3.673 \(\int \frac{x^2 \tan ^{-1}(x)}{(1+x^2)^2} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0470394, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4934, 4884} \[ -\frac{1}{4 \left (x^2+1\right )}-\frac{x \tan ^{-1}(x)}{2 \left (x^2+1\right )}+\frac{1}{4} \tan ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[(x^2*ArcTan[x])/(1 + x^2)^2,x]

[Out]

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

Rule 4934

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^2*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*(d + e*x^2)^(q
+ 1))/(4*c^3*d*(q + 1)^2), x] + (-Dist[1/(2*c^2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x],
x] + Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]))/(2*c^2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && E
qQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -5/2]

Rule 4884

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

Rubi steps

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

Mathematica [A]  time = 0.0262093, size = 28, normalized size = 0.82 \[ \frac{\left (x^2+1\right ) \tan ^{-1}(x)^2-2 x \tan ^{-1}(x)-1}{4 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 29, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{x}^{2}+4}}-{\frac{x\arctan \left ( x \right ) }{2\,{x}^{2}+2}}+{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.41796, size = 54, normalized size = 1.59 \begin{align*} -\frac{1}{2} \,{\left (\frac{x}{x^{2} + 1} - \arctan \left (x\right )\right )} \arctan \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(x^2*arctan(x)/(x^2+1)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.52562, size = 80, normalized size = 2.35 \begin{align*} \frac{{\left (x^{2} + 1\right )} \arctan \left (x\right )^{2} - 2 \, x \arctan \left (x\right ) - 1}{4 \,{\left (x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RecursionError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RecursionError

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \arctan \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(x^2*arctan(x)/(x^2+1)^2,x, algorithm="giac")

[Out]

integrate(x^2*arctan(x)/(x^2 + 1)^2, x)