3.4.66 \(\int \frac {1}{x^8 (1-x^4+x^8)} \, dx\) [366]

Optimal. Leaf size=377 \[ -\frac {1}{7 x^7}-\frac {1}{3 x^3}-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right ) \]

[Out]

-1/7/x^7-1/3/x^3-1/4*arctan((-2*x+1/2*6^(1/2)-1/2*2^(1/2))/(1/2*6^(1/2)+1/2*2^(1/2)))*(1/2*2^(1/2)-1/6*6^(1/2)
)+1/4*arctan((2*x+1/2*6^(1/2)-1/2*2^(1/2))/(1/2*6^(1/2)+1/2*2^(1/2)))*(1/2*2^(1/2)-1/6*6^(1/2))-1/8*ln(1+x^2-x
*(1/2*6^(1/2)+1/2*2^(1/2)))*(1/2*2^(1/2)-1/6*6^(1/2))+1/8*ln(1+x^2+x*(1/2*6^(1/2)+1/2*2^(1/2)))*(1/2*2^(1/2)-1
/6*6^(1/2))+1/4*arctan((-2*x+1/2*6^(1/2)+1/2*2^(1/2))/(1/2*6^(1/2)-1/2*2^(1/2)))*(1/2*2^(1/2)+1/6*6^(1/2))-1/4
*arctan((2*x+1/2*6^(1/2)+1/2*2^(1/2))/(1/2*6^(1/2)-1/2*2^(1/2)))*(1/2*2^(1/2)+1/6*6^(1/2))+1/8*ln(1+x^2-x*(1/2
*6^(1/2)-1/2*2^(1/2)))*(1/2*2^(1/2)+1/6*6^(1/2))-1/8*ln(1+x^2+x*(1/2*6^(1/2)-1/2*2^(1/2)))*(1/2*2^(1/2)+1/6*6^
(1/2))

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Rubi [A]
time = 0.19, antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {1382, 1518, 12, 1387, 1141, 1175, 632, 210, 1178, 642} \begin {gather*} -\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \text {ArcTan}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \text {ArcTan}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \text {ArcTan}\left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \text {ArcTan}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{7 x^7}-\frac {1}{3 x^3}+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(1 - x^4 + x^8)),x]

[Out]

-1/7*1/x^7 - 1/(3*x^3) - (Sqrt[(2 - Sqrt[3])/3]*ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]])/4 + (Sqrt
[(2 + Sqrt[3])/3]*ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]])/4 + (Sqrt[(2 - Sqrt[3])/3]*ArcTan[(Sqrt
[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]])/4 - (Sqrt[(2 + Sqrt[3])/3]*ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 -
Sqrt[3]]])/4 + (Sqrt[(2 + Sqrt[3])/3]*Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2])/8 - (Sqrt[(2 + Sqrt[3])/3]*Log[1 + S
qrt[2 - Sqrt[3]]*x + x^2])/8 - (Sqrt[(2 - Sqrt[3])/3]*Log[1 - Sqrt[2 + Sqrt[3]]*x + x^2])/8 + (Sqrt[(2 - Sqrt[
3])/3]*Log[1 + Sqrt[2 + Sqrt[3]]*x + x^2])/8

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 210

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

Rule 632

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 642

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]

Rule 1141

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

Rule 1175

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

Rule 1178

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

Rule 1382

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

Rule 1387

Int[(x_)^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[
2*q - b/c, 2]}, Dist[1/(2*c*r), Int[x^(m - n/2)/(q - r*x^(n/2) + x^n), x], x] - Dist[1/(2*c*r), Int[x^(m - n/2
)/(q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n/2, 0
] && IGtQ[m, 0] && GeQ[m, n/2] && LtQ[m, 3*(n/2)] && NegQ[b^2 - 4*a*c]

Rule 1518

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Simp[d*(f*x)^(m + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*f*(m + 1))), x] + Dist[1/(a*f^n*(m + 1)), Int[(f*x)^
(m + n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - c*d*(m + 2*n*(p + 1) + 1)*x^n,
x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -
1] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{x^8 \left (1-x^4+x^8\right )} \, dx &=-\frac {1}{7 x^7}+\frac {1}{7} \int \frac {7-7 x^4}{x^4 \left (1-x^4+x^8\right )} \, dx\\ &=-\frac {1}{7 x^7}-\frac {1}{3 x^3}-\frac {1}{21} \int \frac {21 x^4}{1-x^4+x^8} \, dx\\ &=-\frac {1}{7 x^7}-\frac {1}{3 x^3}-\int \frac {x^4}{1-x^4+x^8} \, dx\\ &=-\frac {1}{7 x^7}-\frac {1}{3 x^3}-\frac {\int \frac {x^2}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}+\frac {\int \frac {x^2}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}\\ &=-\frac {1}{7 x^7}-\frac {1}{3 x^3}+\frac {\int \frac {1-x^2}{1-\sqrt {3} x^2+x^4} \, dx}{4 \sqrt {3}}-\frac {\int \frac {1+x^2}{1-\sqrt {3} x^2+x^4} \, dx}{4 \sqrt {3}}-\frac {\int \frac {1-x^2}{1+\sqrt {3} x^2+x^4} \, dx}{4 \sqrt {3}}+\frac {\int \frac {1+x^2}{1+\sqrt {3} x^2+x^4} \, dx}{4 \sqrt {3}}\\ &=-\frac {1}{7 x^7}-\frac {1}{3 x^3}+\frac {\int \frac {1}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}+\frac {\int \frac {1}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}-\frac {\int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}-\frac {\int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}+\frac {\int \frac {\sqrt {2-\sqrt {3}}+2 x}{-1-\sqrt {2-\sqrt {3}} x-x^2} \, dx}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\int \frac {\sqrt {2-\sqrt {3}}-2 x}{-1+\sqrt {2-\sqrt {3}} x-x^2} \, dx}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\int \frac {\sqrt {2+\sqrt {3}}+2 x}{-1-\sqrt {2+\sqrt {3}} x-x^2} \, dx}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\int \frac {\sqrt {2+\sqrt {3}}-2 x}{-1+\sqrt {2+\sqrt {3}} x-x^2} \, dx}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}\\ &=-\frac {1}{7 x^7}-\frac {1}{3 x^3}+\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,-\sqrt {2-\sqrt {3}}+2 x\right )}{4 \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,\sqrt {2-\sqrt {3}}+2 x\right )}{4 \sqrt {3}}+\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )}{4 \sqrt {3}}+\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,\sqrt {2+\sqrt {3}}+2 x\right )}{4 \sqrt {3}}\\ &=-\frac {1}{7 x^7}-\frac {1}{3 x^3}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.01, size = 54, normalized size = 0.14 \begin {gather*} -\frac {1}{7 x^7}-\frac {1}{3 x^3}-\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{-1+2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(1 - x^4 + x^8)),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.03, size = 51, normalized size = 0.14

method	result	size
risch	\(\frac {-\frac {x^{4}}{3}-\frac {1}{7}}{x^{7}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (81 \textit {\_Z}^{8}-9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (18 \textit {\_R}^{5}-\textit {\_R} +x \right )\right )}{4}\)	\(44\)
default	\(-\frac {1}{7 x^{7}}-\frac {1}{3 x^{3}}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(x^8-x^4+1),x,method=_RETURNVERBOSE)

[Out]

-1/7/x^7-1/3/x^3-1/4*sum(_R^4/(2*_R^7-_R^3)*ln(x-_R),_R=RootOf(_Z^8-_Z^4+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(7*x^4 + 3)/x^7 - integrate(x^4/(x^8 - x^4 + 1), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 628 vs. \(2 (265) = 530\).
time = 0.38, size = 628, normalized size = 1.67 \begin {gather*} \frac {56 \, \sqrt {6} \sqrt {2} x^{7} \sqrt {\sqrt {3} + 2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + \frac {1}{18} \, \sqrt {6} \sqrt {3} \sqrt {2} \sqrt {6 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + 36 \, x^{2} + 36} \sqrt {\sqrt {3} + 2} - \sqrt {3} - 2\right ) + 56 \, \sqrt {6} \sqrt {2} x^{7} \sqrt {\sqrt {3} + 2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + \frac {1}{18} \, \sqrt {6} \sqrt {3} \sqrt {2} \sqrt {-6 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + 36 \, x^{2} + 36} \sqrt {\sqrt {3} + 2} + \sqrt {3} + 2\right ) - 28 \, \sqrt {6} \sqrt {2} x^{7} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (-\frac {1}{6} \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + \frac {1}{36} \, \sqrt {6} \sqrt {3} \sqrt {2} \sqrt {3 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + 36 \, x^{2} + 36} \sqrt {-4 \, \sqrt {3} + 8} + \sqrt {3} - 2\right ) - 28 \, \sqrt {6} \sqrt {2} x^{7} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (-\frac {1}{6} \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + \frac {1}{36} \, \sqrt {6} \sqrt {3} \sqrt {2} \sqrt {-3 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + 36 \, x^{2} + 36} \sqrt {-4 \, \sqrt {3} + 8} - \sqrt {3} + 2\right ) - 224 \, x^{4} - 14 \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} x^{7} - 2 \, \sqrt {2} x^{7}\right )} \sqrt {\sqrt {3} + 2} \log \left (6 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + 36 \, x^{2} + 36\right ) + 14 \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} x^{7} - 2 \, \sqrt {2} x^{7}\right )} \sqrt {\sqrt {3} + 2} \log \left (-6 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + 36 \, x^{2} + 36\right ) - 7 \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} x^{7} + 2 \, \sqrt {2} x^{7}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (3 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + 36 \, x^{2} + 36\right ) + 7 \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} x^{7} + 2 \, \sqrt {2} x^{7}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (-3 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + 36 \, x^{2} + 36\right ) - 96}{672 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/672*(56*sqrt(6)*sqrt(2)*x^7*sqrt(sqrt(3) + 2)*arctan(-1/3*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(sqrt(3) + 2) + 1/18
*sqrt(6)*sqrt(3)*sqrt(2)*sqrt(6*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(sqrt(3) + 2) + 36*x^2 + 36)*sqrt(sqrt(3) + 2) -
sqrt(3) - 2) + 56*sqrt(6)*sqrt(2)*x^7*sqrt(sqrt(3) + 2)*arctan(-1/3*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(sqrt(3) +
2) + 1/18*sqrt(6)*sqrt(3)*sqrt(2)*sqrt(-6*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(sqrt(3) + 2) + 36*x^2 + 36)*sqrt(sqrt
(3) + 2) + sqrt(3) + 2) - 28*sqrt(6)*sqrt(2)*x^7*sqrt(-4*sqrt(3) + 8)*arctan(-1/6*sqrt(6)*sqrt(3)*sqrt(2)*x*sq
rt(-4*sqrt(3) + 8) + 1/36*sqrt(6)*sqrt(3)*sqrt(2)*sqrt(3*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(-4*sqrt(3) + 8) + 36*x
^2 + 36)*sqrt(-4*sqrt(3) + 8) + sqrt(3) - 2) - 28*sqrt(6)*sqrt(2)*x^7*sqrt(-4*sqrt(3) + 8)*arctan(-1/6*sqrt(6)
*sqrt(3)*sqrt(2)*x*sqrt(-4*sqrt(3) + 8) + 1/36*sqrt(6)*sqrt(3)*sqrt(2)*sqrt(-3*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(
-4*sqrt(3) + 8) + 36*x^2 + 36)*sqrt(-4*sqrt(3) + 8) - sqrt(3) + 2) - 224*x^4 - 14*sqrt(6)*(sqrt(3)*sqrt(2)*x^7
- 2*sqrt(2)*x^7)*sqrt(sqrt(3) + 2)*log(6*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(sqrt(3) + 2) + 36*x^2 + 36) + 14*sqrt
(6)*(sqrt(3)*sqrt(2)*x^7 - 2*sqrt(2)*x^7)*sqrt(sqrt(3) + 2)*log(-6*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(sqrt(3) + 2)
+ 36*x^2 + 36) - 7*sqrt(6)*(sqrt(3)*sqrt(2)*x^7 + 2*sqrt(2)*x^7)*sqrt(-4*sqrt(3) + 8)*log(3*sqrt(6)*sqrt(3)*s
qrt(2)*x*sqrt(-4*sqrt(3) + 8) + 36*x^2 + 36) + 7*sqrt(6)*(sqrt(3)*sqrt(2)*x^7 + 2*sqrt(2)*x^7)*sqrt(-4*sqrt(3)
+ 8)*log(-3*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(-4*sqrt(3) + 8) + 36*x^2 + 36) - 96)/x^7

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Sympy [A]
time = 1.55, size = 37, normalized size = 0.10 \begin {gather*} \operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (18432 t^{5} - 4 t + x \right )} \right )\right )} + \frac {- 7 x^{4} - 3}{21 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(18432*_t**5 - 4*_t + x))) + (-7*x**4 - 3)/(21*x**7)

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Giac [A]
time = 3.29, size = 265, normalized size = 0.70 \begin {gather*} -\frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {7 \, x^{4} + 3}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2))) - 1/24*(sqrt(6) - 3*sqrt(2))
*arctan((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(2))) - 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x + sqrt(6) + sq
rt(2))/(sqrt(6) - sqrt(2))) - 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2)))
- 1/48*(sqrt(6) - 3*sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1) + 1/48*(sqrt(6) - 3*sqrt(2))*log(x^2 -
1/2*x*(sqrt(6) + sqrt(2)) + 1) - 1/48*(sqrt(6) + 3*sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) + 1/48*(s
qrt(6) + 3*sqrt(2))*log(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/21*(7*x^4 + 3)/x^7

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Mupad [B]
time = 0.06, size = 486, normalized size = 1.29 \begin {gather*} -\frac {\frac {x^4}{3}+\frac {1}{7}}{x^7}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (\frac {\sqrt {3}\,\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}}{4}\right )}+\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {\sqrt {3}\,\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}}{4}\right )}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {\sqrt {3}\,\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}}{4}\right )}-\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (\frac {\sqrt {3}\,\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}}{4}\right )}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{12}-\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{2\,\left (\frac {\sqrt {2}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}}{2}-\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}-\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {\sqrt {2}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}}{2}-\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}-\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {\sqrt {2}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}}{2}-\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}+\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{2\,\left (\frac {\sqrt {2}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}}{2}-\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(x^8 - x^4 + 1)),x)

[Out]

(3^(1/2)*atan((x*(8 - 3^(1/2)*8i)^(1/4))/(2*((3^(1/2)*(8 - 3^(1/2)*8i)^(1/2)*1i)/4 + (8 - 3^(1/2)*8i)^(1/2)/4)
) + (3^(1/2)*x*(8 - 3^(1/2)*8i)^(1/4)*1i)/(2*((3^(1/2)*(8 - 3^(1/2)*8i)^(1/2)*1i)/4 + (8 - 3^(1/2)*8i)^(1/2)/4
)))*(8 - 3^(1/2)*8i)^(1/4)*1i)/12 - (x^4/3 + 1/7)/x^7 + (3^(1/2)*atan((x*(8 - 3^(1/2)*8i)^(1/4)*1i)/(2*((3^(1/
2)*(8 - 3^(1/2)*8i)^(1/2)*1i)/4 + (8 - 3^(1/2)*8i)^(1/2)/4)) - (3^(1/2)*x*(8 - 3^(1/2)*8i)^(1/4))/(2*((3^(1/2)
*(8 - 3^(1/2)*8i)^(1/2)*1i)/4 + (8 - 3^(1/2)*8i)^(1/2)/4)))*(8 - 3^(1/2)*8i)^(1/4))/12 - (2^(3/4)*3^(1/2)*atan
((2^(3/4)*x*(3^(1/2)*1i + 1)^(1/4))/(2*((2^(1/2)*(3^(1/2)*1i + 1)^(1/2))/2 - (2^(1/2)*3^(1/2)*(3^(1/2)*1i + 1)
^(1/2)*1i)/2)) - (2^(3/4)*3^(1/2)*x*(3^(1/2)*1i + 1)^(1/4)*1i)/(2*((2^(1/2)*(3^(1/2)*1i + 1)^(1/2))/2 - (2^(1/
2)*3^(1/2)*(3^(1/2)*1i + 1)^(1/2)*1i)/2)))*(3^(1/2)*1i + 1)^(1/4)*1i)/12 - (2^(3/4)*3^(1/2)*atan((2^(3/4)*x*(3
^(1/2)*1i + 1)^(1/4)*1i)/(2*((2^(1/2)*(3^(1/2)*1i + 1)^(1/2))/2 - (2^(1/2)*3^(1/2)*(3^(1/2)*1i + 1)^(1/2)*1i)/
2)) + (2^(3/4)*3^(1/2)*x*(3^(1/2)*1i + 1)^(1/4))/(2*((2^(1/2)*(3^(1/2)*1i + 1)^(1/2))/2 - (2^(1/2)*3^(1/2)*(3^
(1/2)*1i + 1)^(1/2)*1i)/2)))*(3^(1/2)*1i + 1)^(1/4))/12

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