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

Optimal. Leaf size=21 \[ \frac{1}{6} \log \left (x^2+1\right )-\frac{1}{6} \log \left (x^2+4\right ) \]

[Out]

Log[1 + x^2]/6 - Log[4 + x^2]/6

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Rubi [A]  time = 0.0377833, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{6} \log \left (x^2+1\right )-\frac{1}{6} \log \left (x^2+4\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/((1 + x^2)*(4 + x^2)),x]

[Out]

Log[1 + x^2]/6 - Log[4 + x^2]/6

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Rubi in Sympy [A]  time = 6.70731, size = 15, normalized size = 0.71 \[ \frac{\log{\left (x^{2} + 1 \right )}}{6} - \frac{\log{\left (x^{2} + 4 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2 + 1)/6 - log(x**2 + 4)/6

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Mathematica [A]  time = 0.00697595, size = 21, normalized size = 1. \[ \frac{1}{6} \log \left (x^2+1\right )-\frac{1}{6} \log \left (x^2+4\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[1 + x^2]/6 - Log[4 + x^2]/6

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Maple [A]  time = 0.009, size = 18, normalized size = 0.9 \[{\frac{\ln \left ({x}^{2}+1 \right ) }{6}}-{\frac{\ln \left ({x}^{2}+4 \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^2+1)/(x^2+4),x)

[Out]

1/6*ln(x^2+1)-1/6*ln(x^2+4)

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Maxima [A]  time = 1.33336, size = 23, normalized size = 1.1 \[ -\frac{1}{6} \, \log \left (x^{2} + 4\right ) + \frac{1}{6} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*log(x^2 + 4) + 1/6*log(x^2 + 1)

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Fricas [A]  time = 0.222726, size = 23, normalized size = 1.1 \[ -\frac{1}{6} \, \log \left (x^{2} + 4\right ) + \frac{1}{6} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*log(x^2 + 4) + 1/6*log(x^2 + 1)

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Sympy [A]  time = 0.205388, size = 15, normalized size = 0.71 \[ \frac{\log{\left (x^{2} + 1 \right )}}{6} - \frac{\log{\left (x^{2} + 4 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2 + 1)/6 - log(x**2 + 4)/6

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GIAC/XCAS [A]  time = 0.23364, size = 23, normalized size = 1.1 \[ -\frac{1}{6} \,{\rm ln}\left (x^{2} + 4\right ) + \frac{1}{6} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*ln(x^2 + 4) + 1/6*ln(x^2 + 1)