3.347 \(\int \frac{3+x+x^2+x^3}{\left (1+x^2\right ) \left (3+x^2\right )} \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{2} \log \left (x^2+3\right )+\tan ^{-1}(x) \]

[Out]

ArcTan[x] + Log[3 + x^2]/2

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Rubi [A]  time = 0.166696, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{1}{2} \log \left (x^2+3\right )+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(3 + x + x^2 + x^3)/((1 + x^2)*(3 + x^2)),x]

[Out]

ArcTan[x] + Log[3 + x^2]/2

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Rubi in Sympy [A]  time = 28.9777, size = 10, normalized size = 0.77 \[ \frac{\log{\left (x^{2} + 3 \right )}}{2} + \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3+x**2+x+3)/(x**2+1)/(x**2+3),x)

[Out]

log(x**2 + 3)/2 + atan(x)

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Mathematica [A]  time = 0.0104689, size = 13, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+3\right )+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + x + x^2 + x^3)/((1 + x^2)*(3 + x^2)),x]

[Out]

ArcTan[x] + Log[3 + x^2]/2

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Maple [A]  time = 0.005, size = 12, normalized size = 0.9 \[ \arctan \left ( x \right ) +{\frac{\ln \left ({x}^{2}+3 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+x^2+x+3)/(x^2+1)/(x^2+3),x)

[Out]

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

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Maxima [A]  time = 0.89021, size = 15, normalized size = 1.15 \[ \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x^2 + x + 3)/((x^2 + 3)*(x^2 + 1)),x, algorithm="maxima")

[Out]

arctan(x) + 1/2*log(x^2 + 3)

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Fricas [A]  time = 0.254576, size = 15, normalized size = 1.15 \[ \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x^2 + x + 3)/((x^2 + 3)*(x^2 + 1)),x, algorithm="fricas")

[Out]

arctan(x) + 1/2*log(x^2 + 3)

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Sympy [A]  time = 0.274042, size = 10, normalized size = 0.77 \[ \frac{\log{\left (x^{2} + 3 \right )}}{2} + \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+x**2+x+3)/(x**2+1)/(x**2+3),x)

[Out]

log(x**2 + 3)/2 + atan(x)

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GIAC/XCAS [A]  time = 0.260913, size = 15, normalized size = 1.15 \[ \arctan \left (x\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x^2 + x + 3)/((x^2 + 3)*(x^2 + 1)),x, algorithm="giac")

[Out]

arctan(x) + 1/2*ln(x^2 + 3)