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

Optimal. Leaf size=37 \[ \frac{3}{4} \log \left (x^2+1\right )+x+\frac{5}{2 (1-x)}+\frac{1}{2} \log (1-x)+2 \tan ^{-1}(x) \]

[Out]

5/(2*(1 - x)) + x + 2*ArcTan[x] + Log[1 - x]/2 + (3*Log[1 + x^2])/4

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Rubi [A]  time = 0.0781491, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3}{4} \log \left (x^2+1\right )+x+\frac{5}{2 (1-x)}+\frac{1}{2} \log (1-x)+2 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

5/(2*(1 - x)) + x + 2*ArcTan[x] + Log[1 - x]/2 + (3*Log[1 + x^2])/4

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Rubi in Sympy [A]  time = 34.322, size = 29, normalized size = 0.78 \[ x + \frac{\log{\left (- x + 1 \right )}}{2} + \frac{3 \log{\left (x^{2} + 1 \right )}}{4} + 2 \operatorname{atan}{\left (x \right )} + \frac{5}{2 \left (- x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + log(-x + 1)/2 + 3*log(x**2 + 1)/4 + 2*atan(x) + 5/(2*(-x + 1))

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Mathematica [A]  time = 0.0385096, size = 33, normalized size = 0.89 \[ \frac{3}{4} \log \left (x^2+1\right )+x+\frac{5}{2-2 x}+\frac{1}{2} \log (x-1)+2 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

5/(2 - 2*x) + x + 2*ArcTan[x] + Log[-1 + x]/2 + (3*Log[1 + x^2])/4

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Maple [A]  time = 0.011, size = 28, normalized size = 0.8 \[ x-{\frac{5}{2\,x-2}}+{\frac{\ln \left ( -1+x \right ) }{2}}+{\frac{3\,\ln \left ({x}^{2}+1 \right ) }{4}}+2\,\arctan \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x-5/2/(-1+x)+1/2*ln(-1+x)+3/4*ln(x^2+1)+2*arctan(x)

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Maxima [A]  time = 0.888388, size = 36, normalized size = 0.97 \[ x - \frac{5}{2 \,{\left (x - 1\right )}} + 2 \, \arctan \left (x\right ) + \frac{3}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{2} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 5/2/(x - 1) + 2*arctan(x) + 3/4*log(x^2 + 1) + 1/2*log(x - 1)

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Fricas [A]  time = 0.256245, size = 59, normalized size = 1.59 \[ \frac{4 \, x^{2} + 8 \,{\left (x - 1\right )} \arctan \left (x\right ) + 3 \,{\left (x - 1\right )} \log \left (x^{2} + 1\right ) + 2 \,{\left (x - 1\right )} \log \left (x - 1\right ) - 4 \, x - 10}{4 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(4*x^2 + 8*(x - 1)*arctan(x) + 3*(x - 1)*log(x^2 + 1) + 2*(x - 1)*log(x - 1)
- 4*x - 10)/(x - 1)

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Sympy [A]  time = 0.382463, size = 29, normalized size = 0.78 \[ x + \frac{\log{\left (x - 1 \right )}}{2} + \frac{3 \log{\left (x^{2} + 1 \right )}}{4} + 2 \operatorname{atan}{\left (x \right )} - \frac{5}{2 x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + log(x - 1)/2 + 3*log(x**2 + 1)/4 + 2*atan(x) - 5/(2*x - 2)

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GIAC/XCAS [A]  time = 0.26266, size = 81, normalized size = 2.19 \[ \frac{1}{2} \, \pi - 2 \, \pi \left \lfloor \frac{\pi + 4 \, \arctan \left (x\right )}{4 \, \pi } + \frac{1}{2} \right \rfloor + x - \frac{5}{2 \,{\left (x - 1\right )}} + 2 \, \arctan \left (x\right ) + \frac{3}{4} \,{\rm ln}\left (\frac{2}{x - 1} + \frac{2}{{\left (x - 1\right )}^{2}} + 1\right ) + 2 \,{\rm ln}\left ({\left | x - 1 \right |}\right ) - 1 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*pi - 2*pi*floor(1/4*(pi + 4*arctan(x))/pi + 1/2) + x - 5/2/(x - 1) + 2*arcta
n(x) + 3/4*ln(2/(x - 1) + 2/(x - 1)^2 + 1) + 2*ln(abs(x - 1)) - 1