3.3.38 \(\int \sqrt {1-\cos (x)} \, dx\) [238]

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

[Out]

-2*sin(x)/(1-cos(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 \sin (x)}{\sqrt {1-\cos (x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Cos[x]],x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.29 \begin {gather*} -2 \sqrt {1-\cos (x)} \cot \left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - Cos[x]],x]

[Out]

-2*Sqrt[1 - Cos[x]]*Cot[x/2]

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Maple [A]
time = 0.05, size = 22, normalized size = 1.57

method	result	size
default	\(-\frac {4 \sin \left (\frac {x}{2}\right ) \cos \left (\frac {x}{2}\right ) \sqrt {2}}{\sqrt {2-2 \cos \left (x \right )}}\)	\(22\)
risch	\(-\frac {i \sqrt {2}\, \sqrt {-\left ({\mathrm e}^{i x}-1\right )^{2} {\mathrm e}^{-i x}}\, \left (1+{\mathrm e}^{i x}\right )}{{\mathrm e}^{i x}-1}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 1.93, size = 20, normalized size = 1.43 \begin {gather*} -\frac {2 \, \sqrt {2}}{\sqrt {\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.82, size = 18, normalized size = 1.29 \begin {gather*} -\frac {2 \, {\left (\cos \left (x\right ) + 1\right )} \sqrt {-\cos \left (x\right ) + 1}}{\sin \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(1 - cos(x)), x)

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Giac [A]
time = 0.48, size = 23, normalized size = 1.64 \begin {gather*} -2 \, \sqrt {2} {\left (\cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) - \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.03, size = 12, normalized size = 0.86 \begin {gather*} -\frac {2\,\sin \left (x\right )}{\sqrt {1-\cos \left (x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - cos(x))^(1/2),x)

[Out]

-(2*sin(x))/(1 - cos(x))^(1/2)

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