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