3.3.36 \(\int \sqrt {1-\sin (x)} \, dx\) [236]

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 = {2725} \begin {gather*} \frac {2 \cos (x)}{\sqrt {1-\sin (x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Sin[x]],x]

[Out]

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(42\) vs. \(2(14)=28\).
time = 0.01, size = 42, normalized size = 3.00 \begin {gather*} \frac {2 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \sqrt {1-\sin (x)}}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )} \end {gather*}

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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Maple [A]
time = 0.09, size = 23, normalized size = 1.64

method	result	size
default	\(-\frac {2 \left (\sin \left (x \right )-1\right ) \left (\sin \left (x \right )+1\right )}{\cos \left (x \right ) \sqrt {1-\sin \left (x \right )}}\)	\(23\)
risch	\(\frac {i \sqrt {2-2 \sin \left (x \right )}\, \sqrt {-i \left (2 i {\mathrm e}^{2 i x}-{\mathrm e}^{3 i x}+{\mathrm e}^{i x}\right )}\, \sqrt {2}\, \left ({\mathrm e}^{i x}-i\right ) \left ({\mathrm e}^{i x}+i\right )}{\left (2 i {\mathrm e}^{i x}-{\mathrm e}^{2 i x}+1\right ) \sqrt {i \left ({\mathrm e}^{3 i x}-2 i {\mathrm e}^{2 i x}-{\mathrm e}^{i x}\right )}}\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).
time = 1.27, size = 26, normalized size = 1.86 \begin {gather*} \frac {2 \, {\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )} \sqrt {-\sin \left (x\right ) + 1}}{\cos \left (x\right ) - \sin \left (x\right ) + 1} \end {gather*}

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

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (12) = 24\).
time = 0.46, size = 35, normalized size = 2.50 \begin {gather*} -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 )} \end {gather*}

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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Mupad [B]
time = 0.14, size = 18, normalized size = 1.29 \begin {gather*} \frac {2\,\sqrt {1-\sin \left (x\right )}\,\left (\sin \left (x\right )+1\right )}{\cos \left (x\right )} \end {gather*}

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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