3.4.63 \(\int \frac {1}{x^2 (1-x^4+x^8)} \, dx\) [363]

Optimal. Leaf size=360 \[ -\frac {1}{x}+\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1382, 1520, 1293, 1183, 648, 632, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\text {ArcTan}\left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\text {ArcTan}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\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 )+\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}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(1 - x^4 + x^8)),x]

[Out]

-x^(-1) + ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]]/(4*Sqrt[3*(2 - Sqrt[3])]) - ArcTan[(Sqrt[2 + Sqr
t[3]] - 2*x)/Sqrt[2 - Sqrt[3]]]/(4*Sqrt[3*(2 + Sqrt[3])]) - ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]
]/(4*Sqrt[3*(2 - Sqrt[3])]) + ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/(4*Sqrt[3*(2 + Sqrt[3])]) +
(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 - (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 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 648

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 1183

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

Rule 1293

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[e*f*
(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m -
2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[
{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (Inte
gerQ[p] || IntegerQ[m])

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 1520

Int[(((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Wit
h[{q = Rt[a*c, 2]}, With[{r = Rt[2*c*q - b*c, 2]}, Dist[c/(2*q*r), Int[(f*x)^m*(Simp[d*r - (c*d - e*q)*x^(n/2)
, x]/(q - r*x^(n/2) + c*x^n)), x], x] + Dist[c/(2*q*r), Int[(f*x)^m*(Simp[d*r + (c*d - e*q)*x^(n/2), x]/(q + r
*x^(n/2) + c*x^n)), x], x]] /;  !LtQ[2*c*q - b*c, 0]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && LtQ[b
^2 - 4*a*c, 0] && IntegersQ[m, n/2] && LtQ[0, m, n] && PosQ[a*c]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (1-x^4+x^8\right )} \, dx &=-\frac {1}{x}+\int \frac {x^2 \left (1-x^4\right )}{1-x^4+x^8} \, dx\\ &=-\frac {1}{x}+\frac {\int \frac {x^2 \left (\sqrt {3}-2 x^2\right )}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}+\frac {\int \frac {x^2 \left (\sqrt {3}+2 x^2\right )}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}\\ &=-\frac {1}{x}-\frac {\int \frac {-2+\sqrt {3} x^2}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}-\frac {\int \frac {2+\sqrt {3} x^2}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}\\ &=-\frac {1}{x}-\frac {\int \frac {2 \sqrt {2-\sqrt {3}}-\left (2-\sqrt {3}\right ) x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\int \frac {2 \sqrt {2-\sqrt {3}}+\left (2-\sqrt {3}\right ) x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\int \frac {-2 \sqrt {2+\sqrt {3}}-\left (-2-\sqrt {3}\right ) x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\int \frac {-2 \sqrt {2+\sqrt {3}}+\left (-2-\sqrt {3}\right ) x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}\\ &=-\frac {1}{x}+\frac {1}{8} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \int \frac {\sqrt {2-\sqrt {3}}+2 x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx-\frac {\left (-2+\sqrt {3}\right ) \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}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \int \frac {-\sqrt {2+\sqrt {3}}+2 x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \int \frac {\sqrt {2+\sqrt {3}}+2 x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx-\frac {1}{8} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \int \frac {1}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx-\frac {1}{8} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \int \frac {1}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx\\ &=-\frac {1}{x}+\frac {1}{8} \sqrt {\frac {2}{3}-\frac {1}{\sqrt {3}}} \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 )-\frac {1}{4} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,\sqrt {2+\sqrt {3}}+2 x\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,-\sqrt {2-\sqrt {3}}+2 x\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,\sqrt {2-\sqrt {3}}+2 x\right )\\ &=-\frac {1}{x}+\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 {2}{3}-\frac {1}{\sqrt {3}}} \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 )\\ \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 = 61, normalized size = 0.17 \begin {gather*} -\frac {1}{x}-\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-\text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(1 - x^4 + x^8)),x]

[Out]

-x^(-1) - RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-#1 + 2*#1^5) & ]/4

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.04, size = 52, normalized size = 0.14

method	result	size
risch	\(-\frac {1}{x}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (81 \textit {\_Z}^{8}-9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (-27 \textit {\_R}^{7}+6 \textit {\_R}^{3}+x \right )\right )}{4}\)	\(40\)
default	\(-\frac {1}{x}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{6}-\textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(x^8-x^4+1),x,method=_RETURNVERBOSE)

[Out]

-1/x-1/4*sum((_R^6-_R^2)/(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^2/(x^8-x^4+1),x, algorithm="maxima")

[Out]

-1/x - integrate((x^6 - x^2)/(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. 736 vs. \(2 (268) = 536\).
time = 0.40, size = 736, normalized size = 2.04 \begin {gather*} -\frac {4 \, \sqrt {6} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {1}{36} \, \sqrt {6} \sqrt {3} \sqrt {12 \, x^{2} + \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 12} {\left (2 \, \sqrt {3} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} - \frac {1}{6} \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} - \sqrt {3} - 2\right ) + 4 \, \sqrt {6} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {1}{36} \, \sqrt {6} \sqrt {3} \sqrt {12 \, x^{2} - \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 12} {\left (2 \, \sqrt {3} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} - \frac {1}{6} \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + \sqrt {3} + 2\right ) - 8 \, \sqrt {6} \sqrt {2} x \sqrt {\sqrt {3} + 2} \arctan \left (-\frac {1}{3} \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + \frac {1}{3} \, \sqrt {6 \, x^{2} + \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 6} {\left (2 \, \sqrt {3} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} + \sqrt {3} - 2\right ) - 8 \, \sqrt {6} \sqrt {2} x \sqrt {\sqrt {3} + 2} \arctan \left (-\frac {1}{3} \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + \frac {1}{3} \, \sqrt {6 \, x^{2} - \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 6} {\left (2 \, \sqrt {3} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} - \sqrt {3} + 2\right ) - 2 \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} x - 2 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} \log \left (576 \, x^{2} + 96 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 576\right ) + 2 \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} x - 2 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} \log \left (576 \, x^{2} - 96 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 576\right ) - \sqrt {6} {\left (\sqrt {3} \sqrt {2} x + 2 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (576 \, x^{2} + 48 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 576\right ) + \sqrt {6} {\left (\sqrt {3} \sqrt {2} x + 2 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (576 \, x^{2} - 48 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 576\right ) + 96}{96 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8-x^4+1),x, algorithm="fricas")

[Out]

-1/96*(4*sqrt(6)*sqrt(2)*x*sqrt(-4*sqrt(3) + 8)*arctan(1/36*sqrt(6)*sqrt(3)*sqrt(12*x^2 + sqrt(6)*(2*sqrt(3)*s
qrt(2)*x + 3*sqrt(2)*x)*sqrt(-4*sqrt(3) + 8) + 12)*(2*sqrt(3)*sqrt(2) + 3*sqrt(2))*sqrt(-4*sqrt(3) + 8) - 1/6*
sqrt(6)*(2*sqrt(3)*sqrt(2)*x + 3*sqrt(2)*x)*sqrt(-4*sqrt(3) + 8) - sqrt(3) - 2) + 4*sqrt(6)*sqrt(2)*x*sqrt(-4*
sqrt(3) + 8)*arctan(1/36*sqrt(6)*sqrt(3)*sqrt(12*x^2 - sqrt(6)*(2*sqrt(3)*sqrt(2)*x + 3*sqrt(2)*x)*sqrt(-4*sqr
t(3) + 8) + 12)*(2*sqrt(3)*sqrt(2) + 3*sqrt(2))*sqrt(-4*sqrt(3) + 8) - 1/6*sqrt(6)*(2*sqrt(3)*sqrt(2)*x + 3*sq
rt(2)*x)*sqrt(-4*sqrt(3) + 8) + sqrt(3) + 2) - 8*sqrt(6)*sqrt(2)*x*sqrt(sqrt(3) + 2)*arctan(-1/3*sqrt(6)*(2*sq
rt(3)*sqrt(2)*x - 3*sqrt(2)*x)*sqrt(sqrt(3) + 2) + 1/3*sqrt(6*x^2 + sqrt(6)*(2*sqrt(3)*sqrt(2)*x - 3*sqrt(2)*x
)*sqrt(sqrt(3) + 2) + 6)*(2*sqrt(3)*sqrt(2) - 3*sqrt(2))*sqrt(sqrt(3) + 2) + sqrt(3) - 2) - 8*sqrt(6)*sqrt(2)*
x*sqrt(sqrt(3) + 2)*arctan(-1/3*sqrt(6)*(2*sqrt(3)*sqrt(2)*x - 3*sqrt(2)*x)*sqrt(sqrt(3) + 2) + 1/3*sqrt(6*x^2
- sqrt(6)*(2*sqrt(3)*sqrt(2)*x - 3*sqrt(2)*x)*sqrt(sqrt(3) + 2) + 6)*(2*sqrt(3)*sqrt(2) - 3*sqrt(2))*sqrt(sqr
t(3) + 2) - sqrt(3) + 2) - 2*sqrt(6)*(sqrt(3)*sqrt(2)*x - 2*sqrt(2)*x)*sqrt(sqrt(3) + 2)*log(576*x^2 + 96*sqrt
(6)*(2*sqrt(3)*sqrt(2)*x - 3*sqrt(2)*x)*sqrt(sqrt(3) + 2) + 576) + 2*sqrt(6)*(sqrt(3)*sqrt(2)*x - 2*sqrt(2)*x)
*sqrt(sqrt(3) + 2)*log(576*x^2 - 96*sqrt(6)*(2*sqrt(3)*sqrt(2)*x - 3*sqrt(2)*x)*sqrt(sqrt(3) + 2) + 576) - sqr
t(6)*(sqrt(3)*sqrt(2)*x + 2*sqrt(2)*x)*sqrt(-4*sqrt(3) + 8)*log(576*x^2 + 48*sqrt(6)*(2*sqrt(3)*sqrt(2)*x + 3*
sqrt(2)*x)*sqrt(-4*sqrt(3) + 8) + 576) + sqrt(6)*(sqrt(3)*sqrt(2)*x + 2*sqrt(2)*x)*sqrt(-4*sqrt(3) + 8)*log(57
6*x^2 - 48*sqrt(6)*(2*sqrt(3)*sqrt(2)*x + 3*sqrt(2)*x)*sqrt(-4*sqrt(3) + 8) + 576) + 96)/x

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Sympy [A]
time = 1.49, size = 29, normalized size = 0.08 \begin {gather*} \operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (- 442368 t^{7} + 384 t^{3} + x \right )} \right )\right )} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(x**8-x**4+1),x)

[Out]

RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(-442368*_t**7 + 384*_t**3 + x))) - 1/x

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Giac [A]
time = 4.72, size = 258, normalized size = 0.72 \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 {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(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/x

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Mupad [B]
time = 1.29, size = 253, normalized size = 0.70 \begin {gather*} -\frac {1}{x}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}+\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\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 (-1+\sqrt {3}\,1{}\mathrm {i}\right )}-\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\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}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{3/4}}-\frac {2^{3/4}\,\sqrt {3}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{3/4}}\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\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{3/4}}+\frac {2^{3/4}\,\sqrt {3}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{3/4}}\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^2*(x^8 - x^4 + 1)),x)

[Out]

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

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