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

Optimal. Leaf size=32 \[ -\frac{x}{4 \left (x^2+1\right )}-\frac{\cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac{1}{4} \tan ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0261825, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4931, 199, 203} \[ -\frac{x}{4 \left (x^2+1\right )}-\frac{\cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac{1}{4} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(x*ArcCot[x])/(1 + x^2)^2,x]

[Out]

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

Rule 4931

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 203

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

Rubi steps

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

Mathematica [A]  time = 0.0183403, size = 25, normalized size = 0.78 \[ -\frac{x^2 \tan ^{-1}(x)+x+\tan ^{-1}(x)+2 \cot ^{-1}(x)}{4 x^2+4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*ArcCot[x])/(1 + x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.026, size = 27, normalized size = 0.8 \begin{align*} -{\frac{x}{4\,{x}^{2}+4}}-{\frac{{\rm arccot} \left (x\right )}{2\,{x}^{2}+2}}-{\frac{\arctan \left ( x \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.48701, size = 35, normalized size = 1.09 \begin{align*} -\frac{x}{4 \,{\left (x^{2} + 1\right )}} - \frac{\operatorname{arccot}\left (x\right )}{2 \,{\left (x^{2} + 1\right )}} - \frac{1}{4} \, \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.14481, size = 55, normalized size = 1.72 \begin{align*} \frac{{\left (x^{2} - 1\right )} \operatorname{arccot}\left (x\right ) - x}{4 \,{\left (x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.70656, size = 31, normalized size = 0.97 \begin{align*} \frac{x^{2} \operatorname{acot}{\left (x \right )}}{4 x^{2} + 4} - \frac{x}{4 x^{2} + 4} - \frac{\operatorname{acot}{\left (x \right )}}{4 x^{2} + 4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10231, size = 38, normalized size = 1.19 \begin{align*} -\frac{x}{4 \,{\left (x^{2} + 1\right )}} - \frac{\arctan \left (\frac{1}{x}\right )}{2 \,{\left (x^{2} + 1\right )}} - \frac{1}{4} \, \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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