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

Optimal. Leaf size=42 \[ \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \cot (x)}{\sqrt{\cot ^2(x)-1}}\right )-\tanh ^{-1}\left (\frac{\cot (x)}{\sqrt{\cot ^2(x)-1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0254463, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3661, 402, 217, 206, 377} \[ \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \cot (x)}{\sqrt{\cot ^2(x)-1}}\right )-\tanh ^{-1}\left (\frac{\cot (x)}{\sqrt{\cot ^2(x)-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + Cot[x]^2],x]

[Out]

-ArcTanh[Cot[x]/Sqrt[-1 + Cot[x]^2]] + Sqrt[2]*ArcTanh[(Sqrt[2]*Cot[x])/Sqrt[-1 + Cot[x]^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 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0436697, size = 60, normalized size = 1.43 \[ \frac{\sin (x) \sqrt{\cot ^2(x)-1} \left (\sqrt{2} \log \left (\sqrt{2} \cos (x)+\sqrt{\cos (2 x)}\right )-\tanh ^{-1}\left (\frac{\cos (x)}{\sqrt{\cos (2 x)}}\right )\right )}{\sqrt{\cos (2 x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + Cot[x]^2],x]

[Out]

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

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Maple [A]  time = 0.027, size = 35, normalized size = 0.8 \begin{align*} -\ln \left ( \cot \left ( x \right ) +\sqrt{-1+ \left ( \cot \left ( x \right ) \right ) ^{2}} \right ) +{\it Artanh} \left ({\cot \left ( x \right ) \sqrt{2}{\frac{1}{\sqrt{-1+ \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \right ) \sqrt{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 1.89371, size = 351, normalized size = 8.36 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-2 \, \sqrt{-\frac{\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - 2 \, \cos \left (2 \, x\right ) - 1\right ) - \frac{1}{2} \, \log \left (\frac{\sqrt{2} \sqrt{-\frac{\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) + 1}{\cos \left (2 \, x\right ) + 1}\right ) + \frac{1}{2} \, \log \left (\frac{\sqrt{2} \sqrt{-\frac{\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 1}{\cos \left (2 \, x\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log(-2*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) - 2*cos(2*x) - 1) - 1/2*log((sqrt(2)*sqrt(-cos(2*x)
/(cos(2*x) - 1))*sin(2*x) + cos(2*x) + 1)/(cos(2*x) + 1)) + 1/2*log((sqrt(2)*sqrt(-cos(2*x)/(cos(2*x) - 1))*si
n(2*x) - cos(2*x) - 1)/(cos(2*x) + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(cot(x)**2 - 1), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out