3.279 \(\int \frac{1}{x^5 \left (1+2 x^4+x^8\right )} \, dx\)

Optimal. Leaf size=33 \[ -\frac{1}{4 \left (x^4+1\right )}-\frac{1}{4 x^4}+\frac{1}{2} \log \left (x^4+1\right )-2 \log (x) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0358058, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{1}{4 \left (x^4+1\right )}-\frac{1}{4 x^4}+\frac{1}{2} \log \left (x^4+1\right )-2 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(1 + 2*x^4 + x^8)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.74552, size = 29, normalized size = 0.88 \[ - \frac{\log{\left (x^{4} \right )}}{2} + \frac{\log{\left (x^{4} + 1 \right )}}{2} - \frac{1}{4 \left (x^{4} + 1\right )} - \frac{1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(x**8+2*x**4+1),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0191513, size = 33, normalized size = 1. \[ -\frac{1}{4 \left (x^4+1\right )}-\frac{1}{4 x^4}+\frac{1}{2} \log \left (x^4+1\right )-2 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(1 + 2*x^4 + x^8)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 28, normalized size = 0.9 \[ -{\frac{1}{4\,{x}^{4}}}-{\frac{1}{4\,{x}^{4}+4}}-2\,\ln \left ( x \right ) +{\frac{\ln \left ({x}^{4}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(x^8+2*x^4+1),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.759771, size = 45, normalized size = 1.36 \[ -\frac{2 \, x^{4} + 1}{4 \,{\left (x^{8} + x^{4}\right )}} + \frac{1}{2} \, \log \left (x^{4} + 1\right ) - \frac{1}{2} \, \log \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 2*x^4 + 1)*x^5),x, algorithm="maxima")

[Out]

-1/4*(2*x^4 + 1)/(x^8 + x^4) + 1/2*log(x^4 + 1) - 1/2*log(x^4)

_______________________________________________________________________________________

Fricas [A]  time = 0.252424, size = 59, normalized size = 1.79 \[ -\frac{2 \, x^{4} - 2 \,{\left (x^{8} + x^{4}\right )} \log \left (x^{4} + 1\right ) + 8 \,{\left (x^{8} + x^{4}\right )} \log \left (x\right ) + 1}{4 \,{\left (x^{8} + x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 2*x^4 + 1)*x^5),x, algorithm="fricas")

[Out]

-1/4*(2*x^4 - 2*(x^8 + x^4)*log(x^4 + 1) + 8*(x^8 + x^4)*log(x) + 1)/(x^8 + x^4)

_______________________________________________________________________________________

Sympy [A]  time = 0.428095, size = 29, normalized size = 0.88 \[ - \frac{2 x^{4} + 1}{4 x^{8} + 4 x^{4}} - 2 \log{\left (x \right )} + \frac{\log{\left (x^{4} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(x**8+2*x**4+1),x)

[Out]

-(2*x**4 + 1)/(4*x**8 + 4*x**4) - 2*log(x) + log(x**4 + 1)/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.29694, size = 45, normalized size = 1.36 \[ -\frac{2 \, x^{4} + 1}{4 \,{\left (x^{8} + x^{4}\right )}} + \frac{1}{2} \,{\rm ln}\left (x^{4} + 1\right ) - \frac{1}{2} \,{\rm ln}\left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 2*x^4 + 1)*x^5),x, algorithm="giac")

[Out]

-1/4*(2*x^4 + 1)/(x^8 + x^4) + 1/2*ln(x^4 + 1) - 1/2*ln(x^4)