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

Optimal. Leaf size=28 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{2} \cot (x)}{\sqrt{1-\cot ^2(x)}}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0191643, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3661, 377, 203} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{2} \cot (x)}{\sqrt{1-\cot ^2(x)}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 - Cot[x]^2],x]

[Out]

-(ArcTan[(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 203

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

Mathematica [A]  time = 0.0632248, size = 42, normalized size = 1.5 \[ -\frac{\sqrt{\cos (2 x)} \csc (x) \log \left (\sqrt{2} \cos (x)+\sqrt{\cos (2 x)}\right )}{\sqrt{2-2 \cot ^2(x)}} \]

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 - 2*Cot[x]^2])

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Maple [A]  time = 0.026, size = 31, normalized size = 1.1 \begin{align*}{\frac{\sqrt{2}}{2}\arctan \left ({\frac{\sqrt{2}\cot \left ( x \right ) }{-1+ \left ( \cot \left ( x \right ) \right ) ^{2}}\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*2^(1/2)*arctan(2^(1/2)*(1-cot(x)^2)^(1/2)/(-1+cot(x)^2)*cot(x))

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Maxima [B]  time = 1.67901, size = 122, normalized size = 4.36 \begin{align*} \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 ) \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]

1/4*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))

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Fricas [B]  time = 2.16573, size = 171, normalized size = 6.11 \begin{align*} \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 ) \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]

1/4*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)))

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

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Giac [C]  time = 1.43033, size = 46, normalized size = 1.64 \begin{align*} -\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 )} \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*I*sqrt(2)*log(I*sqrt(2) + I)*sgn(sin(x)) - 1/2*sqrt(2)*arcsin(sqrt(2)*cos(x))/sgn(sin(x))