3.155 \(\int \frac{1}{x (3+4 x^3+x^6)} \, dx\)

Optimal. Leaf size=27 \[ -\frac{1}{6} \log \left (x^3+1\right )+\frac{1}{18} \log \left (x^3+3\right )+\frac{\log (x)}{3} \]

[Out]

Log[x]/3 - Log[1 + x^3]/6 + Log[3 + x^3]/18

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Rubi [A]  time = 0.0189696, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1357, 705, 29, 632, 31} \[ -\frac{1}{6} \log \left (x^3+1\right )+\frac{1}{18} \log \left (x^3+3\right )+\frac{\log (x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(3 + 4*x^3 + x^6)),x]

[Out]

Log[x]/3 - Log[1 + x^3]/6 + Log[3 + x^3]/18

Rule 1357

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

Rule 705

Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 - b*d*e + a*e^2
), Int[1/(d + e*x), x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x]
/; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 29

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

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

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 (3+4 x^3+x^6\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x \left (3+4 x+x^2\right )} \, dx,x,x^3\right )\\ &=\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^3\right )+\frac{1}{9} \operatorname{Subst}\left (\int \frac{-4-x}{3+4 x+x^2} \, dx,x,x^3\right )\\ &=\frac{\log (x)}{3}+\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{3+x} \, dx,x,x^3\right )-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^3\right )\\ &=\frac{\log (x)}{3}-\frac{1}{6} \log \left (1+x^3\right )+\frac{1}{18} \log \left (3+x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.0055941, size = 27, normalized size = 1. \[ -\frac{1}{6} \log \left (x^3+1\right )+\frac{1}{18} \log \left (x^3+3\right )+\frac{\log (x)}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(3 + 4*x^3 + x^6)),x]

[Out]

Log[x]/3 - Log[1 + x^3]/6 + Log[3 + x^3]/18

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Maple [A]  time = 0.013, size = 31, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( x \right ) }{3}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}}+{\frac{\ln \left ({x}^{3}+3 \right ) }{18}}-{\frac{\ln \left ( 1+x \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^6+4*x^3+3),x)

[Out]

1/3*ln(x)-1/6*ln(x^2-x+1)+1/18*ln(x^3+3)-1/6*ln(1+x)

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Maxima [A]  time = 1.19491, size = 31, normalized size = 1.15 \begin{align*} \frac{1}{18} \, \log \left (x^{3} + 3\right ) - \frac{1}{6} \, \log \left (x^{3} + 1\right ) + \frac{1}{9} \, \log \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^6+4*x^3+3),x, algorithm="maxima")

[Out]

1/18*log(x^3 + 3) - 1/6*log(x^3 + 1) + 1/9*log(x^3)

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Fricas [A]  time = 1.43902, size = 69, normalized size = 2.56 \begin{align*} \frac{1}{18} \, \log \left (x^{3} + 3\right ) - \frac{1}{6} \, \log \left (x^{3} + 1\right ) + \frac{1}{3} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^6+4*x^3+3),x, algorithm="fricas")

[Out]

1/18*log(x^3 + 3) - 1/6*log(x^3 + 1) + 1/3*log(x)

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Sympy [A]  time = 0.143223, size = 20, normalized size = 0.74 \begin{align*} \frac{\log{\left (x \right )}}{3} - \frac{\log{\left (x^{3} + 1 \right )}}{6} + \frac{\log{\left (x^{3} + 3 \right )}}{18} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x**6+4*x**3+3),x)

[Out]

log(x)/3 - log(x**3 + 1)/6 + log(x**3 + 3)/18

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Giac [A]  time = 1.15731, size = 32, normalized size = 1.19 \begin{align*} \frac{1}{18} \, \log \left ({\left | x^{3} + 3 \right |}\right ) - \frac{1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) + \frac{1}{3} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^6+4*x^3+3),x, algorithm="giac")

[Out]

1/18*log(abs(x^3 + 3)) - 1/6*log(abs(x^3 + 1)) + 1/3*log(abs(x))