3.178 \(\int \frac{1}{\sqrt{a-a \cos (x)}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \sin (x)}{\sqrt{2} \sqrt{a-a \cos (x)}}\right )}{\sqrt{a}} \]

[Out]

-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[a - a*Cos[x]])])/Sqrt[a])

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Rubi [A]  time = 0.0204732, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2649, 206} \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \sin (x)}{\sqrt{2} \sqrt{a-a \cos (x)}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a - a*Cos[x]],x]

[Out]

-((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[a - a*Cos[x]])])/Sqrt[a])

Rule 2649

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a-a \cos (x)}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \sin (x)}{\sqrt{a-a \cos (x)}}\right )\right )\\ &=-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \sin (x)}{\sqrt{2} \sqrt{a-a \cos (x)}}\right )}{\sqrt{a}}\\ \end{align*}

Mathematica [A]  time = 0.0150268, size = 36, normalized size = 0.97 \[ \frac{2 \sin \left (\frac{x}{2}\right ) \left (\log \left (\sin \left (\frac{x}{4}\right )\right )-\log \left (\cos \left (\frac{x}{4}\right )\right )\right )}{\sqrt{a-a \cos (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a - a*Cos[x]],x]

[Out]

(2*(-Log[Cos[x/4]] + Log[Sin[x/4]])*Sin[x/2])/Sqrt[a - a*Cos[x]]

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Maple [A]  time = 0.889, size = 25, normalized size = 0.7 \begin{align*} -{\sqrt{2}\sin \left ({\frac{x}{2}} \right ){\it Artanh} \left ( \cos \left ({\frac{x}{2}} \right ) \right ){\frac{1}{\sqrt{a \left ( \sin \left ({\frac{x}{2}} \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 2.02647, size = 109, normalized size = 2.95 \begin{align*} -\frac{\sqrt{2} \log \left (\cos \left (\frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right )\right )\right )^{2} + \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right )\right )\right )^{2} + 2 \, \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right )\right )\right ) + 1\right ) - \sqrt{2} \log \left (\cos \left (\frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right )\right )\right )^{2} + \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right )\right )\right )^{2} - 2 \, \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right )\right )\right ) + 1\right )}{2 \, \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(2)*log(cos(1/2*arctan2(sin(x), cos(x)))^2 + sin(1/2*arctan2(sin(x), cos(x)))^2 + 2*cos(1/2*arctan2(
sin(x), cos(x))) + 1) - sqrt(2)*log(cos(1/2*arctan2(sin(x), cos(x)))^2 + sin(1/2*arctan2(sin(x), cos(x)))^2 -
2*cos(1/2*arctan2(sin(x), cos(x))) + 1))/sqrt(a)

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Fricas [A]  time = 1.63184, size = 273, normalized size = 7.38 \begin{align*} \left [\frac{\sqrt{2} \log \left (-\frac{{\left (\cos \left (x\right ) + 3\right )} \sin \left (x\right ) - \frac{2 \, \sqrt{2} \sqrt{-a \cos \left (x\right ) + a}{\left (\cos \left (x\right ) + 1\right )}}{\sqrt{a}}}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right )}{2 \, \sqrt{a}}, \sqrt{2} \sqrt{-\frac{1}{a}} \arctan \left (\frac{\sqrt{2} \sqrt{-a \cos \left (x\right ) + a} \sqrt{-\frac{1}{a}}}{\sin \left (x\right )}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(2)*log(-((cos(x) + 3)*sin(x) - 2*sqrt(2)*sqrt(-a*cos(x) + a)*(cos(x) + 1)/sqrt(a))/((cos(x) - 1)*sin
(x)))/sqrt(a), sqrt(2)*sqrt(-1/a)*arctan(sqrt(2)*sqrt(-a*cos(x) + a)*sqrt(-1/a)/sin(x))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-a*cos(x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-a*cos(x) + a), x)