3.153 \(\int \sqrt{a+a \cos (x)} \, dx\)

Optimal. Leaf size=15 \[ \frac{2 a \sin (x)}{\sqrt{a \cos (x)+a}} \]

[Out]

(2*a*Sin[x])/Sqrt[a + a*Cos[x]]

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Rubi [A]  time = 0.0111368, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2646} \[ \frac{2 a \sin (x)}{\sqrt{a \cos (x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Cos[x]],x]

[Out]

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

Mathematica [A]  time = 0.0082106, size = 18, normalized size = 1.2 \[ 2 \tan \left (\frac{x}{2}\right ) \sqrt{a (\cos (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Cos[x]],x]

[Out]

2*Sqrt[a*(1 + Cos[x])]*Tan[x/2]

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Maple [A]  time = 0.373, size = 25, normalized size = 1.7 \begin{align*} 2\,{\frac{a\cos \left ( x/2 \right ) \sin \left ( x/2 \right ) \sqrt{2}}{\sqrt{ \left ( \cos \left ( x/2 \right ) \right ) ^{2}a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.38141, size = 16, normalized size = 1.07 \begin{align*} 2 \, \sqrt{2} \sqrt{a} \sin \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.56093, size = 57, normalized size = 3.8 \begin{align*} \frac{2 \, \sqrt{a \cos \left (x\right ) + a} \sin \left (x\right )}{\cos \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*cos(x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*cos(x) + a), x)