3.237 \(\int \sqrt {1+\cos (x)} \, dx\)

Optimal. Leaf size=12 \[ \frac {2 \sin (x)}{\sqrt {\cos (x)+1}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2646} \[ \frac {2 \sin (x)}{\sqrt {\cos (x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Cos[x]],x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.33 \[ 2 \sqrt {\cos (x)+1} \tan \left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Cos[x]],x]

[Out]

2*Sqrt[1 + Cos[x]]*Tan[x/2]

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fricas [A]  time = 0.42, size = 10, normalized size = 0.83 \[ \frac {2 \, \sin \relax (x)}{\sqrt {\cos \relax (x) + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sin(x)/sqrt(cos(x) + 1)

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giac [A]  time = 1.04, size = 14, normalized size = 1.17 \[ 2 \, \sqrt {2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 22, normalized size = 1.83 \[ \frac {4 \sqrt {2}\, \cos \left (\frac {x}{2}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {2 \cos \relax (x )+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x)+1)^(1/2),x)

[Out]

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

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maxima [A]  time = 1.16, size = 9, normalized size = 0.75 \[ 2 \, \sqrt {2} \sin \left (\frac {1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(2)*sin(1/2*x)

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mupad [B]  time = 0.15, size = 10, normalized size = 0.83 \[ \frac {2\,\sin \relax (x)}{\sqrt {\cos \relax (x)+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x) + 1)^(1/2),x)

[Out]

(2*sin(x))/(cos(x) + 1)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(cos(x) + 1), x)

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