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

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Rubi [A]  time = 0.04, antiderivative size = 14, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 14, normalized size = 1.00 \[ -2 \sqrt {9-\sin ^2(x)} \]

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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fricas [A]  time = 0.77, size = 10, normalized size = 0.71 \[ -2 \, \sqrt {\cos \relax (x)^{2} + 8} \]

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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giac [A]  time = 0.14, size = 12, normalized size = 0.86 \[ -2 \, \sqrt {-\sin \relax (x)^{2} + 9} \]

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]

-2*sqrt(-sin(x)^2 + 9)

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maple [A]  time = 0.08, size = 13, normalized size = 0.93 \[ -2 \sqrt {9-\left (\sin ^{2}\relax (x )\right )} \]

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 = 0.33, size = 12, normalized size = 0.86 \[ -2 \, \sqrt {-\sin \relax (x)^{2} + 9} \]

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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mupad [B]  time = 0.17, size = 10, normalized size = 0.71 \[ -2\,\sqrt {{\cos \relax (x)}^2+8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x)/(9 - sin(x)^2)^(1/2),x)

[Out]

-2*(cos(x)^2 + 8)^(1/2)

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sympy [A]  time = 1.43, size = 12, normalized size = 0.86 \[ - 2 \sqrt {9 - \sin ^{2}{\relax (x )}} \]

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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