\(\int \frac {1-3 x+2 x^2-x^3}{x (1+x^2)^2} \, dx\) [310]

Optimal result

Integrand size = 26, antiderivative size = 33 \[ \int \frac {1-3 x+2 x^2-x^3}{x \left (1+x^2\right )^2} \, dx=-\frac {1+2 x}{2 \left (1+x^2\right )}-2 \arctan (x)+\log (x)-\frac {1}{2} \log \left (1+x^2\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1819, 815, 649, 209, 266} \[ \int \frac {1-3 x+2 x^2-x^3}{x \left (1+x^2\right )^2} \, dx=-2 \arctan (x)-\frac {2 x+1}{2 \left (x^2+1\right )}-\frac {1}{2} \log \left (x^2+1\right )+\log (x) \]

[In]

Int[(1 - 3*x + 2*x^2 - x^3)/(x*(1 + x^2)^2),x]

[Out]

-1/2*(1 + 2*x)/(1 + x^2) - 2*ArcTan[x] + Log[x] - 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 815

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m*((f + g*x)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 1819

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1-3 x+2 x^2-x^3}{x \left (1+x^2\right )^2} \, dx=\frac {-1-2 x}{2 \left (1+x^2\right )}-2 \arctan (x)+\log (x)-\frac {1}{2} \log \left (1+x^2\right ) \]

[In]

Integrate[(1 - 3*x + 2*x^2 - x^3)/(x*(1 + x^2)^2),x]

[Out]

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

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85

method	result	size
default	\(\ln \left (x \right )-\frac {x +\frac {1}{2}}{x^{2}+1}-\frac {\ln \left (x^{2}+1\right )}{2}-2 \arctan \left (x \right )\)	\(28\)
risch	\(\frac {-x -\frac {1}{2}}{x^{2}+1}+\ln \left (x \right )-\frac {\ln \left (4 x^{2}+4\right )}{2}-2 \arctan \left (x \right )\)	\(31\)
meijerg	\(-\frac {2 x}{2 x^{2}+2}-2 \arctan \left (x \right )+\frac {x^{2}}{x^{2}+1}-\frac {x^{2}}{2 x^{2}+2}-\frac {\ln \left (x^{2}+1\right )}{2}+\frac {1}{2}+\ln \left (x \right )\)	\(64\)
parallelrisch	\(\frac {2 i \ln \left (x -i\right ) x^{2}-2 i \ln \left (x +i\right ) x^{2}+2 \ln \left (x \right ) x^{2}-\ln \left (x -i\right ) x^{2}-\ln \left (x +i\right ) x^{2}-1+2 i \ln \left (x -i\right )-2 i \ln \left (x +i\right )+2 \ln \left (x \right )-\ln \left (x -i\right )-\ln \left (x +i\right )-2 x}{2 x^{2}+2}\)	\(98\)

[In]

int((-x^3+2*x^2-3*x+1)/x/(x^2+1)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {1-3 x+2 x^2-x^3}{x \left (1+x^2\right )^2} \, dx=-\frac {4 \, {\left (x^{2} + 1\right )} \arctan \left (x\right ) + {\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) - 2 \, {\left (x^{2} + 1\right )} \log \left (x\right ) + 2 \, x + 1}{2 \, {\left (x^{2} + 1\right )}} \]

[In]

integrate((-x^3+2*x^2-3*x+1)/x/(x^2+1)^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {1-3 x+2 x^2-x^3}{x \left (1+x^2\right )^2} \, dx=- \frac {2 x + 1}{2 x^{2} + 2} + \log {\left (x \right )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} - 2 \operatorname {atan}{\left (x \right )} \]

[In]

integrate((-x**3+2*x**2-3*x+1)/x/(x**2+1)**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {1-3 x+2 x^2-x^3}{x \left (1+x^2\right )^2} \, dx=-\frac {2 \, x + 1}{2 \, {\left (x^{2} + 1\right )}} - 2 \, \arctan \left (x\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x\right ) \]

[In]

integrate((-x^3+2*x^2-3*x+1)/x/(x^2+1)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {1-3 x+2 x^2-x^3}{x \left (1+x^2\right )^2} \, dx=-\frac {2 \, x + 1}{2 \, {\left (x^{2} + 1\right )}} - 2 \, \arctan \left (x\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \left ({\left | x \right |}\right ) \]

[In]

integrate((-x^3+2*x^2-3*x+1)/x/(x^2+1)^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.59 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1-3 x+2 x^2-x^3}{x \left (1+x^2\right )^2} \, dx=\ln \left (x\right )-\frac {x+\frac {1}{2}}{x^2+1}+\ln \left (x-\mathrm {i}\right )\,\left (-\frac {1}{2}+1{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (-\frac {1}{2}-\mathrm {i}\right ) \]

[In]

int(-(3*x - 2*x^2 + x^3 - 1)/(x*(x^2 + 1)^2),x)

[Out]

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