\(\int \frac {1-x^4+x^8}{x^2 (-1+x^4)^{3/4} (-1-x^4+x^8)} \, dx\) [2434]

Optimal result

Integrand size = 35, antiderivative size = 197 \[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=-\frac {\sqrt [4]{-1+x^4}}{x}+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )+\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right ) \]

[Out]

-(x^4-1)^(1/4)/x+1/10*(-10+10*5^(1/2))^(1/2)*arctan(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4-1)^(1/4))-1/10*(10+10*5^(1
/2))^(1/2)*arctan(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4-1)^(1/4))-1/10*(-10+10*5^(1/2))^(1/2)*arctanh(1/2*(-2+2*5^(1/
2))^(1/2)*x/(x^4-1)^(1/4))+1/10*(10+10*5^(1/2))^(1/2)*arctanh(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4-1)^(1/4))

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.35 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.80, 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} \[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=\frac {4 \sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,\frac {2 x^4}{1-\sqrt {5}}\right )}{3 \sqrt {5} \left (1-\sqrt {5}\right ) \sqrt [4]{1-x^4}}-\frac {4 \sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {2 x^4}{1+\sqrt {5}},x^4\right )}{3 \sqrt {5} \left (1+\sqrt {5}\right ) \sqrt [4]{1-x^4}}-\frac {\sqrt [4]{x^4-1}}{x} \]

[In]

Int[(1 - x^4 + x^8)/(x^2*(-1 + x^4)^(3/4)*(-1 - x^4 + x^8)),x]

[Out]

-((-1 + x^4)^(1/4)/x) + (4*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x^4, (2*x^4)/(1 - Sqrt[5])])/(3*Sq
rt[5]*(1 - Sqrt[5])*(1 - x^4)^(1/4)) - (4*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (2*x^4)/(1 + Sqrt[5
]), x^4])/(3*Sqrt[5]*(1 + Sqrt[5])*(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 {1}{x^2 \left (-1+x^4\right )^{3/4}}+\frac {2 x^2 \sqrt [4]{-1+x^4}}{-1-x^4+x^8}\right ) \, dx \\ & = 2 \int \frac {x^2 \sqrt [4]{-1+x^4}}{-1-x^4+x^8} \, dx-\int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx \\ & = -\frac {\sqrt [4]{-1+x^4}}{x}+2 \int \left (-\frac {2 x^2 \sqrt [4]{-1+x^4}}{\sqrt {5} \left (1+\sqrt {5}-2 x^4\right )}-\frac {2 x^2 \sqrt [4]{-1+x^4}}{\sqrt {5} \left (-1+\sqrt {5}+2 x^4\right )}\right ) \, dx \\ & = -\frac {\sqrt [4]{-1+x^4}}{x}-\frac {4 \int \frac {x^2 \sqrt [4]{-1+x^4}}{1+\sqrt {5}-2 x^4} \, dx}{\sqrt {5}}-\frac {4 \int \frac {x^2 \sqrt [4]{-1+x^4}}{-1+\sqrt {5}+2 x^4} \, dx}{\sqrt {5}} \\ & = -\frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (4 \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{1+\sqrt {5}-2 x^4} \, dx}{\sqrt {5} \sqrt [4]{1-x^4}}-\frac {\left (4 \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{-1+\sqrt {5}+2 x^4} \, dx}{\sqrt {5} \sqrt [4]{1-x^4}} \\ & = -\frac {\sqrt [4]{-1+x^4}}{x}+\frac {4 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-\sqrt {5}}\right )}{3 \sqrt {5} \left (1-\sqrt {5}\right ) \sqrt [4]{1-x^4}}-\frac {4 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+\sqrt {5}},x^4\right )}{3 \sqrt {5} \left (1+\sqrt {5}\right ) \sqrt [4]{1-x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.94 \[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=\frac {1}{10} \left (-\frac {10 \sqrt [4]{-1+x^4}}{x}+\sqrt {10 \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {10 \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {10 \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )+\sqrt {10 \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )\right ) \]

[In]

Integrate[(1 - x^4 + x^8)/(x^2*(-1 + x^4)^(3/4)*(-1 - x^4 + x^8)),x]

[Out]

((-10*(-1 + x^4)^(1/4))/x + Sqrt[10*(-1 + Sqrt[5])]*ArcTan[(Sqrt[(-1 + Sqrt[5])/2]*x)/(-1 + x^4)^(1/4)] - Sqrt
[10*(1 + Sqrt[5])]*ArcTan[(Sqrt[(1 + Sqrt[5])/2]*x)/(-1 + x^4)^(1/4)] - Sqrt[10*(-1 + Sqrt[5])]*ArcTanh[(Sqrt[
(-1 + Sqrt[5])/2]*x)/(-1 + x^4)^(1/4)] + Sqrt[10*(1 + Sqrt[5])]*ArcTanh[(Sqrt[(1 + Sqrt[5])/2]*x)/(-1 + x^4)^(
1/4)])/10

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 11.40 (sec) , antiderivative size = 3552, normalized size of antiderivative = 18.03

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

[In]

int((x^8-x^4+1)/x^2/(x^4-1)^(3/4)/(x^8-x^4-1),x)

[Out]

-(x^4-1)^(1/4)/x+(1/10*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*ln((-32000*RootOf(6400*_Z^4-80*_Z^2-1)
^4*x^12+1600*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^12+160*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^2+400*RootOf(6400*_Z^
4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^9+64000*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^8-15*x^12-6*(x^12-3*
x^8+3*x^4-1)^(1/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*x^9+800*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(
6400*_Z^4-80*_Z^2-1)^2*x^6-3600*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^8+480*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(_Z^2+4
00*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^3-320*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z
^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^5-32000*RootOf(6400*_Z^4-80*_Z^2-1)^4*
x^4-20*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+35*x^8-8*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*
_Z^2-1)^2-5)*x^3+12*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*x^5-800*(x^12-
3*x^8+3*x^4-1)^(1/2)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^2+2400*x^4*RootOf(6400*_Z^4-80*_Z^2-1)^2+160*(x^12-3*x^8+
3*x^4-1)^(1/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x+20*(x^12-3*x^8
+3*x^4-1)^(1/2)*x^2-25*x^4-6*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*x-400
*RootOf(6400*_Z^4-80*_Z^2-1)^2+5)/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*RootOf(6400*_Z^4-80*_Z^2-1)*x^2-1)
/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*RootOf(6400*_Z^4-80*_Z^2-1)*x^2+1)/(1+x)^2/(-1+x)^2/(x^2+1)^2)+2*Ro
otOf(6400*_Z^4-80*_Z^2-1)*ln(-(3200*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^12+80*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^12+3
20*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-6400*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^8+8*RootO
f(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+80*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(6400*_Z^4-80*_Z^2-1
)^2*x^6-200*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^8+960*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3
-640*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+3200*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^4+(x^12
-3*x^8+3*x^4-1)^(1/2)*x^6+4*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3-16*RootOf(6400*_Z^4-80*
_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-80*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^2+160*x^4
*RootOf(6400*_Z^4-80*_Z^2-1)^2+320*RootOf(6400*_Z^4-80*_Z^2-1)^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+8*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x-40*RootOf(6400*_Z^4-80*_Z^2-1)^2)/(8
0*x^4*RootOf(6400*_Z^4-80*_Z^2-1)^2+1)/(1+x)^2/(-1+x)^2/(x^2+1)^2)+8*RootOf(6400*_Z^4-80*_Z^2-1)^2*RootOf(_Z^2
+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*ln(-(-16000*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^12-400*RootOf(6400*_Z^4-80*_
Z^2-1)^2*x^12-240*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4
-80*_Z^2-1)^2*x^9+32000*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^8-(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^2+400*RootOf(64
00*_Z^4-80*_Z^2-1)^2-5)*x^9+400*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^6+1000*RootOf(6400*
_Z^4-80*_Z^2-1)^2*x^8+320*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6
400*_Z^4-80*_Z^2-1)^2*x^3+480*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*Root
Of(6400*_Z^4-80*_Z^2-1)^2*x^5-16000*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^4+5*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+3*(x^12
-3*x^8+3*x^4-1)^(3/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*x^3+2*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf
(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*x^5-400*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^
2-800*x^4*RootOf(6400*_Z^4-80*_Z^2-1)^2-240*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^
2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x-5*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_
Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*x+200*RootOf(6400*_Z^4-80*_Z^2-1)^2)/(80*x^4*RootOf(6400*_Z^4-80*_Z^2
-1)^2+1)/(1+x)^2/(-1+x)^2/(x^2+1)^2)-160*ln((6400*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^12-320*RootOf(6400*_Z^4-80*_
Z^2-1)^2*x^12-1920*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-12800*RootOf(6400*_Z^4-80*_Z^2
-1)^4*x^8+3*x^12+32*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+160*(x^12-3*x^8+3*x^4-1)^(1/2)*
RootOf(6400*_Z^4-80*_Z^2-1)^2*x^6+720*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^8+2560*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^
12-3*x^8+3*x^4-1)^(3/4)*x^3+3840*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+6400*RootOf(6400
*_Z^4-80*_Z^2-1)^4*x^4-4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-7*x^8-56*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4
-1)^(3/4)*x^3-64*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-160*(x^12-3*x^8+3*x^4-1)^(1/2)*Roo
tOf(6400*_Z^4-80*_Z^2-1)^2*x^2-480*x^4*RootOf(6400*_Z^4-80*_Z^2-1)^2-1920*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-
3*x^8+3*x^4-1)^(1/4)*x+4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2+5*x^4+32*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4
-1)^(1/4)*x+80*RootOf(6400*_Z^4-80*_Z^2-1)^2-1)/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*RootOf(6400*_Z^4-80*
_Z^2-1)*x^2-1)/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*RootOf(6400*_Z^4-80*_Z^2-1)*x^2+1)/(1+x)^2/(-1+x)^2/(
x^2+1)^2)*RootOf(6400*_Z^4-80*_Z^2-1)^3+2*ln((6400*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^12-320*RootOf(6400*_Z^4-80*
_Z^2-1)^2*x^12-1920*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-12800*RootOf(6400*_Z^4-80*_Z^
2-1)^4*x^8+3*x^12+32*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+160*(x^12-3*x^8+3*x^4-1)^(1/2)
*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^6+720*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^8+2560*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x
^12-3*x^8+3*x^4-1)^(3/4)*x^3+3840*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+6400*RootOf(640
0*_Z^4-80*_Z^2-1)^4*x^4-4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-7*x^8-56*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^
4-1)^(3/4)*x^3-64*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-160*(x^12-3*x^8+3*x^4-1)^(1/2)*Ro
otOf(6400*_Z^4-80*_Z^2-1)^2*x^2-480*x^4*RootOf(6400*_Z^4-80*_Z^2-1)^2-1920*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12
-3*x^8+3*x^4-1)^(1/4)*x+4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2+5*x^4+32*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^
4-1)^(1/4)*x+80*RootOf(6400*_Z^4-80*_Z^2-1)^2-1)/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*RootOf(6400*_Z^4-80
*_Z^2-1)*x^2-1)/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*RootOf(6400*_Z^4-80*_Z^2-1)*x^2+1)/(1+x)^2/(-1+x)^2/
(x^2+1)^2)*RootOf(6400*_Z^4-80*_Z^2-1))/(x^4-1)^(3/4)*((x^4-1)^3)^(1/4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1271 vs. \(2 (137) = 274\).

Time = 9.48 (sec) , antiderivative size = 1271, normalized size of antiderivative = 6.45 \[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=\text {Too large to display} \]

[In]

integrate((x^8-x^4+1)/x^2/(x^4-1)^(3/4)/(x^8-x^4-1),x, algorithm="fricas")

[Out]

-1/40*(sqrt(10)*x*sqrt(-sqrt(5) + 1)*log((sqrt(10)*sqrt(x^4 - 1)*(5*x^2 + sqrt(5)*(2*x^6 - x^2))*sqrt(-sqrt(5)
+ 1) - sqrt(10)*(5*x^8 - 5*x^4 + sqrt(5)*(2*x^4 - 1))*sqrt(-sqrt(5) + 1) + 10*(2*x^5 + sqrt(5)*x - x)*(x^4 -
1)^(3/4) + 10*(x^7 - 3*x^3 - sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) - sqrt(10)*x*sqrt(-sqrt(5)
+ 1)*log(-(sqrt(10)*sqrt(x^4 - 1)*(5*x^2 + sqrt(5)*(2*x^6 - x^2))*sqrt(-sqrt(5) + 1) - sqrt(10)*(5*x^8 - 5*x^
4 + sqrt(5)*(2*x^4 - 1))*sqrt(-sqrt(5) + 1) - 10*(2*x^5 + sqrt(5)*x - x)*(x^4 - 1)^(3/4) - 10*(x^7 - 3*x^3 - s
qrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) + sqrt(10)*x*sqrt(-sqrt(5) - 1)*log((sqrt(10)*sqrt(x^4 -
1)*(5*x^2 - sqrt(5)*(2*x^6 - x^2))*sqrt(-sqrt(5) - 1) + sqrt(10)*(5*x^8 - 5*x^4 - sqrt(5)*(2*x^4 - 1))*sqrt(-
sqrt(5) - 1) + 10*(2*x^5 - sqrt(5)*x - x)*(x^4 - 1)^(3/4) - 10*(x^7 - 3*x^3 + sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(
1/4))/(x^8 - x^4 - 1)) - sqrt(10)*x*sqrt(-sqrt(5) - 1)*log(-(sqrt(10)*sqrt(x^4 - 1)*(5*x^2 - sqrt(5)*(2*x^6 -
x^2))*sqrt(-sqrt(5) - 1) + sqrt(10)*(5*x^8 - 5*x^4 - sqrt(5)*(2*x^4 - 1))*sqrt(-sqrt(5) - 1) - 10*(2*x^5 - sqr
t(5)*x - x)*(x^4 - 1)^(3/4) + 10*(x^7 - 3*x^3 + sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) + sqrt(
10)*x*sqrt(sqrt(5) - 1)*log((10*(2*x^5 + sqrt(5)*x - x)*(x^4 - 1)^(3/4) + (sqrt(10)*sqrt(x^4 - 1)*(5*x^2 + sqr
t(5)*(2*x^6 - x^2)) + sqrt(10)*(5*x^8 - 5*x^4 + sqrt(5)*(2*x^4 - 1)))*sqrt(sqrt(5) - 1) - 10*(x^7 - 3*x^3 - sq
rt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) - sqrt(10)*x*sqrt(sqrt(5) - 1)*log((10*(2*x^5 + sqrt(5)*x
- x)*(x^4 - 1)^(3/4) - (sqrt(10)*sqrt(x^4 - 1)*(5*x^2 + sqrt(5)*(2*x^6 - x^2)) + sqrt(10)*(5*x^8 - 5*x^4 + sq
rt(5)*(2*x^4 - 1)))*sqrt(sqrt(5) - 1) - 10*(x^7 - 3*x^3 - sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1
)) + sqrt(10)*x*sqrt(sqrt(5) + 1)*log((10*(2*x^5 - sqrt(5)*x - x)*(x^4 - 1)^(3/4) + (sqrt(10)*sqrt(x^4 - 1)*(5
*x^2 - sqrt(5)*(2*x^6 - x^2)) - sqrt(10)*(5*x^8 - 5*x^4 - sqrt(5)*(2*x^4 - 1)))*sqrt(sqrt(5) + 1) + 10*(x^7 -
3*x^3 + sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) - sqrt(10)*x*sqrt(sqrt(5) + 1)*log((10*(2*x^5 -
sqrt(5)*x - x)*(x^4 - 1)^(3/4) - (sqrt(10)*sqrt(x^4 - 1)*(5*x^2 - sqrt(5)*(2*x^6 - x^2)) - sqrt(10)*(5*x^8 -
5*x^4 - sqrt(5)*(2*x^4 - 1)))*sqrt(sqrt(5) + 1) + 10*(x^7 - 3*x^3 + sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8
- x^4 - 1)) + 40*(x^4 - 1)^(1/4))/x

Sympy [F(-1)]

Timed out. \[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate((x**8-x**4+1)/x**2/(x**4-1)**(3/4)/(x**8-x**4-1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=\int { \frac {x^{8} - x^{4} + 1}{{\left (x^{8} - x^{4} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]

[In]

integrate((x^8-x^4+1)/x^2/(x^4-1)^(3/4)/(x^8-x^4-1),x, algorithm="maxima")

[Out]

integrate((x^8 - x^4 + 1)/((x^8 - x^4 - 1)*(x^4 - 1)^(3/4)*x^2), x)

Giac [F]

\[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=\int { \frac {x^{8} - x^{4} + 1}{{\left (x^{8} - x^{4} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]

[In]

integrate((x^8-x^4+1)/x^2/(x^4-1)^(3/4)/(x^8-x^4-1),x, algorithm="giac")

[Out]

integrate((x^8 - x^4 + 1)/((x^8 - x^4 - 1)*(x^4 - 1)^(3/4)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx=\int -\frac {x^8-x^4+1}{x^2\,{\left (x^4-1\right )}^{3/4}\,\left (-x^8+x^4+1\right )} \,d x \]

[In]

int(-(x^8 - x^4 + 1)/(x^2*(x^4 - 1)^(3/4)*(x^4 - x^8 + 1)),x)

[Out]

int(-(x^8 - x^4 + 1)/(x^2*(x^4 - 1)^(3/4)*(x^4 - x^8 + 1)), x)