3.391 \(\int \sqrt{1-\sin (2 x)} \, dx\)

Optimal. Leaf size=17 \[ \frac{\cos (2 x)}{\sqrt{1-\sin (2 x)}} \]

[Out]

Cos[2*x]/Sqrt[1 - Sin[2*x]]

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Rubi [A]  time = 0.0119692, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2646} \[ \frac{\cos (2 x)}{\sqrt{1-\sin (2 x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Sin[2*x]],x]

[Out]

Cos[2*x]/Sqrt[1 - Sin[2*x]]

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \sqrt{1-\sin (2 x)} \, dx &=\frac{\cos (2 x)}{\sqrt{1-\sin (2 x)}}\\ \end{align*}

Mathematica [A]  time = 0.0135429, size = 27, normalized size = 1.59 \[ \frac{\sqrt{1-\sin (2 x)} (\sin (x)+\cos (x))}{\cos (x)-\sin (x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - Sin[2*x]],x]

[Out]

((Cos[x] + Sin[x])*Sqrt[1 - Sin[2*x]])/(Cos[x] - Sin[x])

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Maple [A]  time = 0.033, size = 31, normalized size = 1.8 \begin{align*} -{\frac{ \left ( -1+\sin \left ( 2\,x \right ) \right ) \left ( 1+\sin \left ( 2\,x \right ) \right ) }{\cos \left ( 2\,x \right ) }{\frac{1}{\sqrt{1-\sin \left ( 2\,x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-1+sin(2*x))*(1+sin(2*x))/cos(2*x)/(1-sin(2*x))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-\sin \left (2 \, x\right ) + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-sin(2*x) + 1), x)

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Fricas [B]  time = 1.81164, size = 99, normalized size = 5.82 \begin{align*} \frac{{\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )} \sqrt{-\sin \left (2 \, x\right ) + 1}}{\cos \left (2 \, x\right ) - \sin \left (2 \, x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(cos(2*x) + sin(2*x) + 1)*sqrt(-sin(2*x) + 1)/(cos(2*x) - sin(2*x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{1 - \sin{\left (2 x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(1 - sin(2*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-\sin \left (2 \, x\right ) + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-sin(2*x) + 1), x)