3.43 \(\int \frac{1}{\sqrt{-1+\cot ^2(x)}} \, dx\)

Optimal. Leaf size=26 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \cot (x)}{\sqrt{\cot ^2(x)-1}}\right )}{\sqrt{2}} \]

[Out]

-(ArcTanh[(Sqrt[2]*Cot[x])/Sqrt[-1 + Cot[x]^2]]/Sqrt[2])

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Rubi [A]  time = 0.0172601, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3661, 377, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \cot (x)}{\sqrt{\cot ^2(x)-1}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-1 + Cot[x]^2],x]

[Out]

-(ArcTanh[(Sqrt[2]*Cot[x])/Sqrt[-1 + Cot[x]^2]]/Sqrt[2])

Rule 3661

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[(c*ff)/f, Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

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

Mathematica [A]  time = 0.0333064, size = 45, normalized size = 1.73 \[ -\frac{\sqrt{\cos (2 x)} \csc (x) \log \left (\sqrt{2} \cos (x)+\sqrt{\cos (2 x)}\right )}{\sqrt{2} \sqrt{\cot ^2(x)-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-1 + Cot[x]^2],x]

[Out]

-((Sqrt[Cos[2*x]]*Csc[x]*Log[Sqrt[2]*Cos[x] + Sqrt[Cos[2*x]]])/(Sqrt[2]*Sqrt[-1 + Cot[x]^2]))

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Maple [A]  time = 0.025, size = 21, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\cot \left ( x \right ) \sqrt{2}{\frac{1}{\sqrt{-1+ \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-1+cot(x)^2)^(1/2),x)

[Out]

-1/2*arctanh(cot(x)*2^(1/2)/(-1+cot(x)^2)^(1/2))*2^(1/2)

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Maxima [B]  time = 1.69421, size = 193, normalized size = 7.42 \begin{align*} -\frac{1}{8} \, \sqrt{2}{\left (2 \, \operatorname{arsinh}\left (1\right ) + \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + \sqrt{\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1}{\left (\cos \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )^{2} + \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )^{2}\right )} + 2 \,{\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac{1}{4}}{\left (\cos \left (2 \, x\right ) \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + \sin \left (2 \, x\right ) \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )\right )}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(2)*(2*arcsinh(1) + log(cos(2*x)^2 + sin(2*x)^2 + sqrt(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)*(cos
(1/2*arctan2(sin(4*x), cos(4*x) + 1))^2 + sin(1/2*arctan2(sin(4*x), cos(4*x) + 1))^2) + 2*(cos(4*x)^2 + sin(4*
x)^2 + 2*cos(4*x) + 1)^(1/4)*(cos(2*x)*cos(1/2*arctan2(sin(4*x), cos(4*x) + 1)) + sin(2*x)*sin(1/2*arctan2(sin
(4*x), cos(4*x) + 1)))))

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Fricas [B]  time = 2.10577, size = 177, normalized size = 6.81 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left (2 \, \sqrt{2}{\left (2 \, \sqrt{2} \cos \left (2 \, x\right ) + \sqrt{2}\right )} \sqrt{-\frac{\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - 8 \, \cos \left (2 \, x\right )^{2} - 8 \, \cos \left (2 \, x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(2)*log(2*sqrt(2)*(2*sqrt(2)*cos(2*x) + sqrt(2))*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) - 8*cos(2*x)^
2 - 8*cos(2*x) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\cot ^{2}{\left (x \right )} - 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+cot(x)**2)**(1/2),x)

[Out]

Integral(1/sqrt(cot(x)**2 - 1), x)

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Giac [B]  time = 1.7883, size = 61, normalized size = 2.35 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (\sqrt{2} - 1\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) + \frac{\sqrt{2} \log \left ({\left | -\sqrt{2} \cos \left (x\right ) + \sqrt{2 \, \cos \left (x\right )^{2} - 1} \right |}\right )}{2 \, \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*log(sqrt(2) - 1)*sgn(sin(x)) + 1/2*sqrt(2)*log(abs(-sqrt(2)*cos(x) + sqrt(2*cos(x)^2 - 1)))/sgn(s
in(x))