3.82 \(\int \cot ^{-1}(x) \, dx\)

Optimal. Leaf size=15 \[ \frac{1}{2} \log \left (x^2+1\right )+x \cot ^{-1}(x) \]

[Out]

x*ArcCot[x] + Log[1 + x^2]/2

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Rubi [A]  time = 0.0033438, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4847, 260} \[ \frac{1}{2} \log \left (x^2+1\right )+x \cot ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[x],x]

[Out]

x*ArcCot[x] + Log[1 + x^2]/2

Rule 4847

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCot[c*x])^p, x] + Dist[b*c*p, Int[
(x*(a + b*ArcCot[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

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

Mathematica [A]  time = 0.0014764, size = 15, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+1\right )+x \cot ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[x],x]

[Out]

x*ArcCot[x] + Log[1 + x^2]/2

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Maple [A]  time = 0., size = 14, normalized size = 0.9 \begin{align*} x{\rm arccot} \left (x\right )+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(x),x)

[Out]

x*arccot(x)+1/2*ln(x^2+1)

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Maxima [A]  time = 0.931492, size = 18, normalized size = 1.2 \begin{align*} x \operatorname{arccot}\left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x),x, algorithm="maxima")

[Out]

x*arccot(x) + 1/2*log(x^2 + 1)

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Fricas [A]  time = 0.513946, size = 43, normalized size = 2.87 \begin{align*} x \operatorname{arccot}\left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x),x, algorithm="fricas")

[Out]

x*arccot(x) + 1/2*log(x^2 + 1)

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Sympy [A]  time = 0.18563, size = 12, normalized size = 0.8 \begin{align*} x \operatorname{acot}{\left (x \right )} + \frac{\log{\left (x^{2} + 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(x),x)

[Out]

x*acot(x) + log(x**2 + 1)/2

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Giac [A]  time = 1.08174, size = 20, normalized size = 1.33 \begin{align*} x \arctan \left (\frac{1}{x}\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x),x, algorithm="giac")

[Out]

x*arctan(1/x) + 1/2*log(x^2 + 1)