3.298 \(\int \frac{1}{x^5 (1-2 x^4+x^8)} \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{4 \left (1-x^4\right )}-\frac{1}{4 x^4}-\frac{1}{2} \log \left (1-x^4\right )+2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0185109, antiderivative size = 37, 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, Rules used = {28, 266, 44} \[ \frac{1}{4 \left (1-x^4\right )}-\frac{1}{4 x^4}-\frac{1}{2} \log \left (1-x^4\right )+2 \log (x) \]

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

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^5 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac{1}{x^5 \left (-1+x^4\right )^2} \, dx\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{(-1+x)^2 x^2} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{(-1+x)^2}-\frac{2}{-1+x}+\frac{1}{x^2}+\frac{2}{x}\right ) \, dx,x,x^4\right )\\ &=-\frac{1}{4 x^4}+\frac{1}{4 \left (1-x^4\right )}+2 \log (x)-\frac{1}{2} \log \left (1-x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0126081, size = 35, normalized size = 0.95 \[ -\frac{1}{4 \left (x^4-1\right )}-\frac{1}{4 x^4}-\frac{1}{2} \log \left (1-x^4\right )+2 \log (x) \]

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

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Maple [A]  time = 0.016, size = 54, normalized size = 1.5 \begin{align*}{\frac{1}{8\,{x}^{2}+8}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}-{\frac{1}{4\,{x}^{4}}}+2\,\ln \left ( x \right ) +{\frac{1}{16+16\,x}}-{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{1}{16\,x-16}}-{\frac{\ln \left ( x-1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(x^8-2*x^4+1),x)

[Out]

1/8/(x^2+1)-1/2*ln(x^2+1)-1/4/x^4+2*ln(x)+1/16/(1+x)-1/2*ln(1+x)-1/16/(x-1)-1/2*ln(x-1)

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Maxima [A]  time = 0.974259, size = 47, normalized size = 1.27 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^8-2*x^4+1),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)

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Fricas [A]  time = 1.45873, size = 111, normalized size = 3. \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^8-2*x^4+1),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)

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Sympy [A]  time = 0.160786, size = 29, normalized size = 0.78 \begin{align*} - \frac{2 x^{4} - 1}{4 x^{8} - 4 x^{4}} + 2 \log{\left (x \right )} - \frac{\log{\left (x^{4} - 1 \right )}}{2} \end{align*}

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

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Giac [A]  time = 1.11053, size = 49, normalized size = 1.32 \begin{align*} -\frac{2 \, x^{4} - 1}{4 \,{\left (x^{8} - x^{4}\right )}} + \frac{1}{2} \, \log \left (x^{4}\right ) - \frac{1}{2} \, \log \left ({\left | x^{4} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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