3.100 \(\int \frac {x^2}{(1+x^2)^2} \, dx\)

Optimal. Leaf size=19 \[ \frac {1}{2} \tan ^{-1}(x)-\frac {x}{2 \left (x^2+1\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {288, 203} \[ \frac {1}{2} \tan ^{-1}(x)-\frac {x}{2 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(1 + x^2)^2,x]

[Out]

-x/(2*(1 + x^2)) + ArcTan[x]/2

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])

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {1}{2} \tan ^{-1}(x)-\frac {x}{2 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(1 + x^2)^2,x]

[Out]

-1/2*x/(1 + x^2) + ArcTan[x]/2

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 21, normalized size = 1.11 \[ \frac {{\left (x^{2} + 1\right )} \arctan \relax (x) - x}{2 \, {\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.01, size = 15, normalized size = 0.79 \[ -\frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x/(x^2 + 1) + 1/2*arctan(x)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 16, normalized size = 0.84 \[ -\frac {x}{2 \left (x^{2}+1\right )}+\frac {\arctan \relax (x )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.12, size = 15, normalized size = 0.79 \[ -\frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x/(x^2 + 1) + 1/2*arctan(x)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 17, normalized size = 0.89 \[ \frac {\mathrm {atan}\relax (x)}{2}-\frac {x}{2\,\left (x^2+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^2 + 1)^2,x)

[Out]

atan(x)/2 - x/(2*(x^2 + 1))

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 12, normalized size = 0.63 \[ - \frac {x}{2 x^{2} + 2} + \frac {\operatorname {atan}{\relax (x )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(2*x**2 + 2) + atan(x)/2

________________________________________________________________________________________