3.432 \(\int \frac{24+8 x}{x \left (-4+x^2\right )} \, dx\)

Optimal. Leaf size=17 \[ 5 \log (2-x)-6 \log (x)+\log (x+2) \]

[Out]

5*Log[2 - x] - 6*Log[x] + Log[2 + x]

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Rubi [A]  time = 0.041434, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ 5 \log (2-x)-6 \log (x)+\log (x+2) \]

Antiderivative was successfully verified.

[In]  Int[(24 + 8*x)/(x*(-4 + x^2)),x]

[Out]

5*Log[2 - x] - 6*Log[x] + Log[2 + x]

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Rubi in Sympy [A]  time = 5.6143, size = 15, normalized size = 0.88 \[ - 6 \log{\left (x \right )} + 5 \log{\left (- x + 2 \right )} + \log{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((24+8*x)/x/(x**2-4),x)

[Out]

-6*log(x) + 5*log(-x + 2) + log(x + 2)

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Mathematica [A]  time = 0.00966221, size = 27, normalized size = 1.59 \[ 8 \left (\frac{5}{8} \log (2-x)-\frac{3 \log (x)}{4}+\frac{1}{8} \log (x+2)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(24 + 8*x)/(x*(-4 + x^2)),x]

[Out]

8*((5*Log[2 - x])/8 - (3*Log[x])/4 + Log[2 + x]/8)

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Maple [A]  time = 0.01, size = 16, normalized size = 0.9 \[ \ln \left ( 2+x \right ) -6\,\ln \left ( x \right ) +5\,\ln \left ( x-2 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((24+8*x)/x/(x^2-4),x)

[Out]

ln(2+x)-6*ln(x)+5*ln(x-2)

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Maxima [A]  time = 0.796886, size = 20, normalized size = 1.18 \[ \log \left (x + 2\right ) + 5 \, \log \left (x - 2\right ) - 6 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(8*(x + 3)/((x^2 - 4)*x),x, algorithm="maxima")

[Out]

log(x + 2) + 5*log(x - 2) - 6*log(x)

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Fricas [A]  time = 0.27961, size = 20, normalized size = 1.18 \[ \log \left (x + 2\right ) + 5 \, \log \left (x - 2\right ) - 6 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(8*(x + 3)/((x^2 - 4)*x),x, algorithm="fricas")

[Out]

log(x + 2) + 5*log(x - 2) - 6*log(x)

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Sympy [A]  time = 0.264976, size = 15, normalized size = 0.88 \[ - 6 \log{\left (x \right )} + 5 \log{\left (x - 2 \right )} + \log{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((24+8*x)/x/(x**2-4),x)

[Out]

-6*log(x) + 5*log(x - 2) + log(x + 2)

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GIAC/XCAS [A]  time = 0.259095, size = 24, normalized size = 1.41 \[{\rm ln}\left ({\left | x + 2 \right |}\right ) + 5 \,{\rm ln}\left ({\left | x - 2 \right |}\right ) - 6 \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(8*(x + 3)/((x^2 - 4)*x),x, algorithm="giac")

[Out]

ln(abs(x + 2)) + 5*ln(abs(x - 2)) - 6*ln(abs(x))