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

Optimal. Leaf size=25 \[ -\frac{1}{10} \log \left (x^2+1\right )+\frac{1}{5} \log (x+2)+\frac{2}{5} \tan ^{-1}(x) \]

[Out]

(2*ArcTan[x])/5 + Log[2 + x]/5 - Log[1 + x^2]/10

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Rubi [A]  time = 0.0320511, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{1}{10} \log \left (x^2+1\right )+\frac{1}{5} \log (x+2)+\frac{2}{5} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[1/((2 + x)*(1 + x^2)),x]

[Out]

(2*ArcTan[x])/5 + Log[2 + x]/5 - Log[1 + x^2]/10

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Rubi in Sympy [A]  time = 4.8145, size = 20, normalized size = 0.8 \[ \frac{\log{\left (x + 2 \right )}}{5} - \frac{\log{\left (x^{2} + 1 \right )}}{10} + \frac{2 \operatorname{atan}{\left (x \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x + 2)/5 - log(x**2 + 1)/10 + 2*atan(x)/5

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Mathematica [A]  time = 0.00912464, size = 25, normalized size = 1. \[ -\frac{1}{10} \log \left (x^2+1\right )+\frac{1}{5} \log (x+2)+\frac{2}{5} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTan[x])/5 + Log[2 + x]/5 - Log[1 + x^2]/10

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Maple [A]  time = 0.008, size = 20, normalized size = 0.8 \[{\frac{2\,\arctan \left ( x \right ) }{5}}+{\frac{\ln \left ( 2+x \right ) }{5}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2+x)/(x^2+1),x)

[Out]

2/5*arctan(x)+1/5*ln(2+x)-1/10*ln(x^2+1)

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Maxima [A]  time = 0.889416, size = 26, normalized size = 1.04 \[ \frac{2}{5} \, \arctan \left (x\right ) - \frac{1}{10} \, \log \left (x^{2} + 1\right ) + \frac{1}{5} \, \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*arctan(x) - 1/10*log(x^2 + 1) + 1/5*log(x + 2)

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Fricas [A]  time = 0.260785, size = 26, normalized size = 1.04 \[ \frac{2}{5} \, \arctan \left (x\right ) - \frac{1}{10} \, \log \left (x^{2} + 1\right ) + \frac{1}{5} \, \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*arctan(x) - 1/10*log(x^2 + 1) + 1/5*log(x + 2)

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Sympy [A]  time = 0.287354, size = 20, normalized size = 0.8 \[ \frac{\log{\left (x + 2 \right )}}{5} - \frac{\log{\left (x^{2} + 1 \right )}}{10} + \frac{2 \operatorname{atan}{\left (x \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2+x)/(x**2+1),x)

[Out]

log(x + 2)/5 - log(x**2 + 1)/10 + 2*atan(x)/5

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GIAC/XCAS [A]  time = 0.261556, size = 27, normalized size = 1.08 \[ \frac{2}{5} \, \arctan \left (x\right ) - \frac{1}{10} \,{\rm ln}\left (x^{2} + 1\right ) + \frac{1}{5} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*arctan(x) - 1/10*ln(x^2 + 1) + 1/5*ln(abs(x + 2))