3.1298 \(\int \frac{1}{x (1+x^5)} \, dx\)

Optimal. Leaf size=13 \[ \log (x)-\frac{1}{5} \log \left (x^5+1\right ) \]

[Out]

Log[x] - Log[1 + x^5]/5

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Rubi [A]  time = 0.0047985, antiderivative size = 13, normalized size of antiderivative = 1., 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} \[ \log (x)-\frac{1}{5} \log \left (x^5+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 + x^5)),x]

[Out]

Log[x] - Log[1 + x^5]/5

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0025115, size = 13, normalized size = 1. \[ \log (x)-\frac{1}{5} \log \left (x^5+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + x^5)),x]

[Out]

Log[x] - Log[1 + x^5]/5

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Maple [B]  time = 0.006, size = 29, normalized size = 2.2 \begin{align*} \ln \left ( x \right ) -{\frac{\ln \left ( 1+x \right ) }{5}}-{\frac{\ln \left ({x}^{4}-{x}^{3}+{x}^{2}-x+1 \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^5+1),x)

[Out]

ln(x)-1/5*ln(1+x)-1/5*ln(x^4-x^3+x^2-x+1)

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Maxima [A]  time = 1.02411, size = 20, normalized size = 1.54 \begin{align*} -\frac{1}{5} \, \log \left (x^{5} + 1\right ) + \frac{1}{5} \, \log \left (x^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^5+1),x, algorithm="maxima")

[Out]

-1/5*log(x^5 + 1) + 1/5*log(x^5)

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Fricas [A]  time = 1.69249, size = 38, normalized size = 2.92 \begin{align*} -\frac{1}{5} \, \log \left (x^{5} + 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^5+1),x, algorithm="fricas")

[Out]

-1/5*log(x^5 + 1) + log(x)

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Sympy [A]  time = 0.127759, size = 10, normalized size = 0.77 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{5} + 1 \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x**5+1),x)

[Out]

log(x) - log(x**5 + 1)/5

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Giac [A]  time = 1.15533, size = 18, normalized size = 1.38 \begin{align*} -\frac{1}{5} \, \log \left ({\left | x^{5} + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^5+1),x, algorithm="giac")

[Out]

-1/5*log(abs(x^5 + 1)) + log(abs(x))