3.1.42 \(\int \sqrt {-1+\cot ^2(x)} \, dx\) [42]

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

[Out]

-arctanh(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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Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3742, 399, 223, 212, 385} \begin {gather*} \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 ) \end {gather*}

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 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 223

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 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 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 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*} \int \sqrt {-1+\cot ^2(x)} \, dx &=-\text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=2 \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}{\sqrt {-1+x^2}} \, dx,x,\cot (x)\right )\\ &=2 \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )-\text {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*}

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Mathematica [A]
time = 0.05, size = 60, normalized size = 1.43 \begin {gather*} \frac {\sqrt {-1+\cot ^2(x)} \left (-\tanh ^{-1}\left (\frac {\cos (x)}{\sqrt {\cos (2 x)}}\right )+\sqrt {2} \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right )\right ) \sin (x)}{\sqrt {\cos (2 x)}} \end {gather*}

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 = 0.17, size = 35, normalized size = 0.83

method	result	size
derivativedivides	\(-\ln \left (\cot \left (x \right )+\sqrt {-1+\cot ^{2}\left (x \right )}\right )+\arctanh \left (\frac {\cot \left (x \right ) \sqrt {2}}{\sqrt {-1+\cot ^{2}\left (x \right )}}\right ) \sqrt {2}\)	\(35\)
default	\(-\ln \left (\cot \left (x \right )+\sqrt {-1+\cot ^{2}\left (x \right )}\right )+\arctanh \left (\frac {\cot \left (x \right ) \sqrt {2}}{\sqrt {-1+\cot ^{2}\left (x \right )}}\right ) \sqrt {2}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> ECL says: Error executing code in Maxima: sign: argument cannot be imaginary
; found %i

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (34) = 68\).
time = 2.61, size = 123, normalized size = 2.93 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\cot ^{2}{\left (x \right )} - 1}\, dx \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [B]
time = 0.43, size = 34, normalized size = 0.81 \begin {gather*} \sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\mathrm {cot}\left (x\right )}{\sqrt {{\mathrm {cot}\left (x\right )}^2-1}}\right )-\ln \left (\mathrm {cot}\left (x\right )+\sqrt {{\mathrm {cot}\left (x\right )}^2-1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(x)^2 - 1)^(1/2),x)

[Out]

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

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