3.18 \(\int \frac{\cot (x)}{\sqrt{3-\cos (x)}} \, dx\)

Optimal. Leaf size=44 \[ -\frac{1}{2} \tanh ^{-1}\left (\frac{1}{2} \sqrt{3-\cos (x)}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{3-\cos (x)}}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

-ArcTanh[Sqrt[3 - Cos[x]]/2]/2 - ArcTanh[Sqrt[3 - Cos[x]]/Sqrt[2]]/Sqrt[2]

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Rubi [A]  time = 0.0494116, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2721, 827, 1166, 206} \[ -\frac{1}{2} \tanh ^{-1}\left (\frac{1}{2} \sqrt{3-\cos (x)}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{3-\cos (x)}}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/Sqrt[3 - Cos[x]],x]

[Out]

-ArcTanh[Sqrt[3 - Cos[x]]/2]/2 - ArcTanh[Sqrt[3 - Cos[x]]/Sqrt[2]]/Sqrt[2]

Rule 2721

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

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{\cot (x)}{\sqrt{3-\cos (x)}} \, dx &=-\operatorname{Subst}\left (\int \frac{x}{\sqrt{3+x} \left (1-x^2\right )} \, dx,x,-\cos (x)\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{-3+x^2}{-8+6 x^2-x^4} \, dx,x,\sqrt{3-\cos (x)}\right )\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{3-\cos (x)}\right )-\operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\sqrt{3-\cos (x)}\right )\\ &=-\frac{1}{2} \tanh ^{-1}\left (\frac{1}{2} \sqrt{3-\cos (x)}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{3-\cos (x)}}{\sqrt{2}}\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0717611, size = 44, normalized size = 1. \[ -\frac{1}{2} \tanh ^{-1}\left (\frac{1}{2} \sqrt{3-\cos (x)}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{3-\cos (x)}}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/Sqrt[3 - Cos[x]],x]

[Out]

-ArcTanh[Sqrt[3 - Cos[x]]/2]/2 - ArcTanh[Sqrt[3 - Cos[x]]/Sqrt[2]]/Sqrt[2]

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Maple [B]  time = 2.803, size = 81, normalized size = 1.8 \begin{align*} -{\frac{\sqrt{2}}{4}{\it Artanh} \left ({ \left ( -\sqrt{2}\cos \left ({\frac{x}{2}} \right ) +2\,\sqrt{2} \right ){\frac{1}{\sqrt{2\, \left ( \sin \left ( x/2 \right ) \right ) ^{2}+2}}}} \right ) }-{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\sqrt{2} \left ( 2+\cos \left ({\frac{x}{2}} \right ) \right ){\frac{1}{\sqrt{2\, \left ( \sin \left ( x/2 \right ) \right ) ^{2}+2}}}} \right ) }-{\frac{1}{2}{\it Artanh} \left ( 2\,{\frac{1}{\sqrt{2\, \left ( \sin \left ( x/2 \right ) \right ) ^{2}+2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(3-cos(x))^(1/2),x)

[Out]

-1/4*2^(1/2)*arctanh((-2^(1/2)*cos(1/2*x)+2*2^(1/2))/(2*sin(1/2*x)^2+2)^(1/2))-1/4*2^(1/2)*arctanh(1/(2*sin(1/
2*x)^2+2)^(1/2)*(2+cos(1/2*x))*2^(1/2))-1/2*arctanh(2/(2*sin(1/2*x)^2+2)^(1/2))

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Maxima [A]  time = 1.60343, size = 85, normalized size = 1.93 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{-\cos \left (x\right ) + 3}}{\sqrt{2} + \sqrt{-\cos \left (x\right ) + 3}}\right ) - \frac{1}{4} \, \log \left (\sqrt{-\cos \left (x\right ) + 3} + 2\right ) + \frac{1}{4} \, \log \left (\sqrt{-\cos \left (x\right ) + 3} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(3-cos(x))^(1/2),x, algorithm="maxima")

[Out]

1/4*sqrt(2)*log(-(sqrt(2) - sqrt(-cos(x) + 3))/(sqrt(2) + sqrt(-cos(x) + 3))) - 1/4*log(sqrt(-cos(x) + 3) + 2)
+ 1/4*log(sqrt(-cos(x) + 3) - 2)

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Fricas [B]  time = 1.60545, size = 247, normalized size = 5.61 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left (\frac{\cos \left (x\right )^{2} + 4 \,{\left (\sqrt{2} \cos \left (x\right ) - 5 \, \sqrt{2}\right )} \sqrt{-\cos \left (x\right ) + 3} - 18 \, \cos \left (x\right ) + 49}{\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1}\right ) + \frac{1}{4} \, \log \left (-\frac{4 \, \sqrt{-\cos \left (x\right ) + 3} + \cos \left (x\right ) - 7}{\cos \left (x\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(3-cos(x))^(1/2),x, algorithm="fricas")

[Out]

1/8*sqrt(2)*log((cos(x)^2 + 4*(sqrt(2)*cos(x) - 5*sqrt(2))*sqrt(-cos(x) + 3) - 18*cos(x) + 49)/(cos(x)^2 - 2*c
os(x) + 1)) + 1/4*log(-(4*sqrt(-cos(x) + 3) + cos(x) - 7)/(cos(x) + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (x \right )}}{\sqrt{3 - \cos{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(3-cos(x))**(1/2),x)

[Out]

Integral(cot(x)/sqrt(3 - cos(x)), x)

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Giac [B]  time = 1.40709, size = 92, normalized size = 2.09 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \sqrt{-\cos \left (x\right ) + 3} \right |}}{2 \,{\left (\sqrt{2} + \sqrt{-\cos \left (x\right ) + 3}\right )}}\right ) - \frac{1}{4} \, \log \left (\sqrt{-\cos \left (x\right ) + 3} + 2\right ) + \frac{1}{4} \, \log \left (-\sqrt{-\cos \left (x\right ) + 3} + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(3-cos(x))^(1/2),x, algorithm="giac")

[Out]

1/4*sqrt(2)*log(1/2*abs(-2*sqrt(2) + 2*sqrt(-cos(x) + 3))/(sqrt(2) + sqrt(-cos(x) + 3))) - 1/4*log(sqrt(-cos(x
) + 3) + 2) + 1/4*log(-sqrt(-cos(x) + 3) + 2)