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

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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
]

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*}

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Mathematica [A]  time = 0.02, 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]

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

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fricas [A]  time = 1.34, size = 26, normalized size = 0.76 \[ -\frac {{\left (x^{2} + 1\right )} \operatorname {arccot}\relax (x)^{2} - 2 \, x \operatorname {arccot}\relax (x) + 1}{4 \, {\left (x^{2} + 1\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\relax (x)}{{\left (x^{2} + 1\right )}^{2}}\,{d x} \]

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)

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maple [A]  time = 0.05, size = 35, normalized size = 1.03 \[ \frac {x \,\mathrm {arccot}\relax (x )}{2 x^{2}+2}+\frac {\mathrm {arccot}\relax (x ) \arctan \relax (x )}{2}-\frac {1}{4 \left (x^{2}+1\right )}+\frac {\arctan \relax (x )^{2}}{4} \]

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 = 0.42, size = 38, normalized size = 1.12 \[ \frac {1}{2} \, {\left (\frac {x}{x^{2} + 1} + \arctan \relax (x)\right )} \operatorname {arccot}\relax (x) + \frac {{\left (x^{2} + 1\right )} \arctan \relax (x)^{2} - 1}{4 \, {\left (x^{2} + 1\right )}} \]

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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mupad [B]  time = 0.61, size = 22, normalized size = 0.65 \[ \frac {\frac {x\,\mathrm {acot}\relax (x)}{2}-\frac {1}{4}}{x^2+1}-\frac {{\mathrm {acot}\relax (x)}^2}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(x)/(x^2 + 1)^2,x)

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RecursionError

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