3.1476 \(\int \frac{1}{x (1-x^8)} \, dx\)

Optimal. Leaf size=15 \[ \log (x)-\frac{1}{8} \log \left (1-x^8\right ) \]

[Out]

Log[x] - Log[1 - x^8]/8

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Rubi [A]  time = 0.007262, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 36, 31, 29} \[ \log (x)-\frac{1}{8} \log \left (1-x^8\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 - x^8)),x]

[Out]

Log[x] - Log[1 - x^8]/8

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 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0033479, size = 15, normalized size = 1. \[ \log (x)-\frac{1}{8} \log \left (1-x^8\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 - x^8)),x]

[Out]

Log[x] - Log[1 - x^8]/8

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Maple [B]  time = 0.01, size = 32, normalized size = 2.1 \begin{align*} -{\frac{\ln \left ({x}^{2}+1 \right ) }{8}}+\ln \left ( x \right ) -{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{\ln \left ({x}^{4}+1 \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-x^8+1),x)

[Out]

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

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Maxima [A]  time = 0.954012, size = 20, normalized size = 1.33 \begin{align*} -\frac{1}{8} \, \log \left (x^{8} - 1\right ) + \frac{1}{8} \, \log \left (x^{8}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-x^8+1),x, algorithm="maxima")

[Out]

-1/8*log(x^8 - 1) + 1/8*log(x^8)

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Fricas [A]  time = 1.26036, size = 38, normalized size = 2.53 \begin{align*} -\frac{1}{8} \, \log \left (x^{8} - 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-x^8+1),x, algorithm="fricas")

[Out]

-1/8*log(x^8 - 1) + log(x)

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Sympy [A]  time = 0.111104, size = 10, normalized size = 0.67 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{8} - 1 \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-x**8+1),x)

[Out]

log(x) - log(x**8 - 1)/8

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Giac [A]  time = 1.16158, size = 22, normalized size = 1.47 \begin{align*} \frac{1}{8} \, \log \left (x^{8}\right ) - \frac{1}{8} \, \log \left ({\left | x^{8} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-x^8+1),x, algorithm="giac")

[Out]

1/8*log(x^8) - 1/8*log(abs(x^8 - 1))