*x)^4 + 14*tan(1/2*x)^2 + 9) + 5)