∫ 1_______∘ a − a cos(x) dx [178]

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

-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 = {2728, 212} \begin {gather*} -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {2} \sqrt {a-a \cos (x)}}\right )}{\sqrt {a}} \end {gather*}

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

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]

/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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]

\begin {align*} \int \frac {1}{\sqrt {a-a \cos (x)}} \, dx &=-\left (2 \text {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 \begin {gather*} \frac {2 \left (-\log \left (\cos \left (\frac {x}{4}\right )\right )+\log \left (\sin \left (\frac {x}{4}\right )\right )\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {a-a \cos (x)}} \end {gather*}

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

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method	result	size

default	\(-\frac {\sin \left (\frac {x}{2}\right ) \arctanh \left (\cos \left (\frac {x}{2}\right )\right ) \sqrt {2}}{\sqrt {a \left (\sin ^{2}\left (\frac {x}{2}\right )\right )}}\)	\(25\)

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

-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] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (28) = 56\).
time = 0.56, size = 81, normalized size = 2.19 \begin {gather*} -\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 {gather*}

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

-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 = 0.38, size = 87, normalized size = 2.35 \begin {gather*} \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 {gather*}

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- a \cos {\left (x \right )} + a}}\, dx \end {gather*}

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Giac [A]
time = 0.46, size = 20, normalized size = 0.54 \begin {gather*} \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 )} \end {gather*}

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

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {a-a\,\cos \left (x\right )}} \,d x \end {gather*}

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3.2.78 \(\int \frac {1}{\sqrt {a-a \cos (x)}} \, dx\) [178]