\(\int \frac {1}{\sqrt {1-\cot ^2(x)}} \, dx\) [40]

Optimal result

Integrand size = 12, antiderivative size = 28 \[ \int \frac {1}{\sqrt {1-\cot ^2(x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )}{\sqrt {2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3742, 385, 209} \[ \int \frac {1}{\sqrt {1-\cot ^2(x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )}{\sqrt {2}} \]

[In]

Int[1/Sqrt[1 - Cot[x]^2],x]

[Out]

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 {\arctan \left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {1-\cot ^2(x)}} \, dx=-\frac {\arcsin \left (\frac {\sqrt {2} \cot (x)}{\sqrt {1+\cot ^2(x)}}\right )}{\sqrt {2}} \]

[In]

Integrate[1/Sqrt[1 - Cot[x]^2],x]

[Out]

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

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {1-\cot \left (x \right )^{2}}\, \cot \left (x \right )}{-1+\cot \left (x \right )^{2}}\right )}{2}\)	\(31\)
default	\(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {1-\cot \left (x \right )^{2}}\, \cot \left (x \right )}{-1+\cot \left (x \right )^{2}}\right )}{2}\)	\(31\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (22) = 44\).

Time = 0.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sqrt {1-\cot ^2(x)}} \, dx=\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2}\right )} \sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{4 \, {\left (\cos \left (2 \, x\right )^{2} + \cos \left (2 \, x\right )\right )}}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(1 - cot(x)**2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (22) = 44\).

Time = 0.48 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.21 \[ \int \frac {1}{\sqrt {1-\cot ^2(x)}} \, dx=\frac {1}{4} \, \sqrt {2} \arctan \left ({\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + \sin \left (2 \, x\right ), {\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + \cos \left (2 \, x\right )\right ) \]

[In]

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

[Out]

1/4*sqrt(2)*arctan2((cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*sin(1/2*arctan2(sin(4*x), cos(4*x) + 1))
+ sin(2*x), (cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*cos(1/2*arctan2(sin(4*x), cos(4*x) + 1)) + cos(2*
x))

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\sqrt {1-\cot ^2(x)}} \, dx=-\frac {1}{2} i \, \sqrt {2} \log \left (i \, \sqrt {2} + i\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) - \frac {\sqrt {2} \arcsin \left (\sqrt {2} \cos \left (x\right )\right )}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

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

[Out]

-1/2*I*sqrt(2)*log(I*sqrt(2) + I)*sgn(sin(x)) - 1/2*sqrt(2)*arcsin(sqrt(2)*cos(x))/sgn(sin(x))

Mupad [B] (verification not implemented)

Time = 13.18 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.04 \[ \int \frac {1}{\sqrt {1-\cot ^2(x)}} \, dx=-\frac {\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (-1+\mathrm {cot}\left (x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\sqrt {1-{\mathrm {cot}\left (x\right )}^2}\,1{}\mathrm {i}}{\mathrm {cot}\left (x\right )-\mathrm {i}}\right )\,1{}\mathrm {i}}{4}+\frac {\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (1+\mathrm {cot}\left (x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\sqrt {1-{\mathrm {cot}\left (x\right )}^2}\,1{}\mathrm {i}}{\mathrm {cot}\left (x\right )+1{}\mathrm {i}}\right )\,1{}\mathrm {i}}{4} \]

[In]

int(1/(1 - cot(x)^2)^(1/2),x)

[Out]

(2^(1/2)*log(((2^(1/2)*(cot(x)*1i + 1)*1i)/2 + (1 - cot(x)^2)^(1/2)*1i)/(cot(x) + 1i))*1i)/4 - (2^(1/2)*log(((
2^(1/2)*(cot(x)*1i - 1)*1i)/2 - (1 - cot(x)^2)^(1/2)*1i)/(cot(x) - 1i))*1i)/4