3.35 \(\int \frac{1}{x^4 \left (x-x^3\right )} \, dx\)

Optimal. Leaf size=29 \[ -\frac{1}{4 x^4}-\frac{1}{2 x^2}-\frac{1}{2} \log \left (1-x^2\right )+\log (x) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0351831, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{4 x^4}-\frac{1}{2 x^2}-\frac{1}{2} \log \left (1-x^2\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(x - x^3)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.10786, size = 27, normalized size = 0.93 \[ \frac{\log{\left (x^{2} \right )}}{2} - \frac{\log{\left (- x^{2} + 1 \right )}}{2} - \frac{1}{2 x^{2}} - \frac{1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00605696, size = 29, normalized size = 1. \[ -\frac{1}{4 x^4}-\frac{1}{2 x^2}-\frac{1}{2} \log \left (1-x^2\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(x - x^3)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 26, normalized size = 0.9 \[ -{\frac{1}{4\,{x}^{4}}}-{\frac{1}{2\,{x}^{2}}}+\ln \left ( x \right ) -{\frac{\ln \left ( -1+x \right ) }{2}}-{\frac{\ln \left ( 1+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-x^3+x),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.38804, size = 36, normalized size = 1.24 \[ -\frac{2 \, x^{2} + 1}{4 \, x^{4}} - \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.202468, size = 41, normalized size = 1.41 \[ -\frac{2 \, x^{4} \log \left (x^{2} - 1\right ) - 4 \, x^{4} \log \left (x\right ) + 2 \, x^{2} + 1}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.276597, size = 22, normalized size = 0.76 \[ \log{\left (x \right )} - \frac{\log{\left (x^{2} - 1 \right )}}{2} - \frac{2 x^{2} + 1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(-x**3+x),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220574, size = 45, normalized size = 1.55 \[ -\frac{3 \, x^{4} + 2 \, x^{2} + 1}{4 \, x^{4}} + \frac{1}{2} \,{\rm ln}\left (x^{2}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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