### 3.71 $$\int \frac{x (1+x+x^2)}{(1+x^2)^2} \, dx$$

Optimal. Leaf size=29 $-\frac{x}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )+\frac{1}{2} \tan ^{-1}(x)$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0247806, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.267, Rules used = {1804, 635, 203, 260} $-\frac{x}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )+\frac{1}{2} \tan ^{-1}(x)$

Antiderivative was successfully veriﬁed.

[In]

Int[(x*(1 + x + x^2))/(1 + x^2)^2,x]

[Out]

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

Rule 1804

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x
^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x
], x, 1]}, Simp[((c*x)^m*(a + b*x^2)^(p + 1)*(a*g - b*f*x))/(2*a*b*(p + 1)), x] + Dist[c/(2*a*b*(p + 1)), Int[
(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

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 260

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

Rubi steps

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

Mathematica [A]  time = 0.0084116, size = 23, normalized size = 0.79 $\frac{1}{2} \left (-\frac{x}{x^2+1}+\log \left (x^2+1\right )+\tan ^{-1}(x)\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 24, normalized size = 0.8 \begin{align*} -{\frac{x}{2\,{x}^{2}+2}}+{\frac{\arctan \left ( x \right ) }{2}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.46719, size = 31, normalized size = 1.07 \begin{align*} -\frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.989002, size = 89, normalized size = 3.07 \begin{align*} \frac{{\left (x^{2} + 1\right )} \arctan \left (x\right ) +{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) - x}{2 \,{\left (x^{2} + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.114108, size = 20, normalized size = 0.69 \begin{align*} - \frac{x}{2 x^{2} + 2} + \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{\operatorname{atan}{\left (x \right )}}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(2*x**2 + 2) + log(x**2 + 1)/2 + atan(x)/2

________________________________________________________________________________________

Giac [A]  time = 1.1516, size = 31, normalized size = 1.07 \begin{align*} -\frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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