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]

-1/2*arctanh(1/2*(3-cos(x))^(1/2))-1/2*arctanh(1/2*(3-cos(x))^(1/2)*2^(1/2))*2^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 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 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 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]

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*}

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Mathematica [A]  time = 0.07, size = 44, normalized size = 1.00 \[ -\frac {1}{2} \tanh ^{-1}\left (\frac {1}{2} \sqrt {3-\cos (x)}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3-\cos (x)}}{\sqrt {2}}\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/Sqrt[3 - Cos[x]],x]

[Out]

-1/2*ArcTanh[Sqrt[3 - Cos[x]]/2] - ArcTanh[Sqrt[3 - Cos[x]]/Sqrt[2]]/Sqrt[2]

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fricas [B]  time = 0.86, size = 77, normalized size = 1.75 \[ \frac {1}{8} \, \sqrt {2} \log \left (\frac {\cos \relax (x)^{2} + 4 \, {\left (\sqrt {2} \cos \relax (x) - 5 \, \sqrt {2}\right )} \sqrt {-\cos \relax (x) + 3} - 18 \, \cos \relax (x) + 49}{\cos \relax (x)^{2} - 2 \, \cos \relax (x) + 1}\right ) + \frac {1}{4} \, \log \left (-\frac {4 \, \sqrt {-\cos \relax (x) + 3} + \cos \relax (x) - 7}{\cos \relax (x) + 1}\right ) \]

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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giac [B]  time = 0.69, size = 68, normalized size = 1.55 \[ \frac {1}{4} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {-\cos \relax (x) + 3} \right |}}{2 \, {\left (\sqrt {2} + \sqrt {-\cos \relax (x) + 3}\right )}}\right ) - \frac {1}{4} \, \log \left (\sqrt {-\cos \relax (x) + 3} + 2\right ) + \frac {1}{4} \, \log \left (-\sqrt {-\cos \relax (x) + 3} + 2\right ) \]

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)

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maple [B]  time = 0.49, size = 81, normalized size = 1.84 \[ -\frac {\sqrt {2}\, \arctanh \left (\frac {-\sqrt {2}\, \cos \left (\frac {x}{2}\right )+2 \sqrt {2}}{\sqrt {2 \left (\sin ^{2}\left (\frac {x}{2}\right )\right )+2}}\right )}{4}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2+\cos \left (\frac {x}{2}\right )\right ) \sqrt {2}}{\sqrt {2 \left (\sin ^{2}\left (\frac {x}{2}\right )\right )+2}}\right )}{4}-\frac {\arctanh \left (\frac {2}{\sqrt {2 \left (\sin ^{2}\left (\frac {x}{2}\right )\right )+2}}\right )}{2} \]

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 = 2.54, size = 63, normalized size = 1.43 \[ \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {-\cos \relax (x) + 3}}{\sqrt {2} + \sqrt {-\cos \relax (x) + 3}}\right ) - \frac {1}{4} \, \log \left (\sqrt {-\cos \relax (x) + 3} + 2\right ) + \frac {1}{4} \, \log \left (\sqrt {-\cos \relax (x) + 3} - 2\right ) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {cot}\relax (x)}{\sqrt {3-\cos \relax (x)}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(3 - cos(x))^(1/2),x)

[Out]

int(cot(x)/(3 - cos(x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot {\relax (x )}}{\sqrt {3 - \cos {\relax (x )}}}\, dx \]

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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