3.329 \(\int \frac{-20 x+4 x^2}{9-10 x^2+x^4} \, dx\)

Optimal. Leaf size=31 \[ \log (1-x)-\frac{1}{2} \log (3-x)+\frac{3}{2} \log (x+1)-2 \log (x+3) \]

[Out]

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

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Rubi [A]  time = 0.087748, antiderivative size = 41, normalized size of antiderivative = 1.32, number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{5}{4} \log \left (1-x^2\right )-\frac{5}{4} \log \left (9-x^2\right )-\frac{3}{2} \tanh ^{-1}\left (\frac{x}{3}\right )+\frac{1}{2} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(-20*x + 4*x^2)/(9 - 10*x^2 + x^4),x]

[Out]

(-3*ArcTanh[x/3])/2 + ArcTanh[x]/2 + (5*Log[1 - x^2])/4 - (5*Log[9 - x^2])/4

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Rubi in Sympy [A]  time = 21.913, size = 32, normalized size = 1.03 \[ \frac{5 \log{\left (- x^{2} + 1 \right )}}{4} - \frac{5 \log{\left (- x^{2} + 9 \right )}}{4} - \frac{3 \operatorname{atanh}{\left (\frac{x}{3} \right )}}{2} + \frac{\operatorname{atanh}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4*x**2-20*x)/(x**4-10*x**2+9),x)

[Out]

5*log(-x**2 + 1)/4 - 5*log(-x**2 + 9)/4 - 3*atanh(x/3)/2 + atanh(x)/2

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Mathematica [A]  time = 0.0109546, size = 39, normalized size = 1.26 \[ 4 \left (\frac{1}{4} \log (1-x)-\frac{1}{8} \log (3-x)+\frac{3}{8} \log (x+1)-\frac{1}{2} \log (x+3)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-20*x + 4*x^2)/(9 - 10*x^2 + x^4),x]

[Out]

4*(Log[1 - x]/4 - Log[3 - x]/8 + (3*Log[1 + x])/8 - Log[3 + x]/2)

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Maple [A]  time = 0.013, size = 24, normalized size = 0.8 \[ \ln \left ( -1+x \right ) -{\frac{\ln \left ( -3+x \right ) }{2}}+{\frac{3\,\ln \left ( 1+x \right ) }{2}}-2\,\ln \left ( 3+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4*x^2-20*x)/(x^4-10*x^2+9),x)

[Out]

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

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Maxima [A]  time = 0.817465, size = 31, normalized size = 1. \[ -2 \, \log \left (x + 3\right ) + \frac{3}{2} \, \log \left (x + 1\right ) + \log \left (x - 1\right ) - \frac{1}{2} \, \log \left (x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4*(x^2 - 5*x)/(x^4 - 10*x^2 + 9),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.257986, size = 31, normalized size = 1. \[ -2 \, \log \left (x + 3\right ) + \frac{3}{2} \, \log \left (x + 1\right ) + \log \left (x - 1\right ) - \frac{1}{2} \, \log \left (x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4*(x^2 - 5*x)/(x^4 - 10*x^2 + 9),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.54298, size = 26, normalized size = 0.84 \[ - \frac{\log{\left (x - 3 \right )}}{2} + \log{\left (x - 1 \right )} + \frac{3 \log{\left (x + 1 \right )}}{2} - 2 \log{\left (x + 3 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x**2-20*x)/(x**4-10*x**2+9),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.260978, size = 36, normalized size = 1.16 \[ -2 \,{\rm ln}\left ({\left | x + 3 \right |}\right ) + \frac{3}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) +{\rm ln}\left ({\left | x - 1 \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4*(x^2 - 5*x)/(x^4 - 10*x^2 + 9),x, algorithm="giac")

[Out]

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