3.277 \(\int \frac{1+x+x^3}{4 x+2 x^2+x^4} \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{4} \log \left (x^4+2 x^2+4 x\right ) \]

[Out]

Log[4*x + 2*x^2 + x^4]/4

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Rubi [A]  time = 0.00728313, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{1}{4} \log \left (x^4+2 x^2+4 x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x + x^3)/(4*x + 2*x^2 + x^4),x]

[Out]

Log[4*x + 2*x^2 + x^4]/4

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Rubi in Sympy [A]  time = 6.53486, size = 12, normalized size = 0.71 \[ \frac{\log{\left (x \left (x^{3} + 2 x + 4\right ) \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3+x+1)/(x**4+2*x**2+4*x),x)

[Out]

log(x*(x**3 + 2*x + 4))/4

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Mathematica [A]  time = 0.00861106, size = 20, normalized size = 1.18 \[ \frac{1}{4} \log \left (x^3+2 x+4\right )+\frac{\log (x)}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x + x^3)/(4*x + 2*x^2 + x^4),x]

[Out]

Log[x]/4 + Log[4 + 2*x + x^3]/4

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Maple [A]  time = 0.003, size = 14, normalized size = 0.8 \[{\frac{\ln \left ( x \left ({x}^{3}+2\,x+4 \right ) \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+x+1)/(x^4+2*x^2+4*x),x)

[Out]

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

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Maxima [A]  time = 1.32374, size = 20, normalized size = 1.18 \[ \frac{1}{4} \, \log \left (x^{4} + 2 \, x^{2} + 4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x + 1)/(x^4 + 2*x^2 + 4*x),x, algorithm="maxima")

[Out]

1/4*log(x^4 + 2*x^2 + 4*x)

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Fricas [A]  time = 0.194932, size = 20, normalized size = 1.18 \[ \frac{1}{4} \, \log \left (x^{4} + 2 \, x^{2} + 4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x + 1)/(x^4 + 2*x^2 + 4*x),x, algorithm="fricas")

[Out]

1/4*log(x^4 + 2*x^2 + 4*x)

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Sympy [A]  time = 0.099663, size = 14, normalized size = 0.82 \[ \frac{\log{\left (x^{4} + 2 x^{2} + 4 x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+x+1)/(x**4+2*x**2+4*x),x)

[Out]

log(x**4 + 2*x**2 + 4*x)/4

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GIAC/XCAS [A]  time = 0.219571, size = 24, normalized size = 1.41 \[ \frac{1}{4} \,{\rm ln}\left ({\left | x^{3} + 2 \, x + 4 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x + 1)/(x^4 + 2*x^2 + 4*x),x, algorithm="giac")

[Out]

1/4*ln(abs(x^3 + 2*x + 4)) + 1/4*ln(abs(x))