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

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*Log[1 - x] + Log[1 + x^2]/2

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Rubi [A]  time = 0.0635736, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{1}{2} \log \left (x^2+1\right )+2 \log (1-x)-3 \tan ^{-1}(x) \]

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

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Rubi in Sympy [A]  time = 9.04946, size = 19, normalized size = 0.83 \[ 2 \log{\left (- x + 1 \right )} + \frac{\log{\left (x^{2} + 1 \right )}}{2} - 3 \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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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Mathematica [A]  time = 0.0116205, 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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Maple [A]  time = 0.007, size = 20, normalized size = 0.9 \[ 2\,\ln \left ( -1+x \right ) +{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}-3\,\arctan \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.885305, size = 26, normalized size = 1.13 \[ -3 \, \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) + 2 \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.254199, size = 26, normalized size = 1.13 \[ -3 \, \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) + 2 \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.309548, size = 19, normalized size = 0.83 \[ 2 \log{\left (x - 1 \right )} + \frac{\log{\left (x^{2} + 1 \right )}}{2} - 3 \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.263705, size = 27, normalized size = 1.17 \[ -3 \, \arctan \left (x\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) + 2 \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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