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

Optimal. Leaf size=24 \[ -\frac{1}{2} \log \left (x^2+1\right )+\frac{1}{x-1}+\log (1-x)+\tan ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0344154, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {1629, 635, 203, 260} \[ -\frac{1}{2} \log \left (x^2+1\right )+\frac{1}{x-1}+\log (1-x)+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1629

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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

Mathematica [A]  time = 0.0137617, size = 22, normalized size = 0.92 \[ -\frac{1}{2} \log \left (x^2+1\right )+\frac{1}{x-1}+\log (x-1)+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 21, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}+\arctan \left ( x \right ) +\ln \left ( -1+x \right ) + \left ( -1+x \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.40285, size = 27, normalized size = 1.12 \begin{align*} \frac{1}{x - 1} + \arctan \left (x\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.95175, size = 115, normalized size = 4.79 \begin{align*} \frac{2 \,{\left (x - 1\right )} \arctan \left (x\right ) -{\left (x - 1\right )} \log \left (x^{2} + 1\right ) + 2 \,{\left (x - 1\right )} \log \left (x - 1\right ) + 2}{2 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.125584, size = 20, normalized size = 0.83 \begin{align*} \log{\left (x - 1 \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \operatorname{atan}{\left (x \right )} + \frac{1}{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.06819, size = 63, normalized size = 2.62 \begin{align*} \frac{1}{4} \, \pi - \pi \left \lfloor \frac{\pi + 4 \, \arctan \left (x\right )}{4 \, \pi } + \frac{1}{2} \right \rfloor + \frac{1}{x - 1} + \arctan \left (x\right ) - \frac{1}{2} \, \log \left (\frac{2}{x - 1} + \frac{2}{{\left (x - 1\right )}^{2}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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