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]

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

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Rubi [A]  time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.68, size = 87, normalized size = 2.35 \[ \left [\frac {\sqrt {2} \log \left (-\frac {{\left (\cos \relax (x) + 3\right )} \sin \relax (x) - \frac {2 \, \sqrt {2} \sqrt {-a \cos \relax (x) + a} {\left (\cos \relax (x) + 1\right )}}{\sqrt {a}}}{{\left (\cos \relax (x) - 1\right )} \sin \relax (x)}\right )}{2 \, \sqrt {a}}, \sqrt {2} \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {-a \cos \relax (x) + a} \sqrt {-\frac {1}{a}}}{\sin \relax (x)}\right )\right ] \]

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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giac [A]  time = 0.49, size = 20, normalized size = 0.54 \[ \frac {\sqrt {2} \log \left ({\left | \tan \left (\frac {1}{4} \, x\right ) \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*log(abs(tan(1/4*x)))/(sqrt(a)*sgn(sin(1/2*x)))

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maple [A]  time = 0.18, size = 25, normalized size = 0.68 \[ -\frac {\sin \left (\frac {x}{2}\right ) \arctanh \left (\cos \left (\frac {x}{2}\right )\right ) \sqrt {2}}{\sqrt {a \left (\sin ^{2}\left (\frac {x}{2}\right )\right )}} \]

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 = 1.93, size = 81, normalized size = 2.19 \[ -\frac {\sqrt {2} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x)\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x)\right )\right )^{2} + 2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x)\right )\right ) + 1\right ) - \sqrt {2} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x)\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x)\right )\right )^{2} - 2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x)\right )\right ) + 1\right )}{2 \, \sqrt {a}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {a-a\,\cos \relax (x)}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - a*cos(x))^(1/2),x)

[Out]

int(1/(a - a*cos(x))^(1/2), x)

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

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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