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

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Rubi [A]  time = 0.01, antiderivative size = 15, 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 = {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*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.20 \[ 2 \tan \left (\frac {x}{2}\right ) \sqrt {a (\cos (x)+1)} \]

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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fricas [A]  time = 0.68, size = 18, normalized size = 1.20 \[ \frac {2 \, \sqrt {a \cos \relax (x) + a} \sin \relax (x)}{\cos \relax (x) + 1} \]

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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giac [A]  time = 1.37, size = 17, normalized size = 1.13 \[ 2 \, \sqrt {2} \sqrt {a} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 25, normalized size = 1.67 \[ \frac {2 a \cos \left (\frac {x}{2}\right ) \sin \left (\frac {x}{2}\right ) \sqrt {2}}{\sqrt {a \left (\cos ^{2}\left (\frac {x}{2}\right )\right )}} \]

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

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maxima [A]  time = 1.48, size = 12, normalized size = 0.80 \[ 2 \, \sqrt {2} \sqrt {a} \sin \left (\frac {1}{2} \, x\right ) \]

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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mupad [B]  time = 0.29, size = 34, normalized size = 2.27 \[ \frac {2\,\sqrt {a}\,\sqrt {\cos \relax (x)+1}\,\left (\cos \relax (x)-1+\sin \relax (x)\,1{}\mathrm {i}\right )}{\cos \relax (x)\,1{}\mathrm {i}-\sin \relax (x)+1{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*cos(x))^(1/2),x)

[Out]

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

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

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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