3.72 \(\int \frac{\cot ^{-1}(x)}{(1+x^2)^2} \, dx\)

Optimal. Leaf size=34 \[ -\frac{1}{4 \left (x^2+1\right )}+\frac{x \cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac{1}{4} \cot ^{-1}(x)^2 \]

[Out]

-1/(4*(1 + x^2)) + (x*ArcCot[x])/(2*(1 + x^2)) - ArcCot[x]^2/4

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Rubi [A]  time = 0.0145949, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4893, 261} \[ -\frac{1}{4 \left (x^2+1\right )}+\frac{x \cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac{1}{4} \cot ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[x]/(1 + x^2)^2,x]

[Out]

-1/(4*(1 + x^2)) + (x*ArcCot[x])/(2*(1 + x^2)) - ArcCot[x]^2/4

Rule 4893

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

Rule 261

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

Rubi steps

\begin{align*} \int \frac{\cot ^{-1}(x)}{\left (1+x^2\right )^2} \, dx &=\frac{x \cot ^{-1}(x)}{2 \left (1+x^2\right )}-\frac{1}{4} \cot ^{-1}(x)^2+\frac{1}{2} \int \frac{x}{\left (1+x^2\right )^2} \, dx\\ &=-\frac{1}{4 \left (1+x^2\right )}+\frac{x \cot ^{-1}(x)}{2 \left (1+x^2\right )}-\frac{1}{4} \cot ^{-1}(x)^2\\ \end{align*}

Mathematica [A]  time = 0.0127626, size = 28, normalized size = 0.82 \[ -\frac{\left (x^2+1\right ) \cot ^{-1}(x)^2-2 x \cot ^{-1}(x)+1}{4 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[x]/(1 + x^2)^2,x]

[Out]

-(1 - 2*x*ArcCot[x] + (1 + x^2)*ArcCot[x]^2)/(4*(1 + x^2))

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Maple [A]  time = 0.034, size = 35, normalized size = 1. \begin{align*}{\frac{x{\rm arccot} \left (x\right )}{2\,{x}^{2}+2}}+{\frac{{\rm arccot} \left (x\right )\arctan \left ( x \right ) }{2}}-{\frac{1}{4\,{x}^{2}+4}}+{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*arccot(x)/(x^2+1)+1/2*arccot(x)*arctan(x)-1/4/(x^2+1)+1/4*arctan(x)^2

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Maxima [A]  time = 1.46364, size = 51, normalized size = 1.5 \begin{align*} \frac{1}{2} \,{\left (\frac{x}{x^{2} + 1} + \arctan \left (x\right )\right )} \operatorname{arccot}\left (x\right ) + \frac{{\left (x^{2} + 1\right )} \arctan \left (x\right )^{2} - 1}{4 \,{\left (x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x/(x^2 + 1) + arctan(x))*arccot(x) + 1/4*((x^2 + 1)*arctan(x)^2 - 1)/(x^2 + 1)

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Fricas [A]  time = 2.1514, size = 81, normalized size = 2.38 \begin{align*} -\frac{{\left (x^{2} + 1\right )} \operatorname{arccot}\left (x\right )^{2} - 2 \, x \operatorname{arccot}\left (x\right ) + 1}{4 \,{\left (x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((x^2 + 1)*arccot(x)^2 - 2*x*arccot(x) + 1)/(x^2 + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(x)/(x**2+1)**2,x)

[Out]

Integral(acot(x)/(x**2 + 1)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (x\right )}{{\left (x^{2} + 1\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccot(x)/(x^2 + 1)^2, x)