\(\int \frac {x (1+x+x^2)}{(1+x^2)^2} \, dx\) [71]

Optimal result

Integrand size = 15, antiderivative size = 29 \[ \int \frac {x \left (1+x+x^2\right )}{\left (1+x^2\right )^2} \, dx=-\frac {x}{2 \left (1+x^2\right )}+\frac {\arctan (x)}{2}+\frac {1}{2} \log \left (1+x^2\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1818, 649, 209, 266} \[ \int \frac {x \left (1+x+x^2\right )}{\left (1+x^2\right )^2} \, dx=\frac {\arctan (x)}{2}-\frac {x}{2 \left (x^2+1\right )}+\frac {1}{2} \log \left (x^2+1\right ) \]

[In]

Int[(x*(1 + x + x^2))/(1 + x^2)^2,x]

[Out]

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

Rule 209

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

Rule 266

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]

Rule 649

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 1818

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]

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {x \left (1+x+x^2\right )}{\left (1+x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {x}{1+x^2}+\arctan (x)+\log \left (1+x^2\right )\right ) \]

[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] (verified)

Time = 0.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83

method	result	size
default	\(-\frac {x}{2 \left (x^{2}+1\right )}+\frac {\arctan \left (x \right )}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\)	\(24\)
meijerg	\(-\frac {x}{2 \left (x^{2}+1\right )}+\frac {\arctan \left (x \right )}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\)	\(24\)
risch	\(-\frac {x}{2 \left (x^{2}+1\right )}+\frac {\arctan \left (x \right )}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\)	\(24\)
parallelrisch	\(-\frac {i \ln \left (x -i\right ) x^{2}-i \ln \left (x +i\right ) x^{2}-2 \ln \left (x -i\right ) x^{2}-2 \ln \left (x +i\right ) x^{2}+i \ln \left (x -i\right )-i \ln \left (x +i\right )-2 \ln \left (x -i\right )-2 \ln \left (x +i\right )+2 x}{4 \left (x^{2}+1\right )}\)	\(86\)

[In]

int(x*(x^2+x+1)/(x^2+1)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.58 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {x \left (1+x+x^2\right )}{\left (1+x^2\right )^2} \, dx=\frac {{\left (x^{2} + 1\right )} \arctan \left (x\right ) + {\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) - x}{2 \, {\left (x^{2} + 1\right )}} \]

[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] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {x \left (1+x+x^2\right )}{\left (1+x^2\right )^2} \, dx=- \frac {x}{2 x^{2} + 2} + \frac {\log {\left (x^{2} + 1 \right )}}{2} + \frac {\operatorname {atan}{\left (x \right )}}{2} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {x \left (1+x+x^2\right )}{\left (1+x^2\right )^2} \, dx=-\frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {x \left (1+x+x^2\right )}{\left (1+x^2\right )^2} \, dx=-\frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {x \left (1+x+x^2\right )}{\left (1+x^2\right )^2} \, dx=\frac {\ln \left (x^2+1\right )}{2}+\frac {\mathrm {atan}\left (x\right )}{2}-\frac {x}{2\,\left (x^2+1\right )} \]

[In]

int((x*(x + x^2 + 1))/(x^2 + 1)^2,x)

[Out]

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