3.409 \(\int \frac{1}{x (1+x^5+x^{10})} \, dx\)

Optimal. Leaf size=39 \[ -\frac{1}{10} \log \left (x^{10}+x^5+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{3}}\right )}{5 \sqrt{3}}+\log (x) \]

[Out]

-ArcTan[(1 + 2*x^5)/Sqrt[3]]/(5*Sqrt[3]) + Log[x] - Log[1 + x^5 + x^10]/10

________________________________________________________________________________________

Rubi [A]  time = 0.0353901, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1357, 705, 29, 634, 618, 204, 628} \[ -\frac{1}{10} \log \left (x^{10}+x^5+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{3}}\right )}{5 \sqrt{3}}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 + x^5 + x^10)),x]

[Out]

-ArcTan[(1 + 2*x^5)/Sqrt[3]]/(5*Sqrt[3]) + Log[x] - Log[1 + x^5 + x^10]/10

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 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), 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 618

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

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

Rubi steps

\begin{align*} \int \frac{1}{x \left (1+x^5+x^{10}\right )} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x \left (1+x+x^2\right )} \, 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-x}{1+x+x^2} \, dx,x,x^5\right )\\ &=\log (x)-\frac{1}{10} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,x^5\right )-\frac{1}{10} \operatorname{Subst}\left (\int \frac{1+2 x}{1+x+x^2} \, dx,x,x^5\right )\\ &=\log (x)-\frac{1}{10} \log \left (1+x^5+x^{10}\right )+\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x^5\right )\\ &=-\frac{\tan ^{-1}\left (\frac{1+2 x^5}{\sqrt{3}}\right )}{5 \sqrt{3}}+\log (x)-\frac{1}{10} \log \left (1+x^5+x^{10}\right )\\ \end{align*}

Mathematica [C]  time = 0.0362244, size = 197, normalized size = 5.05 \[ -\frac{1}{5} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^7+\text{$\#$1}^5-\text{$\#$1}^4+\text{$\#$1}^3-\text{$\#$1}+1\& ,\frac{4 \text{$\#$1}^7 \log (x-\text{$\#$1})-3 \text{$\#$1}^6 \log (x-\text{$\#$1})-\text{$\#$1}^5 \log (x-\text{$\#$1})+3 \text{$\#$1}^4 \log (x-\text{$\#$1})-\text{$\#$1}^3 \log (x-\text{$\#$1})+2 \text{$\#$1}^2 \log (x-\text{$\#$1})-\text{$\#$1} \log (x-\text{$\#$1})}{8 \text{$\#$1}^7-7 \text{$\#$1}^6+5 \text{$\#$1}^4-4 \text{$\#$1}^3+3 \text{$\#$1}^2-1}\& \right ]-\frac{1}{10} \log \left (x^2+x+1\right )+\log (x)+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{5 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + x^5 + x^10)),x]

[Out]

ArcTan[(1 + 2*x)/Sqrt[3]]/(5*Sqrt[3]) + Log[x] - Log[1 + x + x^2]/10 - RootSum[1 - #1 + #1^3 - #1^4 + #1^5 - #
1^7 + #1^8 & , (-(Log[x - #1]*#1) + 2*Log[x - #1]*#1^2 - Log[x - #1]*#1^3 + 3*Log[x - #1]*#1^4 - Log[x - #1]*#
1^5 - 3*Log[x - #1]*#1^6 + 4*Log[x - #1]*#1^7)/(-1 + 3*#1^2 - 4*#1^3 + 5*#1^4 - 7*#1^6 + 8*#1^7) & ]/5

________________________________________________________________________________________

Maple [B]  time = 0.037, size = 66, normalized size = 1.7 \begin{align*} -{\frac{\ln \left ({x}^{2}+x+1 \right ) }{10}}-{\frac{\ln \left ( 4\,{x}^{8}-4\,{x}^{7}+4\,{x}^{5}-4\,{x}^{4}+4\,{x}^{3}-4\,x+4 \right ) }{10}}-{\frac{\sqrt{3}}{15}\arctan \left ({\frac{2\,\sqrt{3}{x}^{5}}{3}}+{\frac{\sqrt{3}}{3}} \right ) }+\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^10+x^5+1),x)

[Out]

-1/10*ln(x^2+x+1)-1/10*ln(4*x^8-4*x^7+4*x^5-4*x^4+4*x^3-4*x+4)-1/15*3^(1/2)*arctan(2/3*3^(1/2)*x^5+1/3*3^(1/2)
)+ln(x)

________________________________________________________________________________________

Maxima [A]  time = 1.51733, size = 49, normalized size = 1.26 \begin{align*} -\frac{1}{15} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{5} + 1\right )}\right ) - \frac{1}{10} \, \log \left (x^{10} + 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^10+x^5+1),x, algorithm="maxima")

[Out]

-1/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^5 + 1)) - 1/10*log(x^10 + x^5 + 1) + 1/5*log(x^5)

________________________________________________________________________________________

Fricas [A]  time = 1.70077, size = 112, normalized size = 2.87 \begin{align*} -\frac{1}{15} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{5} + 1\right )}\right ) - \frac{1}{10} \, \log \left (x^{10} + 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^10+x^5+1),x, algorithm="fricas")

[Out]

-1/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^5 + 1)) - 1/10*log(x^10 + x^5 + 1) + log(x)

________________________________________________________________________________________

Sympy [A]  time = 0.164685, size = 41, normalized size = 1.05 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{10} + x^{5} + 1 \right )}}{10} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{5}}{3} + \frac{\sqrt{3}}{3} \right )}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x**10+x**5+1),x)

[Out]

log(x) - log(x**10 + x**5 + 1)/10 - sqrt(3)*atan(2*sqrt(3)*x**5/3 + sqrt(3)/3)/15

________________________________________________________________________________________

Giac [A]  time = 1.11495, size = 45, normalized size = 1.15 \begin{align*} -\frac{1}{15} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{5} + 1\right )}\right ) - \frac{1}{10} \, \log \left (x^{10} + x^{5} + 1\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^10+x^5+1),x, algorithm="giac")

[Out]

-1/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^5 + 1)) - 1/10*log(x^10 + x^5 + 1) + log(abs(x))