3.356 \(\int \frac{1}{x^5 (1-x^4+x^8)} \, dx\)

Optimal. Leaf size=48 \[ -\frac{1}{4 x^4}-\frac{1}{8} \log \left (x^8-x^4+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}}+\log (x) \]

[Out]

-1/(4*x^4) + ArcTan[(1 - 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) + Log[x] - Log[1 - x^4 + x^8]/8

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Rubi [A]  time = 0.0536337, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {1357, 709, 800, 634, 618, 204, 628} \[ -\frac{1}{4 x^4}-\frac{1}{8} \log \left (x^8-x^4+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(1 - x^4 + x^8)),x]

[Out]

-1/(4*x^4) + ArcTan[(1 - 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) + Log[x] - Log[1 - x^4 + x^8]/8

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 709

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

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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

Mathematica [C]  time = 0.0134839, size = 51, normalized size = 1.06 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\& \right ]-\frac{1}{4 x^4}+\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(1 - x^4 + x^8)),x]

[Out]

-1/(4*x^4) + Log[x] - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1^4)/(-1 + 2*#1^4) & ]/4

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Maple [A]  time = 0.005, size = 40, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{x}^{4}}}+\ln \left ( x \right ) -{\frac{\ln \left ({x}^{8}-{x}^{4}+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,{x}^{4}-1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(x^8-x^4+1),x)

[Out]

-1/4/x^4+ln(x)-1/8*ln(x^8-x^4+1)-1/12*3^(1/2)*arctan(1/3*(2*x^4-1)*3^(1/2))

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Maxima [A]  time = 1.5457, size = 58, normalized size = 1.21 \begin{align*} -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) - \frac{1}{4 \, x^{4}} - \frac{1}{8} \, \log \left (x^{8} - 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^5/(x^8-x^4+1),x, algorithm="maxima")

[Out]

-1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1)) - 1/4/x^4 - 1/8*log(x^8 - x^4 + 1) + 1/4*log(x^4)

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Fricas [A]  time = 1.50353, size = 143, normalized size = 2.98 \begin{align*} -\frac{2 \, \sqrt{3} x^{4} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) + 3 \, x^{4} \log \left (x^{8} - x^{4} + 1\right ) - 24 \, x^{4} \log \left (x\right ) + 6}{24 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^8-x^4+1),x, algorithm="fricas")

[Out]

-1/24*(2*sqrt(3)*x^4*arctan(1/3*sqrt(3)*(2*x^4 - 1)) + 3*x^4*log(x^8 - x^4 + 1) - 24*x^4*log(x) + 6)/x^4

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Sympy [A]  time = 0.190079, size = 48, normalized size = 1. \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{8} - x^{4} + 1 \right )}}{8} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{4}}{3} - \frac{\sqrt{3}}{3} \right )}}{12} - \frac{1}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(x**8-x**4+1),x)

[Out]

log(x) - log(x**8 - x**4 + 1)/8 - sqrt(3)*atan(2*sqrt(3)*x**4/3 - sqrt(3)/3)/12 - 1/(4*x**4)

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Giac [A]  time = 1.13813, size = 65, normalized size = 1.35 \begin{align*} -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) - \frac{x^{4} + 1}{4 \, x^{4}} - \frac{1}{8} \, \log \left (x^{8} - 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^5/(x^8-x^4+1),x, algorithm="giac")

[Out]

-1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1)) - 1/4*(x^4 + 1)/x^4 - 1/8*log(x^8 - x^4 + 1) + 1/4*log(x^4)