3.4.91 \(\int \sqrt {1-\sin (2 x)} \, dx\) [391]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = {2725} \begin {gather*} \frac {\cos (2 x)}{\sqrt {1-\sin (2 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Sin[2*x]],x]

[Out]

Cos[2*x]/Sqrt[1 - Sin[2*x]]

Rule 2725

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*}

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Mathematica [A]
time = 0.01, size = 27, normalized size = 1.59 \begin {gather*} \frac {(\cos (x)+\sin (x)) \sqrt {1-\sin (2 x)}}{\cos (x)-\sin (x)} \end {gather*}

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 = 0.10, size = 31, normalized size = 1.82

method	result	size
default	\(-\frac {\left (\sin \left (2 x \right )-1\right ) \left (1+\sin \left (2 x \right )\right )}{\cos \left (2 x \right ) \sqrt {1-\sin \left (2 x \right )}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (15) = 30\).
time = 0.95, size = 35, normalized size = 2.06 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {1 - \sin {\left (2 x \right )}}\, dx \end {gather*}

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 [A]
time = 1.36, size = 29, normalized size = 1.71 \begin {gather*} -\sqrt {2} {\left (\cos \left (-\frac {1}{4} \, \pi + x\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + x\right )\right ) - \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + x\right )\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*(cos(-1/4*pi + x)*sgn(sin(-1/4*pi + x)) - sgn(sin(-1/4*pi + x)))

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Mupad [B]
time = 0.22, size = 23, normalized size = 1.35 \begin {gather*} \frac {\sqrt {1-\sin \left (2\,x\right )}\,\left (\sin \left (2\,x\right )+1\right )}{\cos \left (2\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - sin(2*x))^(1/2),x)

[Out]

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

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