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

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

[Out]

(2*Sin[x])/Sqrt[1 + Cos[x]]

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Rubi [A]  time = 0.007023, antiderivative size = 12, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0062206, 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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Maple [B]  time = 0.027, size = 22, normalized size = 1.8 \begin{align*} 2\,{\frac{\cos \left ( x/2 \right ) \sin \left ( x/2 \right ) \sqrt{2}}{\sqrt{ \left ( \cos \left ( x/2 \right ) \right ) ^{2}}}} \end{align*}

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 = 1.589, size = 12, normalized size = 1. \begin{align*} 2 \, \sqrt{2} \sin \left (\frac{1}{2} \, x\right ) \end{align*}

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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Fricas [A]  time = 1.49828, size = 36, normalized size = 3. \begin{align*} \frac{2 \, \sin \left (x\right )}{\sqrt{\cos \left (x\right ) + 1}} \end{align*}

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

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\cos \left (x\right ) + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(cos(x) + 1), x)