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

Optimal. Leaf size=14 \[ \frac {2 \cos (x)}{\sqrt {1-\sin (x)}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2646} \[ \frac {2 \cos (x)}{\sqrt {1-\sin (x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Sin[x]],x]

[Out]

(2*Cos[x])/Sqrt[1 - Sin[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 (x)} \, dx &=\frac {2 \cos (x)}{\sqrt {1-\sin (x)}}\\ \end {align*}

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Mathematica [B]  time = 0.01, size = 42, normalized size = 3.00 \[ \frac {2 \sqrt {1-\sin (x)} \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - Sin[x]],x]

[Out]

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

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fricas [B]  time = 0.43, size = 26, normalized size = 1.86 \[ \frac {2 \, {\left (\cos \relax (x) + \sin \relax (x) + 1\right )} \sqrt {-\sin \relax (x) + 1}}{\cos \relax (x) - \sin \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.20, size = 35, normalized size = 2.50 \[ -2 \, \sqrt {2} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, x\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )\right ) - \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 23, normalized size = 1.64 \[ -\frac {2 \left (\sin \relax (x )-1\right ) \left (\sin \relax (x )+1\right )}{\sqrt {-\sin \relax (x )+1}\, \cos \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-\sin \relax (x) + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-sin(x) + 1), x)

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mupad [B]  time = 0.14, size = 18, normalized size = 1.29 \[ \frac {2\,\sqrt {1-\sin \relax (x)}\,\left (\sin \relax (x)+1\right )}{\cos \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - sin(x))^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {1 - \sin {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(1 - sin(x)), x)

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