3.358 \(\int \frac {5-4 x+3 x^2}{(-1+x) (1+x^2)} \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{2} \log \left (x^2+1\right )+2 \log (1-x)-3 \tan ^{-1}(x) \]

[Out]

-3*arctan(x)+2*ln(1-x)+1/2*ln(x^2+1)

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Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1629, 635, 203, 260} \[ \frac {1}{2} \log \left (x^2+1\right )+2 \log (1-x)-3 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(5 - 4*x + 3*x^2)/((-1 + x)*(1 + x^2)),x]

[Out]

-3*ArcTan[x] + 2*Log[1 - x] + Log[1 + x^2]/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 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]

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 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]

Rubi steps

\begin {align*} \int \frac {5-4 x+3 x^2}{(-1+x) \left (1+x^2\right )} \, dx &=\int \left (\frac {2}{-1+x}+\frac {-3+x}{1+x^2}\right ) \, dx\\ &=2 \log (1-x)+\int \frac {-3+x}{1+x^2} \, dx\\ &=2 \log (1-x)-3 \int \frac {1}{1+x^2} \, dx+\int \frac {x}{1+x^2} \, dx\\ &=-3 \tan ^{-1}(x)+2 \log (1-x)+\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.22 \[ \frac {1}{2} \log \left ((x-1)^2+2 (x-1)+2\right )+2 \log (x-1)-3 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - 4*x + 3*x^2)/((-1 + x)*(1 + x^2)),x]

[Out]

-3*ArcTan[x] + Log[2 + 2*(-1 + x) + (-1 + x)^2]/2 + 2*Log[-1 + x]

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fricas [A]  time = 0.71, size = 19, normalized size = 0.83 \[ -3 \, \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) + 2 \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-4*x+5)/(-1+x)/(x^2+1),x, algorithm="fricas")

[Out]

-3*arctan(x) + 1/2*log(x^2 + 1) + 2*log(x - 1)

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giac [A]  time = 0.31, size = 20, normalized size = 0.87 \[ -3 \, \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) + 2 \, \log \left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-4*x+5)/(-1+x)/(x^2+1),x, algorithm="giac")

[Out]

-3*arctan(x) + 1/2*log(x^2 + 1) + 2*log(abs(x - 1))

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maple [A]  time = 0.00, size = 20, normalized size = 0.87 \[ -3 \arctan \relax (x )+2 \ln \left (x -1\right )+\frac {\ln \left (x^{2}+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2-4*x+5)/(x-1)/(x^2+1),x)

[Out]

2*ln(x-1)+1/2*ln(x^2+1)-3*arctan(x)

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maxima [A]  time = 1.56, size = 19, normalized size = 0.83 \[ -3 \, \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) + 2 \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-4*x+5)/(-1+x)/(x^2+1),x, algorithm="maxima")

[Out]

-3*arctan(x) + 1/2*log(x^2 + 1) + 2*log(x - 1)

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mupad [B]  time = 0.05, size = 25, normalized size = 1.09 \[ 2\,\ln \left (x-1\right )+\ln \left (x-\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {3}{2}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (\frac {1}{2}-\frac {3}{2}{}\mathrm {i}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2 - 4*x + 5)/((x^2 + 1)*(x - 1)),x)

[Out]

2*log(x - 1) + log(x - 1i)*(1/2 + 3i/2) + log(x + 1i)*(1/2 - 3i/2)

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sympy [A]  time = 0.14, size = 19, normalized size = 0.83 \[ 2 \log {\left (x - 1 \right )} + \frac {\log {\left (x^{2} + 1 \right )}}{2} - 3 \operatorname {atan}{\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2-4*x+5)/(-1+x)/(x**2+1),x)

[Out]

2*log(x - 1) + log(x**2 + 1)/2 - 3*atan(x)

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