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 = 0.0140171, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 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
]

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]

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*}

Mathematica [B]  time = 0.0126543, size = 28, normalized size = 5.6 \[ \sin (x) \sqrt{\csc ^2(x)} \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right ) \]

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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Maple [A]  time = 0.021, size = 6, normalized size = 1.2 \begin{align*} -{\it Arcsinh} \left ( \cot \left ( x \right ) \right ) \end{align*}

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 = 1.64863, size = 47, normalized size = 9.4 \begin{align*} -\frac{1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (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="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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Fricas [B]  time = 1.85152, size = 163, normalized size = 32.6 \begin{align*} -\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 ) \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*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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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 [A]  time = 1.24676, size = 14, normalized size = 2.8 \begin{align*} \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(tan(1/2*x)))*sgn(sin(x))