\(\int \frac {\sqrt [4]{-1+x^4} (-1+x^4+2 x^8)}{x^6 (1-x^4+x^8)} \, dx\) [1069]

Optimal result

Integrand size = 35, antiderivative size = 80 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^4+2 x^8\right )}{x^6 \left (1-x^4+x^8\right )} \, dx=\frac {\left (1-x^4\right ) \sqrt [4]{-1+x^4}}{5 x^5}+\frac {3}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.06, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6860, 270, 1542, 525, 524} \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^4+2 x^8\right )}{x^6 \left (1-x^4+x^8\right )} \, dx=\frac {2 \sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,\frac {2 x^4}{1-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+i\right ) \sqrt [4]{1-x^4}}-\frac {2 \sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {2 x^4}{1+i \sqrt {3}},x^4\right )}{\sqrt {3} \left (-\sqrt {3}+i\right ) \sqrt [4]{1-x^4}}-\frac {\left (x^4-1\right )^{5/4}}{5 x^5} \]

[In]

Int[((-1 + x^4)^(1/4)*(-1 + x^4 + 2*x^8))/(x^6*(1 - x^4 + x^8)),x]

[Out]

-1/5*(-1 + x^4)^(5/4)/x^5 + (2*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x^4, (2*x^4)/(1 - I*Sqrt[3])])
/(Sqrt[3]*(I + Sqrt[3])*(1 - x^4)^(1/4)) - (2*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (2*x^4)/(1 + I*
Sqrt[3]), x^4])/(Sqrt[3]*(I - Sqrt[3])*(1 - x^4)^(1/4))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 1542

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

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt [4]{-1+x^4}}{x^6}+\frac {3 x^2 \sqrt [4]{-1+x^4}}{1-x^4+x^8}\right ) \, dx \\ & = 3 \int \frac {x^2 \sqrt [4]{-1+x^4}}{1-x^4+x^8} \, dx-\int \frac {\sqrt [4]{-1+x^4}}{x^6} \, dx \\ & = -\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+3 \int \left (\frac {2 i x^2 \sqrt [4]{-1+x^4}}{\sqrt {3} \left (1+i \sqrt {3}-2 x^4\right )}+\frac {2 i x^2 \sqrt [4]{-1+x^4}}{\sqrt {3} \left (-1+i \sqrt {3}+2 x^4\right )}\right ) \, dx \\ & = -\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+\left (2 i \sqrt {3}\right ) \int \frac {x^2 \sqrt [4]{-1+x^4}}{1+i \sqrt {3}-2 x^4} \, dx+\left (2 i \sqrt {3}\right ) \int \frac {x^2 \sqrt [4]{-1+x^4}}{-1+i \sqrt {3}+2 x^4} \, dx \\ & = -\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+\frac {\left (2 i \sqrt {3} \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{1+i \sqrt {3}-2 x^4} \, dx}{\sqrt [4]{1-x^4}}+\frac {\left (2 i \sqrt {3} \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{-1+i \sqrt {3}+2 x^4} \, dx}{\sqrt [4]{1-x^4}} \\ & = -\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+\frac {2 x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,\frac {2 x^4}{1-i \sqrt {3}}\right )}{\sqrt {3} \left (i+\sqrt {3}\right ) \sqrt [4]{1-x^4}}-\frac {2 x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {2 x^4}{1+i \sqrt {3}},x^4\right )}{\sqrt {3} \left (i-\sqrt {3}\right ) \sqrt [4]{1-x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^4+2 x^8\right )}{x^6 \left (1-x^4+x^8\right )} \, dx=\frac {-4 \left (-1+x^4\right )^{5/4}+15 x^5 \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^4}\&\right ]}{20 x^5} \]

[In]

Integrate[((-1 + x^4)^(1/4)*(-1 + x^4 + 2*x^8))/(x^6*(1 - x^4 + x^8)),x]

[Out]

(-4*(-1 + x^4)^(5/4) + 15*x^5*RootSum[1 - #1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-1 + x^4)^(1/4) - x*#1]*#1)/(-1
+ 2*#1^4) & ])/(20*x^5)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 1.

Time = 0.02 (sec) , antiderivative size = 3975, normalized size of antiderivative = 49.69

\[\text {output too large to display}\]

[In]

int((x^4-1)^(1/4)*(2*x^8+x^4-1)/x^6/(x^8-x^4+1),x)

[Out]

-1/5*(x^8-2*x^4+1)/x^5/(x^4-1)^(3/4)+(331776*ln(-(127401984*RootOf(5308416*_Z^8-2304*_Z^4+1)^10*x^12-254803968
*RootOf(5308416*_Z^8-2304*_Z^4+1)^10*x^8-165888*x^12*RootOf(5308416*_Z^8-2304*_Z^4+1)^6-9216*RootOf(5308416*_Z
^8-2304*_Z^4+1)^5*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+127401984*RootOf(5308416*_Z^8-2304*_Z^4+1)^10*x^4+387072*x^8*
RootOf(5308416*_Z^8-2304*_Z^4+1)^6+48*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*x^12-442368*(x^12-3*x^8+3*x^4-1)^(3/4
)*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x^3+2304*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^
6+18432*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-4*RootOf(5308416*_Z^8-2304*_Z^4+1)*(
x^12-3*x^8+3*x^4-1)^(1/4)*x^9-276480*x^4*RootOf(5308416*_Z^8-2304*_Z^4+1)^6-120*x^8*RootOf(5308416*_Z^8-2304*_
Z^4+1)^2+384*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3-2304*(x^12-3*x^8+3*x^4-1)^(1/2)
*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^2-(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-9216*RootOf(5308416*_Z^8-2304*_Z^4+1)^5
*(x^12-3*x^8+3*x^4-1)^(1/4)*x+8*RootOf(5308416*_Z^8-2304*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+55296*RootOf(5
308416*_Z^8-2304*_Z^4+1)^6+96*x^4*RootOf(5308416*_Z^8-2304*_Z^4+1)^2+(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-4*RootOf(5
308416*_Z^8-2304*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x-24*RootOf(5308416*_Z^8-2304*_Z^4+1)^2)/(442368*RootOf(53
08416*_Z^8-2304*_Z^4+1)^7*x-384*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x+1)/(18432*RootOf(5308416*_Z^8-2304*_Z^4+1
)^5*x-4*x*RootOf(5308416*_Z^8-2304*_Z^4+1)-1)/(442368*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x-384*RootOf(5308416*
_Z^8-2304*_Z^4+1)^3*x-1)/(18432*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x-4*x*RootOf(5308416*_Z^8-2304*_Z^4+1)+1)/(
110592*RootOf(5308416*_Z^8-2304*_Z^4+1)^6*x+1)^2/(x-1)^2/(110592*RootOf(5308416*_Z^8-2304*_Z^4+1)^6*x-1)^2/(1+
x)^2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^7+331776*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*ln(-(13824*RootOf(5308416*_
Z^8-2304*_Z^4+1)^5*x^12+2304*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^9-27648*RootOf(53
08416*_Z^8-2304*_Z^4+1)^5*x^8+6*RootOf(5308416*_Z^8-2304*_Z^4+1)*x^12+110592*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf
(5308416*_Z^8-2304*_Z^4+1)^6*x^3+576*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x^6-4608*(x
^12-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^5+(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+13824*RootOf(53
08416*_Z^8-2304*_Z^4+1)^5*x^4-18*RootOf(5308416*_Z^8-2304*_Z^4+1)*x^8+48*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(530
8416*_Z^8-2304*_Z^4+1)^2*x^3-576*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x^2+2304*(x^12-
3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x-2*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+18*RootOf(5308416*_
Z^8-2304*_Z^4+1)*x^4+(x^12-3*x^8+3*x^4-1)^(1/4)*x-6*RootOf(5308416*_Z^8-2304*_Z^4+1))/(9216*RootOf(5308416*_Z^
8-2304*_Z^4+1)^5*x-8*x*RootOf(5308416*_Z^8-2304*_Z^4+1)-1)/(442368*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x+192*Ro
otOf(5308416*_Z^8-2304*_Z^4+1)^3*x-1)/(9216*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x-8*x*RootOf(5308416*_Z^8-2304*
_Z^4+1)+1)/(442368*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x+192*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x+1)/(110592*Ro
otOf(5308416*_Z^8-2304*_Z^4+1)^6*x+1)^2/(x-1)^2/(110592*RootOf(5308416*_Z^8-2304*_Z^4+1)^6*x-1)^2/(1+x)^2)+691
2*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*ln((127401984*RootOf(5308416*_Z^8-2304*_Z^4+1)^10*x^12-254803968*RootOf(5
308416*_Z^8-2304*_Z^4+1)^10*x^8-165888*x^12*RootOf(5308416*_Z^8-2304*_Z^4+1)^6+884736*RootOf(5308416*_Z^8-2304
*_Z^4+1)^7*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+127401984*RootOf(5308416*_Z^8-2304*_Z^4+1)^10*x^4+387072*x^8*RootOf(
5308416*_Z^8-2304*_Z^4+1)^6+48*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*x^12-1769472*RootOf(5308416*_Z^8-2304*_Z^4+1
)^7*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-192*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-2304*
(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^6+18432*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*(x^
12-3*x^8+3*x^4-1)^(3/4)*x^3-276480*x^4*RootOf(5308416*_Z^8-2304*_Z^4+1)^6-120*x^8*RootOf(5308416*_Z^8-2304*_Z^
4+1)^2+884736*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*(x^12-3*x^8+3*x^4-1)^(1/4)*x+384*RootOf(5308416*_Z^8-2304*_Z^
4+1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+2304*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^2+(
x^12-3*x^8+3*x^4-1)^(1/2)*x^6-4*RootOf(5308416*_Z^8-2304*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3+55296*RootOf(5
308416*_Z^8-2304*_Z^4+1)^6+96*x^4*RootOf(5308416*_Z^8-2304*_Z^4+1)^2-192*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*(x
^12-3*x^8+3*x^4-1)^(1/4)*x-(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-24*RootOf(5308416*_Z^8-2304*_Z^4+1)^2)/(442368*RootO
f(5308416*_Z^8-2304*_Z^4+1)^7*x-384*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x+1)/(18432*RootOf(5308416*_Z^8-2304*_Z
^4+1)^5*x-4*x*RootOf(5308416*_Z^8-2304*_Z^4+1)-1)/(442368*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x-384*RootOf(5308
416*_Z^8-2304*_Z^4+1)^3*x-1)/(18432*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x-4*x*RootOf(5308416*_Z^8-2304*_Z^4+1)+
1)/(110592*RootOf(5308416*_Z^8-2304*_Z^4+1)^6*x+1)^2/(x-1)^2/(110592*RootOf(5308416*_Z^8-2304*_Z^4+1)^6*x-1)^2
/(1+x)^2)-144*ln(-(127401984*RootOf(5308416*_Z^8-2304*_Z^4+1)^10*x^12-254803968*RootOf(5308416*_Z^8-2304*_Z^4+
1)^10*x^8-165888*x^12*RootOf(5308416*_Z^8-2304*_Z^4+1)^6-9216*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*(x^12-3*x^8+3
*x^4-1)^(1/4)*x^9+127401984*RootOf(5308416*_Z^8-2304*_Z^4+1)^10*x^4+387072*x^8*RootOf(5308416*_Z^8-2304*_Z^4+1
)^6+48*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*x^12-442368*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(5308416*_Z^8-2304*_Z^4
+1)^7*x^3+2304*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^6+18432*RootOf(5308416*_Z^8-230
4*_Z^4+1)^5*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-4*RootOf(5308416*_Z^8-2304*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-2
76480*x^4*RootOf(5308416*_Z^8-2304*_Z^4+1)^6-120*x^8*RootOf(5308416*_Z^8-2304*_Z^4+1)^2+384*RootOf(5308416*_Z^
8-2304*_Z^4+1)^3*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3-2304*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+
1)^4*x^2-(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-9216*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*(x^12-3*x^8+3*x^4-1)^(1/4)*x+8
*RootOf(5308416*_Z^8-2304*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+55296*RootOf(5308416*_Z^8-2304*_Z^4+1)^6+96*x
^4*RootOf(5308416*_Z^8-2304*_Z^4+1)^2+(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-4*RootOf(5308416*_Z^8-2304*_Z^4+1)*(x^12-
3*x^8+3*x^4-1)^(1/4)*x-24*RootOf(5308416*_Z^8-2304*_Z^4+1)^2)/(442368*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x-384
*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x+1)/(18432*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x-4*x*RootOf(5308416*_Z^8-2
304*_Z^4+1)-1)/(442368*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x-384*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x-1)/(18432
*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x-4*x*RootOf(5308416*_Z^8-2304*_Z^4+1)+1)/(110592*RootOf(5308416*_Z^8-2304
*_Z^4+1)^6*x+1)^2/(x-1)^2/(110592*RootOf(5308416*_Z^8-2304*_Z^4+1)^6*x-1)^2/(1+x)^2)*RootOf(5308416*_Z^8-2304*
_Z^4+1)^3-3*RootOf(5308416*_Z^8-2304*_Z^4+1)*ln((6912*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x^12+110592*(x^12-3*x
^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^6*x^9-13824*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x^8+3*RootOf
(5308416*_Z^8-2304*_Z^4+1)*x^12-221184*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^6*x^5-24*(x
^12-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*x^9-288*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*
_Z^8-2304*_Z^4+1)^3*x^6-1152*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^3+6912*RootOf(530
8416*_Z^8-2304*_Z^4+1)^5*x^4-9*RootOf(5308416*_Z^8-2304*_Z^4+1)*x^8+110592*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(5
308416*_Z^8-2304*_Z^4+1)^6*x+48*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*x^5+288*(x^12-3*
x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x^2+(x^12-3*x^8+3*x^4-1)^(3/4)*x^3+9*RootOf(5308416*_Z^8
-2304*_Z^4+1)*x^4-24*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*x-3*RootOf(5308416*_Z^8-230
4*_Z^4+1))/(9216*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x-8*x*RootOf(5308416*_Z^8-2304*_Z^4+1)-1)/(442368*RootOf(5
308416*_Z^8-2304*_Z^4+1)^7*x+192*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x-1)/(9216*RootOf(5308416*_Z^8-2304*_Z^4+1
)^5*x-8*x*RootOf(5308416*_Z^8-2304*_Z^4+1)+1)/(442368*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x+192*RootOf(5308416*
_Z^8-2304*_Z^4+1)^3*x+1)/(110592*RootOf(5308416*_Z^8-2304*_Z^4+1)^6*x+1)^2/(x-1)^2/(110592*RootOf(5308416*_Z^8
-2304*_Z^4+1)^6*x-1)^2/(1+x)^2))/(x^4-1)^(3/4)*((x^4-1)^3)^(1/4)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 4.74 (sec) , antiderivative size = 1549, normalized size of antiderivative = 19.36 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^4+2 x^8\right )}{x^6 \left (1-x^4+x^8\right )} \, dx=\text {Too large to display} \]

[In]

integrate((x^4-1)^(1/4)*(2*x^8+x^4-1)/x^6/(x^8-x^4+1),x, algorithm="fricas")

[Out]

-1/80*(5*sqrt(3)*sqrt(2)*x^5*sqrt(sqrt(2)*sqrt(sqrt(-3) + 1))*log(3*(6*sqrt(2)*(2*x^7 + sqrt(-3)*x^3 - x^3)*(x
^4 - 1)^(1/4)*sqrt(sqrt(-3) + 1) + sqrt(3)*(2*sqrt(2)*sqrt(x^4 - 1)*(3*x^2 - sqrt(-3)*(2*x^6 - x^2)) + (3*x^8
- 9*x^4 + sqrt(-3)*(3*x^8 - x^4 - 1) + 3)*sqrt(sqrt(-3) + 1))*sqrt(sqrt(2)*sqrt(sqrt(-3) + 1)) - 12*(2*x^5 + s
qrt(-3)*x - x)*(x^4 - 1)^(3/4))/(x^8 - x^4 + 1)) - 5*sqrt(3)*sqrt(2)*x^5*sqrt(sqrt(2)*sqrt(sqrt(-3) + 1))*log(
3*(6*sqrt(2)*(2*x^7 + sqrt(-3)*x^3 - x^3)*(x^4 - 1)^(1/4)*sqrt(sqrt(-3) + 1) - sqrt(3)*(2*sqrt(2)*sqrt(x^4 - 1
)*(3*x^2 - sqrt(-3)*(2*x^6 - x^2)) + (3*x^8 - 9*x^4 + sqrt(-3)*(3*x^8 - x^4 - 1) + 3)*sqrt(sqrt(-3) + 1))*sqrt
(sqrt(2)*sqrt(sqrt(-3) + 1)) - 12*(2*x^5 + sqrt(-3)*x - x)*(x^4 - 1)^(3/4))/(x^8 - x^4 + 1)) - 5*sqrt(3)*sqrt(
2)*x^5*sqrt(-sqrt(2)*sqrt(sqrt(-3) + 1))*log(-3*(6*sqrt(2)*(2*x^7 + sqrt(-3)*x^3 - x^3)*(x^4 - 1)^(1/4)*sqrt(s
qrt(-3) + 1) + sqrt(3)*(2*sqrt(2)*sqrt(x^4 - 1)*(3*x^2 - sqrt(-3)*(2*x^6 - x^2)) - (3*x^8 - 9*x^4 + sqrt(-3)*(
3*x^8 - x^4 - 1) + 3)*sqrt(sqrt(-3) + 1))*sqrt(-sqrt(2)*sqrt(sqrt(-3) + 1)) + 12*(2*x^5 + sqrt(-3)*x - x)*(x^4
- 1)^(3/4))/(x^8 - x^4 + 1)) + 5*sqrt(3)*sqrt(2)*x^5*sqrt(-sqrt(2)*sqrt(sqrt(-3) + 1))*log(-3*(6*sqrt(2)*(2*x
^7 + sqrt(-3)*x^3 - x^3)*(x^4 - 1)^(1/4)*sqrt(sqrt(-3) + 1) - sqrt(3)*(2*sqrt(2)*sqrt(x^4 - 1)*(3*x^2 - sqrt(-
3)*(2*x^6 - x^2)) - (3*x^8 - 9*x^4 + sqrt(-3)*(3*x^8 - x^4 - 1) + 3)*sqrt(sqrt(-3) + 1))*sqrt(-sqrt(2)*sqrt(sq
rt(-3) + 1)) + 12*(2*x^5 + sqrt(-3)*x - x)*(x^4 - 1)^(3/4))/(x^8 - x^4 + 1)) - 5*sqrt(2)*x^5*sqrt(-sqrt(2)*sqr
t(-9*sqrt(-3) + 9))*log(-(6*sqrt(2)*(2*x^7 - sqrt(-3)*x^3 - x^3)*(x^4 - 1)^(1/4)*sqrt(-9*sqrt(-3) + 9) + 36*(2
*x^5 - sqrt(-3)*x - x)*(x^4 - 1)^(3/4) + (6*sqrt(2)*sqrt(x^4 - 1)*(3*x^2 + sqrt(-3)*(2*x^6 - x^2)) - (3*x^8 -
9*x^4 - sqrt(-3)*(3*x^8 - x^4 - 1) + 3)*sqrt(-9*sqrt(-3) + 9))*sqrt(-sqrt(2)*sqrt(-9*sqrt(-3) + 9)))/(x^8 - x^
4 + 1)) + 5*sqrt(2)*x^5*sqrt(-sqrt(2)*sqrt(-9*sqrt(-3) + 9))*log(-(6*sqrt(2)*(2*x^7 - sqrt(-3)*x^3 - x^3)*(x^4
- 1)^(1/4)*sqrt(-9*sqrt(-3) + 9) + 36*(2*x^5 - sqrt(-3)*x - x)*(x^4 - 1)^(3/4) - (6*sqrt(2)*sqrt(x^4 - 1)*(3*
x^2 + sqrt(-3)*(2*x^6 - x^2)) - (3*x^8 - 9*x^4 - sqrt(-3)*(3*x^8 - x^4 - 1) + 3)*sqrt(-9*sqrt(-3) + 9))*sqrt(-
sqrt(2)*sqrt(-9*sqrt(-3) + 9)))/(x^8 - x^4 + 1)) + 5*sqrt(2)*x^5*sqrt(sqrt(2)*sqrt(-9*sqrt(-3) + 9))*log((6*sq
rt(2)*(2*x^7 - sqrt(-3)*x^3 - x^3)*(x^4 - 1)^(1/4)*sqrt(-9*sqrt(-3) + 9) - 36*(2*x^5 - sqrt(-3)*x - x)*(x^4 -
1)^(3/4) + (6*sqrt(2)*sqrt(x^4 - 1)*(3*x^2 + sqrt(-3)*(2*x^6 - x^2)) + (3*x^8 - 9*x^4 - sqrt(-3)*(3*x^8 - x^4
- 1) + 3)*sqrt(-9*sqrt(-3) + 9))*sqrt(sqrt(2)*sqrt(-9*sqrt(-3) + 9)))/(x^8 - x^4 + 1)) - 5*sqrt(2)*x^5*sqrt(sq
rt(2)*sqrt(-9*sqrt(-3) + 9))*log((6*sqrt(2)*(2*x^7 - sqrt(-3)*x^3 - x^3)*(x^4 - 1)^(1/4)*sqrt(-9*sqrt(-3) + 9)
- 36*(2*x^5 - sqrt(-3)*x - x)*(x^4 - 1)^(3/4) - (6*sqrt(2)*sqrt(x^4 - 1)*(3*x^2 + sqrt(-3)*(2*x^6 - x^2)) + (
3*x^8 - 9*x^4 - sqrt(-3)*(3*x^8 - x^4 - 1) + 3)*sqrt(-9*sqrt(-3) + 9))*sqrt(sqrt(2)*sqrt(-9*sqrt(-3) + 9)))/(x
^8 - x^4 + 1)) + 16*(x^4 - 1)^(5/4))/x^5

Sympy [N/A]

Not integrable

Time = 18.98 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^4+2 x^8\right )}{x^6 \left (1-x^4+x^8\right )} \, dx=\int \frac {\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + 1\right ) \left (2 x^{4} - 1\right )}{x^{6} \left (x^{8} - x^{4} + 1\right )}\, dx \]

[In]

integrate((x**4-1)**(1/4)*(2*x**8+x**4-1)/x**6/(x**8-x**4+1),x)

[Out]

Integral(((x - 1)*(x + 1)*(x**2 + 1))**(1/4)*(x**4 + 1)*(2*x**4 - 1)/(x**6*(x**8 - x**4 + 1)), x)

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^4+2 x^8\right )}{x^6 \left (1-x^4+x^8\right )} \, dx=\int { \frac {{\left (2 \, x^{8} + x^{4} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (x^{8} - x^{4} + 1\right )} x^{6}} \,d x } \]

[In]

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

[Out]

integrate((2*x^8 + x^4 - 1)*(x^4 - 1)^(1/4)/((x^8 - x^4 + 1)*x^6), x)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.29 (sec) , antiderivative size = 372, normalized size of antiderivative = 4.65 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^4+2 x^8\right )}{x^6 \left (1-x^4+x^8\right )} \, dx=\frac {1}{8} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} - \sqrt {2} + \frac {4 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{8} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} - \sqrt {2} - \frac {4 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{8} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} + \sqrt {2} + \frac {4 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{8} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} + \sqrt {2} - \frac {4 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{16} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\sqrt {6} + \sqrt {2}\right )}}{2 \, x} + \frac {\sqrt {x^{4} - 1}}{x^{2}} + 1\right ) - \frac {1}{16} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (-\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\sqrt {6} + \sqrt {2}\right )}}{2 \, x} + \frac {\sqrt {x^{4} - 1}}{x^{2}} + 1\right ) + \frac {1}{16} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )}}{2 \, x} + \frac {\sqrt {x^{4} - 1}}{x^{2}} + 1\right ) - \frac {1}{16} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (-\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )}}{2 \, x} + \frac {\sqrt {x^{4} - 1}}{x^{2}} + 1\right ) + \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} - 1\right )}}{5 \, x} \]

[In]

integrate((x^4-1)^(1/4)*(2*x^8+x^4-1)/x^6/(x^8-x^4+1),x, algorithm="giac")

[Out]

1/8*(sqrt(6) - 3*sqrt(2))*arctan((sqrt(6) - sqrt(2) + 4*(x^4 - 1)^(1/4)/x)/(sqrt(6) + sqrt(2))) + 1/8*(sqrt(6)
- 3*sqrt(2))*arctan(-(sqrt(6) - sqrt(2) - 4*(x^4 - 1)^(1/4)/x)/(sqrt(6) + sqrt(2))) + 1/8*(sqrt(6) + 3*sqrt(2
))*arctan((sqrt(6) + sqrt(2) + 4*(x^4 - 1)^(1/4)/x)/(sqrt(6) - sqrt(2))) + 1/8*(sqrt(6) + 3*sqrt(2))*arctan(-(
sqrt(6) + sqrt(2) - 4*(x^4 - 1)^(1/4)/x)/(sqrt(6) - sqrt(2))) + 1/16*(sqrt(6) - 3*sqrt(2))*log(1/2*(x^4 - 1)^(
1/4)*(sqrt(6) + sqrt(2))/x + sqrt(x^4 - 1)/x^2 + 1) - 1/16*(sqrt(6) - 3*sqrt(2))*log(-1/2*(x^4 - 1)^(1/4)*(sqr
t(6) + sqrt(2))/x + sqrt(x^4 - 1)/x^2 + 1) + 1/16*(sqrt(6) + 3*sqrt(2))*log(1/2*(x^4 - 1)^(1/4)*(sqrt(6) - sqr
t(2))/x + sqrt(x^4 - 1)/x^2 + 1) - 1/16*(sqrt(6) + 3*sqrt(2))*log(-1/2*(x^4 - 1)^(1/4)*(sqrt(6) - sqrt(2))/x +
sqrt(x^4 - 1)/x^2 + 1) + 1/5*(x^4 - 1)^(1/4)*(1/x^4 - 1)/x

Mupad [N/A]

Not integrable

Time = 5.97 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^4+2 x^8\right )}{x^6 \left (1-x^4+x^8\right )} \, dx=\int \frac {{\left (x^4-1\right )}^{1/4}\,\left (2\,x^8+x^4-1\right )}{x^6\,\left (x^8-x^4+1\right )} \,d x \]

[In]

int(((x^4 - 1)^(1/4)*(x^4 + 2*x^8 - 1))/(x^6*(x^8 - x^4 + 1)),x)

[Out]

int(((x^4 - 1)^(1/4)*(x^4 + 2*x^8 - 1))/(x^6*(x^8 - x^4 + 1)), x)