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

Optimal. Leaf size=43 \[ -\frac{1-2 x}{5 \left (x^2+1\right )}-\frac{14}{25} \log \left (x^2+1\right )-\frac{47}{25} \log (2-x)-\frac{46}{25} \tan ^{-1}(x) \]

[Out]

-(1 - 2*x)/(5*(1 + x^2)) - (46*ArcTan[x])/25 - (47*Log[2 - x])/25 - (14*Log[1 +
x^2])/25

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Rubi [A]  time = 0.114839, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{1-2 x}{5 \left (x^2+1\right )}-\frac{14}{25} \log \left (x^2+1\right )-\frac{47}{25} \log (2-x)-\frac{46}{25} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-(1 - 2*x)/(5*(1 + x^2)) - (46*ArcTan[x])/25 - (47*Log[2 - x])/25 - (14*Log[1 +
x^2])/25

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.036632, size = 57, normalized size = 1.33 \[ \frac{2 (x-2)+3}{5 \left ((x-2)^2+4 (x-2)+5\right )}-\frac{14}{25} \log \left ((x-2)^2+4 (x-2)+5\right )-\frac{47}{25} \log (x-2)-\frac{46}{25} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(3 + 2*(-2 + x))/(5*(5 + 4*(-2 + x) + (-2 + x)^2)) - (46*ArcTan[x])/25 - (14*Log
[5 + 4*(-2 + x) + (-2 + x)^2])/25 - (47*Log[-2 + x])/25

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Maple [A]  time = 0.013, size = 34, normalized size = 0.8 \[ -{\frac{2}{25\,{x}^{2}+25} \left ( -5\,x+{\frac{5}{2}} \right ) }-{\frac{14\,\ln \left ({x}^{2}+1 \right ) }{25}}-{\frac{46\,\arctan \left ( x \right ) }{25}}-{\frac{47\,\ln \left ( x-2 \right ) }{25}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/25*(-5*x+5/2)/(x^2+1)-14/25*ln(x^2+1)-46/25*arctan(x)-47/25*ln(x-2)

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Maxima [A]  time = 0.90292, size = 45, normalized size = 1.05 \[ \frac{2 \, x - 1}{5 \,{\left (x^{2} + 1\right )}} - \frac{46}{25} \, \arctan \left (x\right ) - \frac{14}{25} \, \log \left (x^{2} + 1\right ) - \frac{47}{25} \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(2*x - 1)/(x^2 + 1) - 46/25*arctan(x) - 14/25*log(x^2 + 1) - 47/25*log(x - 2
)

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Fricas [A]  time = 0.274598, size = 63, normalized size = 1.47 \[ -\frac{46 \,{\left (x^{2} + 1\right )} \arctan \left (x\right ) + 14 \,{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 47 \,{\left (x^{2} + 1\right )} \log \left (x - 2\right ) - 10 \, x + 5}{25 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/25*(46*(x^2 + 1)*arctan(x) + 14*(x^2 + 1)*log(x^2 + 1) + 47*(x^2 + 1)*log(x -
2) - 10*x + 5)/(x^2 + 1)

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Sympy [A]  time = 0.421405, size = 36, normalized size = 0.84 \[ \frac{2 x - 1}{5 x^{2} + 5} - \frac{47 \log{\left (x - 2 \right )}}{25} - \frac{14 \log{\left (x^{2} + 1 \right )}}{25} - \frac{46 \operatorname{atan}{\left (x \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*x - 1)/(5*x**2 + 5) - 47*log(x - 2)/25 - 14*log(x**2 + 1)/25 - 46*atan(x)/25

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GIAC/XCAS [A]  time = 0.261812, size = 46, normalized size = 1.07 \[ \frac{2 \, x - 1}{5 \,{\left (x^{2} + 1\right )}} - \frac{46}{25} \, \arctan \left (x\right ) - \frac{14}{25} \,{\rm ln}\left (x^{2} + 1\right ) - \frac{47}{25} \,{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(2*x - 1)/(x^2 + 1) - 46/25*arctan(x) - 14/25*ln(x^2 + 1) - 47/25*ln(abs(x -
2))