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

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

[Out]

-1/(4*x) - 1/(4*(2 + x)) + ArcTanh[1 + x]/2

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Rubi [A]  time = 0.0274699, antiderivative size = 31, normalized size of antiderivative = 1.24, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{1}{4 x}-\frac{1}{4 (x+2)}-\frac{\log (x)}{4}+\frac{1}{4} \log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*x) - 1/(4*(2 + x)) - Log[x]/4 + Log[2 + x]/4

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Rubi in Sympy [A]  time = 15.5941, size = 22, normalized size = 0.88 \[ - \frac{\log{\left (x \right )}}{4} + \frac{\log{\left (x + 2 \right )}}{4} - \frac{1}{4 \left (x + 2\right )} - \frac{1}{4 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.03099, size = 26, normalized size = 1.04 \[ \frac{1}{4} \left (-\frac{2 (x+1)}{x (x+2)}-\log (x)+\log (x+2)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-2*(1 + x))/(x*(2 + x)) - Log[x] + Log[2 + x])/4

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Maple [A]  time = 0.013, size = 24, normalized size = 1. \[ -{\frac{1}{4\,x}}-{\frac{1}{8+4\,x}}-{\frac{\ln \left ( x \right ) }{4}}+{\frac{\ln \left ( 2+x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4+4*x^3+4*x^2),x)

[Out]

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

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Maxima [A]  time = 0.780537, size = 34, normalized size = 1.36 \[ -\frac{x + 1}{2 \,{\left (x^{2} + 2 \, x\right )}} + \frac{1}{4} \, \log \left (x + 2\right ) - \frac{1}{4} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.276176, size = 53, normalized size = 2.12 \[ \frac{{\left (x^{2} + 2 \, x\right )} \log \left (x + 2\right ) -{\left (x^{2} + 2 \, x\right )} \log \left (x\right ) - 2 \, x - 2}{4 \,{\left (x^{2} + 2 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.235672, size = 22, normalized size = 0.88 \[ - \frac{x + 1}{2 x^{2} + 4 x} - \frac{\log{\left (x \right )}}{4} + \frac{\log{\left (x + 2 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.260364, size = 36, normalized size = 1.44 \[ -\frac{x + 1}{2 \,{\left (x^{2} + 2 \, x\right )}} + \frac{1}{4} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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