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

Optimal. Leaf size=5 \[ -\sinh ^{-1}(\cot (x)) \]

[Out]

-arcsinh(cot(x))

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Rubi [A]  time = 0.01, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3657, 4122, 215} \[ -\sinh ^{-1}(\cot (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Cot[x]^2],x]

[Out]

-ArcSinh[Cot[x]]

Rule 215

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

Rule 3657

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rubi steps

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

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Mathematica [B]  time = 0.01, size = 28, normalized size = 5.60 \[ \sin (x) \sqrt {\csc ^2(x)} \left (\log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Cot[x]^2],x]

[Out]

Sqrt[Csc[x]^2]*(-Log[Cos[x/2]] + Log[Sin[x/2]])*Sin[x]

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fricas [B]  time = 0.52, size = 53, normalized size = 10.60 \[ -\frac {1}{2} \, \log \left (\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) + \frac {1}{2} \, \log \left (-\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 0.23, size = 6, normalized size = 1.20 \[ -\arcsinh \left (\cot \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsinh(cot(x))

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maxima [B]  time = 1.00, size = 35, normalized size = 7.00 \[ -\frac {1}{2} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \frac {1}{2} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 1/2*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)

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mupad [B]  time = 0.32, size = 5, normalized size = 1.00 \[ -\mathrm {asinh}\left (\mathrm {cot}\relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(x)^2 + 1)^(1/2),x)

[Out]

-asinh(cot(x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cot ^{2}{\relax (x )} + 1}\, dx \]

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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