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

Optimal. Leaf size=37 \[ \frac{1}{6} \log \left (x^2+1\right )-\frac{1}{6} \log \left (x^2+4\right )-\frac{1}{3} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{2}{3} \tan ^{-1}(x) \]

[Out]

-ArcTan[x/2]/3 + (2*ArcTan[x])/3 + Log[1 + x^2]/6 - Log[4 + x^2]/6

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Rubi [A]  time = 0.0796073, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{6} \log \left (x^2+1\right )-\frac{1}{6} \log \left (x^2+4\right )-\frac{1}{3} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{2}{3} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[x/2]/3 + (2*ArcTan[x])/3 + Log[1 + x^2]/6 - Log[4 + x^2]/6

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Rubi in Sympy [A]  time = 17.1073, size = 29, normalized size = 0.78 \[ \frac{\log{\left (x^{2} + 1 \right )}}{6} - \frac{\log{\left (x^{2} + 4 \right )}}{6} - \frac{\operatorname{atan}{\left (\frac{x}{2} \right )}}{3} + \frac{2 \operatorname{atan}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2 + 1)/6 - log(x**2 + 4)/6 - atan(x/2)/3 + 2*atan(x)/3

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Mathematica [A]  time = 0.0124736, size = 37, normalized size = 1. \[ \frac{1}{6} \log \left (x^2+1\right )-\frac{1}{6} \log \left (x^2+4\right )-\frac{1}{3} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{2}{3} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[x/2]/3 + (2*ArcTan[x])/3 + Log[1 + x^2]/6 - Log[4 + x^2]/6

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Maple [A]  time = 0.005, size = 28, normalized size = 0.8 \[ -{\frac{1}{3}\arctan \left ({\frac{x}{2}} \right ) }+{\frac{2\,\arctan \left ( x \right ) }{3}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{6}}-{\frac{\ln \left ({x}^{2}+4 \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+x)/(x^2+1)/(x^2+4),x)

[Out]

-1/3*arctan(1/2*x)+2/3*arctan(x)+1/6*ln(x^2+1)-1/6*ln(x^2+4)

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Maxima [A]  time = 0.894388, size = 36, normalized size = 0.97 \[ -\frac{1}{3} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{2}{3} \, \arctan \left (x\right ) - \frac{1}{6} \, \log \left (x^{2} + 4\right ) + \frac{1}{6} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*arctan(1/2*x) + 2/3*arctan(x) - 1/6*log(x^2 + 4) + 1/6*log(x^2 + 1)

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Fricas [A]  time = 0.251885, size = 36, normalized size = 0.97 \[ -\frac{1}{3} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{2}{3} \, \arctan \left (x\right ) - \frac{1}{6} \, \log \left (x^{2} + 4\right ) + \frac{1}{6} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*arctan(1/2*x) + 2/3*arctan(x) - 1/6*log(x^2 + 4) + 1/6*log(x^2 + 1)

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Sympy [A]  time = 0.480808, size = 29, normalized size = 0.78 \[ \frac{\log{\left (x^{2} + 1 \right )}}{6} - \frac{\log{\left (x^{2} + 4 \right )}}{6} - \frac{\operatorname{atan}{\left (\frac{x}{2} \right )}}{3} + \frac{2 \operatorname{atan}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2 + 1)/6 - log(x**2 + 4)/6 - atan(x/2)/3 + 2*atan(x)/3

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GIAC/XCAS [A]  time = 0.262499, size = 36, normalized size = 0.97 \[ -\frac{1}{3} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{2}{3} \, \arctan \left (x\right ) - \frac{1}{6} \,{\rm ln}\left (x^{2} + 4\right ) + \frac{1}{6} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*arctan(1/2*x) + 2/3*arctan(x) - 1/6*ln(x^2 + 4) + 1/6*ln(x^2 + 1)