3.374 \(\int \frac{1}{x (1+3 x^4+x^8)} \, dx\)

Optimal. Leaf size=57 \[ -\frac{1}{40} \left (5+3 \sqrt{5}\right ) \log \left (2 x^4-\sqrt{5}+3\right )-\frac{1}{40} \left (5-3 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right )+\log (x) \]

[Out]

Log[x] - ((5 + 3*Sqrt[5])*Log[3 - Sqrt[5] + 2*x^4])/40 - ((5 - 3*Sqrt[5])*Log[3 + Sqrt[5] + 2*x^4])/40

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Rubi [A]  time = 0.0342272, antiderivative size = 57, 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} \[ -\frac{1}{40} \left (5+3 \sqrt{5}\right ) \log \left (2 x^4-\sqrt{5}+3\right )-\frac{1}{40} \left (5-3 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right )+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 + 3*x^4 + x^8)),x]

[Out]

Log[x] - ((5 + 3*Sqrt[5])*Log[3 - Sqrt[5] + 2*x^4])/40 - ((5 - 3*Sqrt[5])*Log[3 + Sqrt[5] + 2*x^4])/40

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 (1+3 x^4+x^8\right )} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x \left (1+3 x+x^2\right )} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^4\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{-3-x}{1+3 x+x^2} \, dx,x,x^4\right )\\ &=\log (x)+\frac{1}{40} \left (-5+3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x} \, dx,x,x^4\right )-\frac{1}{40} \left (5+3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x} \, dx,x,x^4\right )\\ &=\log (x)-\frac{1}{40} \left (5+3 \sqrt{5}\right ) \log \left (3-\sqrt{5}+2 x^4\right )-\frac{1}{40} \left (5-3 \sqrt{5}\right ) \log \left (3+\sqrt{5}+2 x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0292216, size = 55, normalized size = 0.96 \[ \frac{1}{40} \left (-5-3 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}-3\right )+\frac{1}{40} \left (3 \sqrt{5}-5\right ) \log \left (2 x^4+\sqrt{5}+3\right )+\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + 3*x^4 + x^8)),x]

[Out]

Log[x] + ((-5 - 3*Sqrt[5])*Log[-3 + Sqrt[5] - 2*x^4])/40 + ((-5 + 3*Sqrt[5])*Log[3 + Sqrt[5] + 2*x^4])/40

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Maple [A]  time = 0.007, size = 35, normalized size = 0.6 \begin{align*} -{\frac{\ln \left ({x}^{8}+3\,{x}^{4}+1 \right ) }{8}}+{\frac{3\,\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{4}+3 \right ) \sqrt{5}}{5}} \right ) }+\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^8+3*x^4+1),x)

[Out]

-1/8*ln(x^8+3*x^4+1)+3/20*arctanh(1/5*(2*x^4+3)*5^(1/2))*5^(1/2)+ln(x)

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Maxima [A]  time = 1.48879, size = 69, normalized size = 1.21 \begin{align*} -\frac{3}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) - \frac{1}{8} \, \log \left (x^{8} + 3 \, x^{4} + 1\right ) + \frac{1}{4} \, \log \left (x^{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^8+3*x^4+1),x, algorithm="maxima")

[Out]

-3/40*sqrt(5)*log((2*x^4 - sqrt(5) + 3)/(2*x^4 + sqrt(5) + 3)) - 1/8*log(x^8 + 3*x^4 + 1) + 1/4*log(x^4)

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Fricas [A]  time = 1.48148, size = 155, normalized size = 2.72 \begin{align*} \frac{3}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{8} + 6 \, x^{4} + \sqrt{5}{\left (2 \, x^{4} + 3\right )} + 7}{x^{8} + 3 \, x^{4} + 1}\right ) - \frac{1}{8} \, \log \left (x^{8} + 3 \, x^{4} + 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^8+3*x^4+1),x, algorithm="fricas")

[Out]

3/40*sqrt(5)*log((2*x^8 + 6*x^4 + sqrt(5)*(2*x^4 + 3) + 7)/(x^8 + 3*x^4 + 1)) - 1/8*log(x^8 + 3*x^4 + 1) + log
(x)

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Sympy [A]  time = 0.152717, size = 58, normalized size = 1.02 \begin{align*} \log{\left (x \right )} + \left (- \frac{3 \sqrt{5}}{40} - \frac{1}{8}\right ) \log{\left (x^{4} - \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} + \left (- \frac{1}{8} + \frac{3 \sqrt{5}}{40}\right ) \log{\left (x^{4} + \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x**8+3*x**4+1),x)

[Out]

log(x) + (-3*sqrt(5)/40 - 1/8)*log(x**4 - sqrt(5)/2 + 3/2) + (-1/8 + 3*sqrt(5)/40)*log(x**4 + sqrt(5)/2 + 3/2)

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Giac [A]  time = 1.24926, size = 69, normalized size = 1.21 \begin{align*} -\frac{3}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) - \frac{1}{8} \, \log \left (x^{8} + 3 \, x^{4} + 1\right ) + \frac{1}{4} \, \log \left (x^{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^8+3*x^4+1),x, algorithm="giac")

[Out]

-3/40*sqrt(5)*log((2*x^4 - sqrt(5) + 3)/(2*x^4 + sqrt(5) + 3)) - 1/8*log(x^8 + 3*x^4 + 1) + 1/4*log(x^4)