3.25.34 \(\int \frac {1-x^4+x^8}{x^2 (-1+x^4)^{3/4} (-1-x^4+x^8)} \, dx\) [2434]
Optimal. Leaf size=197 \[ -\frac {\sqrt [4]{-1+x^4}}{x}+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )+\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right ) \]
[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] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 0.38, antiderivative size = 157, normalized size of antiderivative = 0.80, number of steps
used = 9, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6860, 270,
1542, 525, 524} \begin {gather*} \frac {4 \sqrt [4]{x^4-1} x^3 F_1\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 F_1\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} \end {gather*}
Antiderivative was successfully verified.
[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*} \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 \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} F_1\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} F_1\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]
time = 0.68, size = 185, normalized size = 0.94 \begin {gather*} \frac {1}{10} \left (-\frac {10 \sqrt [4]{-1+x^4}}{x}+\sqrt {10 \left (-1+\sqrt {5}\right )} \text {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 {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 )} \tanh ^{-1}\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 )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[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] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 12.20, size = 3502, normalized size = 17.78 \[\text {output too large to display}\]
Verification of antiderivative is not currently implemented for this CAS.
[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+(-2*RootOf(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^12-320*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*RootOf(6400*_Z^4+80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-80*RootOf(6400*_Z^4+80*_Z^2-1)^2
*(x^12-3*x^8+3*x^4-1)^(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+(x^12-3*x^8+3*x^4-1)
^(1/2)*x^6-3200*RootOf(6400*_Z^4+80*_Z^2-1)^4*x^4-4*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*RootOf(6400*_Z^4+80*_Z^2-1)^2*(x^12-3*x^8+3*
x^4-1)^(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(64
00*_Z^4+80*_Z^2-1)^2)/(640*RootOf(6400*_Z^4+80*_Z^2-1)^3*x^2+4*RootOf(6400*_Z^4+80*_Z^2-1)*x^2+1)/(640*RootOf(
6400*_Z^4+80*_Z^2-1)^3*x^2+4*RootOf(6400*_Z^4+80*_Z^2-1)*x^2-1)/(x^2+1)^2/(-1+x)^2/(1+x)^2)+1/10*RootOf(_Z^2+4
00*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*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*RootOf(6400*_Z^4+80*_Z^2-1)^2*(x^12-3*x^8+3*
x^4-1)^(1/4)*x^9+64000*RootOf(6400*_Z^4+80*_Z^2-1)^4*x^8-15*x^12-6*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)
^2+5)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+800*RootOf(6400*_Z^4+80*_Z^2-1)^2*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+3600*Roo
tOf(6400*_Z^4+80*_Z^2-1)^2*x^8+480*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*RootOf(6400*_Z^4+80*_Z^2-1
)^2*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3+320*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*RootOf(6400*_Z^4+80*_Z
^2-1)^2*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+20*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-32000*RootOf(6400*_Z^4+80*_Z^2-1)^4*x
^4+35*x^8+8*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3+12*RootOf(_Z^2+400
*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-800*RootOf(6400*_Z^4+80*_Z^2-1)^2*(x^12-3*x^8
+3*x^4-1)^(1/2)*x^2-2400*x^4*RootOf(6400*_Z^4+80*_Z^2-1)^2-160*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5
)*RootOf(6400*_Z^4+80*_Z^2-1)^2*(x^12-3*x^8+3*x^4-1)^(1/4)*x-20*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-25*x^4-6*RootOf
(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*(x^12-3*x^8+3*x^4-1)^(1/4)*x+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)/(
80*x^4*RootOf(6400*_Z^4+80*_Z^2-1)^2+x^4+1)/(x^2+1)^2/(-1+x)^2/(1+x)^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*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*RootOf(6400*_Z^4+80*_Z^2-1)^2*(x^12-3*x
^8+3*x^4-1)^(1/4)*x^9-32000*RootOf(6400*_Z^4+80*_Z^2-1)^4*x^8-RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)
*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-400*RootOf(6400*_Z^4+80*_Z^2-1)^2*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+1000*RootOf(6
400*_Z^4+80*_Z^2-1)^2*x^8+320*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*RootOf(6400*_Z^4+80*_Z^2-1)^2*(
x^12-3*x^8+3*x^4-1)^(3/4)*x^3-480*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*RootOf(6400*_Z^4+80*_Z^2-1)
^2*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+5*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+16000*RootOf(6400*_Z^4+80*_Z^2-1)^4*x^4-3*R
ootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3+2*RootOf(_Z^2+400*RootOf(6400*_
Z^4+80*_Z^2-1)^2+5)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+400*RootOf(6400*_Z^4+80*_Z^2-1)^2*(x^12-3*x^8+3*x^4-1)^(1/2
)*x^2-800*x^4*RootOf(6400*_Z^4+80*_Z^2-1)^2+240*RootOf(_Z^2+400*RootOf(6400*_Z^4+80*_Z^2-1)^2+5)*RootOf(6400*_
Z^4+80*_Z^2-1)^2*(x^12-3*x^8+3*x^4-1)^(1/4)*x-5*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-RootOf(_Z^2+400*RootOf(6400*_Z^
4+80*_Z^2-1)^2+5)*(x^12-3*x^8+3*x^4-1)^(1/4)*x+200*RootOf(6400*_Z^4+80*_Z^2-1)^2)/(640*RootOf(6400*_Z^4+80*_Z^
2-1)^3*x^2+4*RootOf(6400*_Z^4+80*_Z^2-1)*x^2+1)/(640*RootOf(6400*_Z^4+80*_Z^2-1)^3*x^2+4*RootOf(6400*_Z^4+80*_
Z^2-1)*x^2-1)/(x^2+1)^2/(-1+x)^2/(1+x)^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+8
0*_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*RootOf(6400*_Z^4+80*_
Z^2-1)^2*(x^12-3*x^8+3*x^4-1)^(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^5-4*(x^12-3*
x^8+3*x^4-1)^(1/2)*x^6-6400*RootOf(6400*_Z^4+80*_Z^2-1)^4*x^4+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*RootOf(6400*_Z^4+80*_Z^2
-1)^2*(x^12-3*x^8+3*x^4-1)^(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^...
________________________________________________________________________________________
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((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)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 971 vs.
\(2 (137) = 274\).
time = 9.74, size = 971, normalized size = 4.93 \begin {gather*} -\frac {4 \, \sqrt {10} x \sqrt {\sqrt {5} + 1} \arctan \left (-\frac {\sqrt {2} {\left (2 \, \sqrt {10} {\left (5 \, x^{6} - \sqrt {5} {\left (x^{6} + 2 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} - \sqrt {10} {\left (5 \, x^{8} + 5 \, x^{4} - \sqrt {5} {\left (5 \, x^{8} - 3 \, x^{4} - 1\right )} - 5\right )}\right )} {\left (\sqrt {5} + 1\right )} + 4 \, {\left (\sqrt {10} {\left (5 \, x^{5} - \sqrt {5} {\left (x^{5} + 2 \, x\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - \sqrt {10} {\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (5 \, x^{3} - \sqrt {5} {\left (2 \, x^{7} - x^{3}\right )}\right )}\right )} \sqrt {\sqrt {5} + 1}}{40 \, {\left (x^{8} - x^{4} - 1\right )}}\right ) + 4 \, \sqrt {10} x \sqrt {\sqrt {5} - 1} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {10} {\left (5 \, x^{6} + \sqrt {5} {\left (x^{6} + 2 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} + \sqrt {10} {\left (5 \, x^{8} + 5 \, x^{4} + \sqrt {5} {\left (5 \, x^{8} - 3 \, x^{4} - 1\right )} - 5\right )}\right )} {\left (\sqrt {5} - 1\right )} - 4 \, {\left (\sqrt {10} {\left (5 \, x^{5} + \sqrt {5} {\left (x^{5} + 2 \, x\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + \sqrt {10} {\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (5 \, x^{3} + \sqrt {5} {\left (2 \, x^{7} - x^{3}\right )}\right )}\right )} \sqrt {\sqrt {5} - 1}}{40 \, {\left (x^{8} - x^{4} - 1\right )}}\right ) + \sqrt {10} x \sqrt {\sqrt {5} - 1} \log \left (\frac {10 \, {\left (2 \, x^{5} + \sqrt {5} x - x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + {\left (\sqrt {10} \sqrt {x^{4} - 1} {\left (5 \, x^{2} + \sqrt {5} {\left (2 \, x^{6} - x^{2}\right )}\right )} + \sqrt {10} {\left (5 \, x^{8} - 5 \, x^{4} + \sqrt {5} {\left (2 \, x^{4} - 1\right )}\right )}\right )} \sqrt {\sqrt {5} - 1} - 10 \, {\left (x^{7} - 3 \, x^{3} - \sqrt {5} {\left (x^{7} - x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{8} - x^{4} - 1}\right ) - \sqrt {10} x \sqrt {\sqrt {5} - 1} \log \left (\frac {10 \, {\left (2 \, x^{5} + \sqrt {5} x - x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - {\left (\sqrt {10} \sqrt {x^{4} - 1} {\left (5 \, x^{2} + \sqrt {5} {\left (2 \, x^{6} - x^{2}\right )}\right )} + \sqrt {10} {\left (5 \, x^{8} - 5 \, x^{4} + \sqrt {5} {\left (2 \, x^{4} - 1\right )}\right )}\right )} \sqrt {\sqrt {5} - 1} - 10 \, {\left (x^{7} - 3 \, x^{3} - \sqrt {5} {\left (x^{7} - x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{8} - x^{4} - 1}\right ) + \sqrt {10} x \sqrt {\sqrt {5} + 1} \log \left (\frac {10 \, {\left (2 \, x^{5} - \sqrt {5} x - x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + {\left (\sqrt {10} \sqrt {x^{4} - 1} {\left (5 \, x^{2} - \sqrt {5} {\left (2 \, x^{6} - x^{2}\right )}\right )} - \sqrt {10} {\left (5 \, x^{8} - 5 \, x^{4} - \sqrt {5} {\left (2 \, x^{4} - 1\right )}\right )}\right )} \sqrt {\sqrt {5} + 1} + 10 \, {\left (x^{7} - 3 \, x^{3} + \sqrt {5} {\left (x^{7} - x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{8} - x^{4} - 1}\right ) - \sqrt {10} x \sqrt {\sqrt {5} + 1} \log \left (\frac {10 \, {\left (2 \, x^{5} - \sqrt {5} x - x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - {\left (\sqrt {10} \sqrt {x^{4} - 1} {\left (5 \, x^{2} - \sqrt {5} {\left (2 \, x^{6} - x^{2}\right )}\right )} - \sqrt {10} {\left (5 \, x^{8} - 5 \, x^{4} - \sqrt {5} {\left (2 \, x^{4} - 1\right )}\right )}\right )} \sqrt {\sqrt {5} + 1} + 10 \, {\left (x^{7} - 3 \, x^{3} + \sqrt {5} {\left (x^{7} - x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{8} - x^{4} - 1}\right ) + 40 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{40 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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*(4*sqrt(10)*x*sqrt(sqrt(5) + 1)*arctan(-1/40*(sqrt(2)*(2*sqrt(10)*(5*x^6 - sqrt(5)*(x^6 + 2*x^2))*sqrt(x
^4 - 1) - sqrt(10)*(5*x^8 + 5*x^4 - sqrt(5)*(5*x^8 - 3*x^4 - 1) - 5))*(sqrt(5) + 1) + 4*(sqrt(10)*(5*x^5 - sqr
t(5)*(x^5 + 2*x))*(x^4 - 1)^(3/4) - sqrt(10)*(x^4 - 1)^(1/4)*(5*x^3 - sqrt(5)*(2*x^7 - x^3)))*sqrt(sqrt(5) + 1
))/(x^8 - x^4 - 1)) + 4*sqrt(10)*x*sqrt(sqrt(5) - 1)*arctan(1/40*(sqrt(2)*(2*sqrt(10)*(5*x^6 + sqrt(5)*(x^6 +
2*x^2))*sqrt(x^4 - 1) + sqrt(10)*(5*x^8 + 5*x^4 + sqrt(5)*(5*x^8 - 3*x^4 - 1) - 5))*(sqrt(5) - 1) - 4*(sqrt(10
)*(5*x^5 + sqrt(5)*(x^5 + 2*x))*(x^4 - 1)^(3/4) + sqrt(10)*(x^4 - 1)^(1/4)*(5*x^3 + sqrt(5)*(2*x^7 - x^3)))*sq
rt(sqrt(5) - 1))/(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*sqr
t(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)) - 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 - sq
rt(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 + s
qrt(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
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**8-x**4+1)/x**2/(x**4-1)**(3/4)/(x**8-x**4-1),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________