\(\int \frac {1}{\sqrt {-1+\cot ^2(x)}} \, dx\) [43]

Optimal result

Integrand size = 10, antiderivative size = 26 \[ \int \frac {1}{\sqrt {-1+\cot ^2(x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )}{\sqrt {2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3742, 385, 212} \[ \int \frac {1}{\sqrt {-1+\cot ^2(x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {\cot ^2(x)-1}}\right )}{\sqrt {2}} \]

[In]

Int[1/Sqrt[-1 + Cot[x]^2],x]

[Out]

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

Rule 212

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])

Rule 385

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 3742

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])

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (1+x^2\right )} \, dx,x,\cot (x)\right ) \\ & = -\text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\cot (x)}{\sqrt {-1+\cot ^2(x)}}\right ) \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {1}{\sqrt {-1+\cot ^2(x)}} \, dx=-\frac {\arcsin \left (\frac {\sqrt {2} \cot (x)}{\sqrt {1+\cot ^2(x)}}\right ) \sqrt {1-\cot ^2(x)}}{\sqrt {2} \sqrt {-1+\cot ^2(x)}} \]

[In]

Integrate[1/Sqrt[-1 + Cot[x]^2],x]

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(-\frac {\operatorname {arctanh}\left (\frac {\cot \left (x \right ) \sqrt {2}}{\sqrt {-1+\cot \left (x \right )^{2}}}\right ) \sqrt {2}}{2}\)	\(21\)
default	\(-\frac {\operatorname {arctanh}\left (\frac {\cot \left (x \right ) \sqrt {2}}{\sqrt {-1+\cot \left (x \right )^{2}}}\right ) \sqrt {2}}{2}\)	\(21\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).

Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {1}{\sqrt {-1+\cot ^2(x)}} \, dx=\frac {1}{8} \, \sqrt {2} \log \left (2 \, \sqrt {2} {\left (2 \, \sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2}\right )} \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - 8 \, \cos \left (2 \, x\right )^{2} - 8 \, \cos \left (2 \, x\right ) - 1\right ) \]

[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)

Sympy [F]

\[ \int \frac {1}{\sqrt {-1+\cot ^2(x)}} \, dx=\int \frac {1}{\sqrt {\cot ^{2}{\left (x \right )} - 1}}\, dx \]

[In]

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

[Out]

Integral(1/sqrt(cot(x)**2 - 1), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (20) = 40\).

Time = 0.53 (sec) , antiderivative size = 143, normalized size of antiderivative = 5.50 \[ \int \frac {1}{\sqrt {-1+\cot ^2(x)}} \, dx=-\frac {1}{8} \, \sqrt {2} {\left (2 \, \operatorname {arsinh}\left (1\right ) + \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + \sqrt {\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1} {\left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )^{2}\right )} + 2 \, {\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {1}{4}} {\left (\cos \left (2 \, x\right ) \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + \sin \left (2 \, x\right ) \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )\right )}\right )\right )} \]

[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)))))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (20) = 40\).

Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int \frac {1}{\sqrt {-1+\cot ^2(x)}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (\sqrt {2} - 1\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) + \frac {\sqrt {2} \log \left ({\left | -\sqrt {2} \cos \left (x\right ) + \sqrt {2 \, \cos \left (x\right )^{2} - 1} \right |}\right )}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[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))

Mupad [B] (verification not implemented)

Time = 13.91 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\sqrt {-1+\cot ^2(x)}} \, dx=-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\mathrm {cot}\left (x\right )}{\sqrt {{\mathrm {cot}\left (x\right )}^2-1}}\right )}{2} \]

[In]

int(1/(cot(x)^2 - 1)^(1/2),x)

[Out]

-(2^(1/2)*atanh((2^(1/2)*cot(x))/(cot(x)^2 - 1)^(1/2)))/2