3.1495 \(\int \frac {1}{x (1+x^8)} \, dx\)

Optimal. Leaf size=13 \[ \log (x)-\frac {1}{8} \log \left (x^8+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {266, 36, 29, 31} \[ \log (x)-\frac {1}{8} \log \left (x^8+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 + x^8)),x]

[Out]

Log[x] - Log[1 + x^8]/8

Rule 29

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

Rule 31

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

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 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]]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ \log (x)-\frac {1}{8} \log \left (x^8+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + x^8)),x]

[Out]

Log[x] - Log[1 + x^8]/8

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fricas [A]  time = 0.76, size = 11, normalized size = 0.85 \[ -\frac {1}{8} \, \log \left (x^{8} + 1\right ) + \log \relax (x) \]

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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giac [A]  time = 0.17, size = 15, normalized size = 1.15 \[ -\frac {1}{8} \, \log \left (x^{8} + 1\right ) + \frac {1}{8} \, \log \left (x^{8}\right ) \]

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) + 1/8*log(x^8)

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maple [A]  time = 0.00, size = 12, normalized size = 0.92 \[ \ln \relax (x )-\frac {\ln \left (x^{8}+1\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.90, size = 15, normalized size = 1.15 \[ -\frac {1}{8} \, \log \left (x^{8} + 1\right ) + \frac {1}{8} \, \log \left (x^{8}\right ) \]

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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mupad [B]  time = 0.04, size = 11, normalized size = 0.85 \[ \ln \relax (x)-\frac {\ln \left (x^8+1\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^8 + 1)),x)

[Out]

log(x) - log(x^8 + 1)/8

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sympy [A]  time = 0.14, size = 10, normalized size = 0.77 \[ \log {\relax (x )} - \frac {\log {\left (x^{8} + 1 \right )}}{8} \]

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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