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

Optimal. Leaf size=15 \[ \log (x+4)-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{2}\right ) \]

[Out]

-ArcTanh[x/2]/2 + Log[4 + x]

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Rubi [A]  time = 0.0849651, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \log (x+4)-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTanh[x/2]/2 + Log[4 + x]

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Rubi in Sympy [A]  time = 34.0988, size = 10, normalized size = 0.67 \[ \log{\left (x + 4 \right )} - \frac{\operatorname{atanh}{\left (\frac{x}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x + 4) - atanh(x/2)/2

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Mathematica [A]  time = 0.00925871, size = 23, normalized size = 1.53 \[ \frac{1}{4} \log (2-x)-\frac{1}{4} \log (x+2)+\log (x+4) \]

Antiderivative was successfully verified.

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

[Out]

Log[2 - x]/4 - Log[2 + x]/4 + Log[4 + x]

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Maple [A]  time = 0.012, size = 18, normalized size = 1.2 \[ -{\frac{\ln \left ( 2+x \right ) }{4}}+\ln \left ( 4+x \right ) +{\frac{\ln \left ( x-2 \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+x)/(4+x)/(x^2-4),x)

[Out]

-1/4*ln(2+x)+ln(4+x)+1/4*ln(x-2)

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Maxima [A]  time = 0.879987, size = 23, normalized size = 1.53 \[ \log \left (x + 4\right ) - \frac{1}{4} \, \log \left (x + 2\right ) + \frac{1}{4} \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x + 4) - 1/4*log(x + 2) + 1/4*log(x - 2)

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Fricas [A]  time = 0.275746, size = 23, normalized size = 1.53 \[ \log \left (x + 4\right ) - \frac{1}{4} \, \log \left (x + 2\right ) + \frac{1}{4} \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x + 4) - 1/4*log(x + 2) + 1/4*log(x - 2)

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Sympy [A]  time = 0.291078, size = 17, normalized size = 1.13 \[ \frac{\log{\left (x - 2 \right )}}{4} - \frac{\log{\left (x + 2 \right )}}{4} + \log{\left (x + 4 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x - 2)/4 - log(x + 2)/4 + log(x + 4)

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GIAC/XCAS [A]  time = 0.261533, size = 27, normalized size = 1.8 \[{\rm ln}\left ({\left | x + 4 \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(x + 4)) - 1/4*ln(abs(x + 2)) + 1/4*ln(abs(x - 2))