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

Optimal. Leaf size=25 \[ -\frac{1}{4} \log \left (x^2+1\right )+\frac{1}{2} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]

[Out]

ArcTan[x]/2 + Log[1 + x]/2 - Log[1 + x^2]/4

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Rubi [A]  time = 0.0310307, 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 \[ -\frac{1}{4} \log \left (x^2+1\right )+\frac{1}{2} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[1/((1 + x)*(1 + x^2)),x]

[Out]

ArcTan[x]/2 + Log[1 + x]/2 - Log[1 + x^2]/4

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Rubi in Sympy [A]  time = 5.12394, size = 19, normalized size = 0.76 \[ \frac{\log{\left (x + 1 \right )}}{2} - \frac{\log{\left (x^{2} + 1 \right )}}{4} + \frac{\operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00840627, size = 25, normalized size = 1. \[ -\frac{1}{4} \log \left (x^2+1\right )+\frac{1}{2} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[x]/2 + Log[1 + x]/2 - Log[1 + x^2]/4

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Maple [A]  time = 0.006, size = 20, normalized size = 0.8 \[{\frac{\arctan \left ( x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x)/(x^2+1),x)

[Out]

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

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Maxima [A]  time = 0.883932, size = 26, normalized size = 1.04 \[ \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{2} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.266685, size = 26, normalized size = 1.04 \[ \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{2} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.261416, size = 19, normalized size = 0.76 \[ \frac{\log{\left (x + 1 \right )}}{2} - \frac{\log{\left (x^{2} + 1 \right )}}{4} + \frac{\operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+x)/(x**2+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.259575, size = 27, normalized size = 1.08 \[ \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \,{\rm ln}\left (x^{2} + 1\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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