3.4.64 \(\int \frac {1}{x^4 (1-x^4+x^8)} \, dx\) [364]

Optimal. Leaf size=370 \[ -\frac {1}{3 x^3}-\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right ) \]

[Out]

-1/3/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*ar
ctan((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 = 0.16, antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1382, 1435, 1183, 648, 632, 210, 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}{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^4*(1 - x^4 + x^8)),x]

[Out]

-1/3*1/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 + 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 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 1435

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[-2*(d/e) -
b/c, 2]}, Dist[e/(2*c*q), Int[(q - 2*x^(n/2))/Simp[d/e + q*x^(n/2) - x^n, x], x], x] + Dist[e/(2*c*q), Int[(q
+ 2*x^(n/2))/Simp[d/e - q*x^(n/2) - x^n, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2
- 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] &&  !GtQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {1}{x^4 \left (1-x^4+x^8\right )} \, dx &=-\frac {1}{3 x^3}+\frac {1}{3} \int \frac {3-3 x^4}{1-x^4+x^8} \, dx\\ &=-\frac {1}{3 x^3}-\frac {\int \frac {\sqrt {3}+2 x^2}{-1-\sqrt {3} x^2-x^4} \, dx}{2 \sqrt {3}}-\frac {\int \frac {\sqrt {3}-2 x^2}{-1+\sqrt {3} x^2-x^4} \, dx}{2 \sqrt {3}}\\ &=-\frac {1}{3 x^3}+\frac {\int \frac {\sqrt {3 \left (2-\sqrt {3}\right )}-\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 {\sqrt {3 \left (2-\sqrt {3}\right )}+\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 {\sqrt {3 \left (2+\sqrt {3}\right )}-\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 {\sqrt {3 \left (2+\sqrt {3}\right )}+\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}{3 x^3}-\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}{3 x^3}+\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 {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}{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}{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 {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 )\\ \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 = 65, normalized size = 0.18 \begin {gather*} -\frac {1}{3 x^3}-\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(1 - x^4 + x^8)),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.03, size = 50, normalized size = 0.14

method	result	size
risch	\(-\frac {1}{3 x^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (81 \textit {\_Z}^{8}-9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (-9 \textit {\_R}^{5}+2 \textit {\_R} +x \right )\right )}{4}\)	\(38\)
default	\(-\frac {1}{3 x^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/3/x^3 - integrate((x^4 - 1)/(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. 760 vs. \(2 (260) = 520\).
time = 0.45, size = 760, normalized size = 2.05 \begin {gather*} \frac {4 \, \sqrt {6} \sqrt {2} x^{3} \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^{3} \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^{3} \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^{3} \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^{3} - 2 \, \sqrt {2} x^{3}\right )} \sqrt {\sqrt {3} + 2} \log \left (144 \, x^{2} + 24 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 144\right ) - 2 \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} x^{3} - 2 \, \sqrt {2} x^{3}\right )} \sqrt {\sqrt {3} + 2} \log \left (144 \, x^{2} - 24 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 144\right ) + \sqrt {6} {\left (\sqrt {3} \sqrt {2} x^{3} + 2 \, \sqrt {2} x^{3}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (144 \, x^{2} + 12 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 144\right ) - \sqrt {6} {\left (\sqrt {3} \sqrt {2} x^{3} + 2 \, \sqrt {2} x^{3}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (144 \, x^{2} - 12 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 144\right ) - 32}{96 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(4*sqrt(6)*sqrt(2)*x^3*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*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^3*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*
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) - 8*sqrt(6)*sqrt(2)*x^3*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(sqrt(3) + 2) + sqrt(3) - 2) - 8*sqrt(6)*sqr
t(2)*x^3*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*sqr
t(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))*s
qrt(sqrt(3) + 2) - sqrt(3) + 2) + 2*sqrt(6)*(sqrt(3)*sqrt(2)*x^3 - 2*sqrt(2)*x^3)*sqrt(sqrt(3) + 2)*log(144*x^
2 + 24*sqrt(6)*(2*sqrt(3)*sqrt(2)*x - 3*sqrt(2)*x)*sqrt(sqrt(3) + 2) + 144) - 2*sqrt(6)*(sqrt(3)*sqrt(2)*x^3 -
2*sqrt(2)*x^3)*sqrt(sqrt(3) + 2)*log(144*x^2 - 24*sqrt(6)*(2*sqrt(3)*sqrt(2)*x - 3*sqrt(2)*x)*sqrt(sqrt(3) +
2) + 144) + sqrt(6)*(sqrt(3)*sqrt(2)*x^3 + 2*sqrt(2)*x^3)*sqrt(-4*sqrt(3) + 8)*log(144*x^2 + 12*sqrt(6)*(2*sqr
t(3)*sqrt(2)*x + 3*sqrt(2)*x)*sqrt(-4*sqrt(3) + 8) + 144) - sqrt(6)*(sqrt(3)*sqrt(2)*x^3 + 2*sqrt(2)*x^3)*sqrt
(-4*sqrt(3) + 8)*log(144*x^2 - 12*sqrt(6)*(2*sqrt(3)*sqrt(2)*x + 3*sqrt(2)*x)*sqrt(-4*sqrt(3) + 8) + 144) - 32
)/x^3

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Sympy [A]
time = 1.51, size = 31, normalized size = 0.08 \begin {gather*} \operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (- 9216 t^{5} + 8 t + x \right )} \right )\right )} - \frac {1}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(-9216*_t**5 + 8*_t + x))) - 1/(3*x**3)

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Giac [A]
time = 4.16, size = 258, 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 {1}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(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) + sqr
t(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*(sq
rt(6) - 3*sqrt(2))*log(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/3/x^3

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Mupad [B]
time = 1.29, size = 213, normalized size = 0.58 \begin {gather*} -\frac {1}{3\,x^3}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}-\frac {\sqrt {3}\,x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{1/4}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}-\frac {2^{1/4}\,\sqrt {3}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\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^{1/4}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}+\frac {2^{1/4}\,\sqrt {3}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\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^4*(x^8 - x^4 + 1)),x)

[Out]

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

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