\(\int \frac {-2 x+x^2}{(1-x+x^2) \sqrt [4]{1+x^4}} \, dx\) [1965]

Optimal result

Integrand size = 27, antiderivative size = 139 \[ \int \frac {-2 x+x^2}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\arctan \left (\frac {\left (-\sqrt {2}+\sqrt {2} x\right ) \sqrt [4]{1+x^4}}{-1+2 x-x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\left (-\sqrt {2}+\sqrt {2} x\right ) \sqrt [4]{1+x^4}}{1-2 x+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \]

[Out]

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

Rubi [C] (warning: unable to verify)

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

Time = 1.17 (sec) , antiderivative size = 851, normalized size of antiderivative = 6.12, number of steps used = 53, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {1607, 6860, 246, 218, 212, 209, 2184, 1254, 385, 214, 211, 524, 1262, 760, 408, 504, 1227, 551, 455, 65, 304} \[ \int \frac {-2 x+x^2}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=-\frac {1}{6} \left (1+i \sqrt {3}\right ) \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right ) x^3-\frac {1}{6} \left (1-i \sqrt {3}\right ) \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right ) x^3+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{4} \left (1+i \sqrt {3}\right ) \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \arctan \left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x^4+1}}\right )-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1-i \sqrt {3}}}\right )-\frac {1}{2} \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1+i \sqrt {3}}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{4} \left (1+i \sqrt {3}\right ) \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \text {arctanh}\left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x^4+1}}\right )-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1-i \sqrt {3}}}\right )+\frac {1}{2} \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1+i \sqrt {3}}}\right )-\frac {i \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-i-\sqrt {3}\right ),\arcsin \left (\sqrt [4]{x^4+1}\right ),-1\right )}{2 x^2}+\frac {i \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{2} \left (i-\sqrt {3}\right ),\arcsin \left (\sqrt [4]{x^4+1}\right ),-1\right )}{2 x^2}+\frac {i \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}},\arcsin \left (\sqrt [4]{x^4+1}\right ),-1\right )}{2 x^2}-\frac {i \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}},\arcsin \left (\sqrt [4]{x^4+1}\right ),-1\right )}{2 x^2} \]

[In]

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

[Out]

-1/6*((1 + I*Sqrt[3])*x^3*AppellF1[3/4, 1/4, 1, 7/4, -x^4, (-2*x^4)/(1 - I*Sqrt[3])]) - ((1 - I*Sqrt[3])*x^3*A
ppellF1[3/4, 1/4, 1, 7/4, -x^4, (-2*x^4)/(1 + I*Sqrt[3])])/6 + ArcTan[x/(1 + x^4)^(1/4)]/2 - ((1 + I*Sqrt[3])*
(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*ArcTan[x/((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*(1 + x^4)^(1/4))])/4 -
((-((I - Sqrt[3])/(I + Sqrt[3])))^(3/4)*ArcTan[((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*x)/(1 + x^4)^(1/4)])/2
- (((1 - I*Sqrt[3])/2)^(3/4)*ArcTan[(2^(1/4)*(1 + x^4)^(1/4))/(1 - I*Sqrt[3])^(1/4)])/2 - (((1 + I*Sqrt[3])/2
)^(3/4)*ArcTan[(2^(1/4)*(1 + x^4)^(1/4))/(1 + I*Sqrt[3])^(1/4)])/2 + ArcTanh[x/(1 + x^4)^(1/4)]/2 - ((1 + I*Sq
rt[3])*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*ArcTanh[x/((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*(1 + x^4)^(1/4
))])/4 - ((-((I - Sqrt[3])/(I + Sqrt[3])))^(3/4)*ArcTanh[((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*x)/(1 + x^4)^
(1/4)])/2 + (((1 - I*Sqrt[3])/2)^(3/4)*ArcTanh[(2^(1/4)*(1 + x^4)^(1/4))/(1 - I*Sqrt[3])^(1/4)])/2 + (((1 + I*
Sqrt[3])/2)^(3/4)*ArcTanh[(2^(1/4)*(1 + x^4)^(1/4))/(1 + I*Sqrt[3])^(1/4)])/2 - ((I/2)*Sqrt[-x^4]*EllipticPi[(
-I - Sqrt[3])/2, ArcSin[(1 + x^4)^(1/4)], -1])/x^2 + ((I/2)*Sqrt[-x^4]*EllipticPi[(I - Sqrt[3])/2, ArcSin[(1 +
x^4)^(1/4)], -1])/x^2 + ((I/2)*Sqrt[-x^4]*EllipticPi[1/Sqrt[(1 - I*Sqrt[3])/2], ArcSin[(1 + x^4)^(1/4)], -1])
/x^2 - ((I/2)*Sqrt[-x^4]*EllipticPi[1/Sqrt[(1 + I*Sqrt[3])/2], ArcSin[(1 + x^4)^(1/4)], -1])/x^2

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 385

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

Rule 408

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Dist[2*(Sqrt[(-b)*(x^2/a)]/x), Subst[I
nt[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d*x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0]

Rule 455

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

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

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 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 760

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/4)), x_Symbol] :> Dist[d, Int[1/((d^2 - e^2*x^2)*(a + c*x^
2)^(1/4)), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*(a + c*x^2)^(1/4)), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ
[c*d^2 + a*e^2, 0]

Rule 1227

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c],
Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c,
0]

Rule 1254

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^4)^p, (d/
(d^2 - e^2*x^4) - e*(x^2/(d^2 - e^2*x^4)))^(-q), x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&&  !IntegerQ[p] && ILtQ[q, 0]

Rule 1262

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2184

Int[((c_) + (d_.)*(x_)^(n_.))^(q_)*((a_) + (b_.)*(x_)^(nn_.))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^
nn)^p, (c/(c^2 - d^2*x^(2*n)) - d*(x^n/(c^2 - d^2*x^(2*n))))^(-q), x], x] /; FreeQ[{a, b, c, d, n, nn, p}, x]
&&  !IntegerQ[p] && ILtQ[q, 0] && IGtQ[Log[2, nn/n], 0]

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 \frac {(-2+x) x}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}} \, dx \\ & = \int \left (\frac {1}{\sqrt [4]{1+x^4}}-\frac {1+x}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}}\right ) \, dx \\ & = \int \frac {1}{\sqrt [4]{1+x^4}} \, dx-\int \frac {1+x}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}} \, dx \\ & = -\int \left (\frac {1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [4]{1+x^4}}+\frac {1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [4]{1+x^4}}\right ) \, dx+\text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\left (1-i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [4]{1+x^4}} \, dx-\left (1+i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [4]{1+x^4}} \, dx \\ & = \frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\left (1-i \sqrt {3}\right ) \int \left (\frac {i-\sqrt {3}}{2 \left (i+\sqrt {3}+2 i x^2\right ) \sqrt [4]{1+x^4}}+\frac {x}{\left (1-i \sqrt {3}+2 x^2\right ) \sqrt [4]{1+x^4}}\right ) \, dx-\left (1+i \sqrt {3}\right ) \int \left (\frac {-i-\sqrt {3}}{2 \left (-i+\sqrt {3}-2 i x^2\right ) \sqrt [4]{1+x^4}}+\frac {x}{\left (1+i \sqrt {3}+2 x^2\right ) \sqrt [4]{1+x^4}}\right ) \, dx \\ & = \frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+2 i \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt [4]{1+x^4}} \, dx-2 i \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt [4]{1+x^4}} \, dx-\left (1-i \sqrt {3}\right ) \int \frac {x}{\left (1-i \sqrt {3}+2 x^2\right ) \sqrt [4]{1+x^4}} \, dx-\left (1+i \sqrt {3}\right ) \int \frac {x}{\left (1+i \sqrt {3}+2 x^2\right ) \sqrt [4]{1+x^4}} \, dx \\ & = \frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+2 i \int \left (\frac {1+i \sqrt {3}}{2 \left (i+\sqrt {3}+2 i x^4\right ) \sqrt [4]{1+x^4}}+\frac {i x^2}{\sqrt [4]{1+x^4} \left (1-i \sqrt {3}+2 x^4\right )}\right ) \, dx-2 i \int \left (\frac {1-i \sqrt {3}}{2 \left (-i+\sqrt {3}-2 i x^4\right ) \sqrt [4]{1+x^4}}-\frac {i x^2}{\sqrt [4]{1+x^4} \left (1+i \sqrt {3}+2 x^4\right )}\right ) \, dx-\frac {1}{2} \left (1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [4]{1+x^2}} \, dx,x,x^2\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [4]{1+x^2}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-2 \int \frac {x^2}{\sqrt [4]{1+x^4} \left (1-i \sqrt {3}+2 x^4\right )} \, dx-2 \int \frac {x^2}{\sqrt [4]{1+x^4} \left (1+i \sqrt {3}+2 x^4\right )} \, dx+\left (i-\sqrt {3}\right ) \int \frac {1}{\left (i+\sqrt {3}+2 i x^4\right ) \sqrt [4]{1+x^4}} \, dx-\left (-1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt [4]{1+x^2}} \, dx,x,x^2\right )-\left (-1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {x}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt [4]{1+x^2}} \, dx,x,x^2\right )-\left (-1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {x}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt [4]{1+x^2}} \, dx,x,x^2\right )-\frac {1}{2} \left (1+i \sqrt {3}\right )^2 \text {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt [4]{1+x^2}} \, dx,x,x^2\right )-\left (i+\sqrt {3}\right ) \int \frac {1}{\left (-i+\sqrt {3}-2 i x^4\right ) \sqrt [4]{1+x^4}} \, dx \\ & = -\frac {2 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 )}{3 \left (1-i \sqrt {3}\right )}-\frac {2 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 )}{3 \left (1+i \sqrt {3}\right )}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\left (i-\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{i+\sqrt {3}-\left (-i+\sqrt {3}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \left (-1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x\right ) \sqrt [4]{1+x}} \, dx,x,x^4\right )-\frac {1}{2} \left (-1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x\right ) \sqrt [4]{1+x}} \, dx,x,x^4\right )-\left (i+\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{-i+\sqrt {3}-\left (i+\sqrt {3}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\left (2 \left (-1-i \sqrt {3}\right ) \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (4+\left (1-i \sqrt {3}\right )^2-4 x^4\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1+x^4}\right )}{x^2}-\frac {\left (\left (1+i \sqrt {3}\right )^2 \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (4+\left (1+i \sqrt {3}\right )^2-4 x^4\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1+x^4}\right )}{x^2} \\ & = -\frac {2 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 )}{3 \left (1-i \sqrt {3}\right )}-\frac {2 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 )}{3 \left (1+i \sqrt {3}\right )}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\left (2 \left (1-i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {x^2}{4+\left (1-i \sqrt {3}\right )^2-4 x^4} \, dx,x,\sqrt [4]{1+x^4}\right )+\left (2 \left (1+i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {x^2}{4+\left (1+i \sqrt {3}\right )^2-4 x^4} \, dx,x,\sqrt [4]{1+x^4}\right )+\frac {\left (i-\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}-\sqrt {-i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {i+\sqrt {3}}}+\frac {\left (i-\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}+\sqrt {-i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {i+\sqrt {3}}}-\frac {\left (i+\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-i+\sqrt {3}}-\sqrt {i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {-i+\sqrt {3}}}-\frac {\left (i+\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-i+\sqrt {3}}+\sqrt {i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {-i+\sqrt {3}}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2} x^2}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2} x^2}-\frac {\left (\left (1+i \sqrt {3}\right )^2 \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1+x^4}\right )}{4 \sqrt {2} x^2}+\frac {\left (\left (1+i \sqrt {3}\right )^2 \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1+x^4}\right )}{4 \sqrt {2} x^2} \\ & = -\frac {2 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 )}{3 \left (1-i \sqrt {3}\right )}-\frac {2 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 )}{3 \left (1+i \sqrt {3}\right )}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\arctan \left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )}{2 \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4}}-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\text {arctanh}\left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )}{2 \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4}}-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )+\frac {\left (1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-i \sqrt {3}}-\sqrt {2} x^2} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2}}-\frac {\left (1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-i \sqrt {3}}+\sqrt {2} x^2} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2}}+\frac {\left (1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+i \sqrt {3}}-\sqrt {2} x^2} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2}}-\frac {\left (1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+i \sqrt {3}}+\sqrt {2} x^2} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x^2\right )} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2} x^2}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x^2\right )} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2} x^2}-\frac {\left (\left (1+i \sqrt {3}\right )^2 \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x^2\right )} \, dx,x,\sqrt [4]{1+x^4}\right )}{4 \sqrt {2} x^2}+\frac {\left (\left (1+i \sqrt {3}\right )^2 \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x^2\right )} \, dx,x,\sqrt [4]{1+x^4}\right )}{4 \sqrt {2} x^2} \\ & = -\frac {2 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 )}{3 \left (1-i \sqrt {3}\right )}-\frac {2 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 )}{3 \left (1+i \sqrt {3}\right )}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\arctan \left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )}{2 \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4}}-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{1+x^4}}{\sqrt [4]{1-i \sqrt {3}}}\right )-\frac {1}{2} \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{1+x^4}}{\sqrt [4]{1+i \sqrt {3}}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\text {arctanh}\left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )}{2 \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4}}-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{1+x^4}}{\sqrt [4]{1-i \sqrt {3}}}\right )+\frac {1}{2} \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{1+x^4}}{\sqrt [4]{1+i \sqrt {3}}}\right )-\frac {\left (1+i \sqrt {3}\right ) \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-i-\sqrt {3}\right ),\arcsin \left (\sqrt [4]{1+x^4}\right ),-1\right )}{2 \sqrt {2} \sqrt {1-i \sqrt {3}} x^2}+\frac {\left (1+i \sqrt {3}\right )^{3/2} \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{2} \left (i-\sqrt {3}\right ),\arcsin \left (\sqrt [4]{1+x^4}\right ),-1\right )}{4 \sqrt {2} x^2}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}},\arcsin \left (\sqrt [4]{1+x^4}\right ),-1\right )}{2 \sqrt {2} \sqrt {1-i \sqrt {3}} x^2}-\frac {\left (1+i \sqrt {3}\right )^{3/2} \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}},\arcsin \left (\sqrt [4]{1+x^4}\right ),-1\right )}{4 \sqrt {2} x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.02 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.87 \[ \int \frac {-2 x+x^2}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\frac {1}{2} \left (\arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {2} (-1+x) \sqrt [4]{1+x^4}}{-1+2 x-x^2+\sqrt {1+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} (-1+x) \sqrt [4]{1+x^4}}{1-2 x+x^2+\sqrt {1+x^4}}\right )\right ) \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 7.81 (sec) , antiderivative size = 593, normalized size of antiderivative = 4.27

method	result	size
trager	\(\text {Expression too large to display}\)	\(593\)

[In]

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

[Out]

1/4*RootOf(_Z^2+1)*ln(-2*RootOf(_Z^2+1)*(x^4+1)^(1/2)*x^2+2*RootOf(_Z^2+1)*x^4+2*(x^4+1)^(3/4)*x-2*x^3*(x^4+1)
^(1/4)+RootOf(_Z^2+1))-1/4*ln(2*(x^4+1)^(3/4)*x-2*(x^4+1)^(1/2)*x^2+2*x^3*(x^4+1)^(1/4)-2*x^4-1)+1/2*RootOf(_Z
^2+RootOf(_Z^2+1))*ln((-(x^4+1)^(1/2)*RootOf(_Z^2+RootOf(_Z^2+1))*RootOf(_Z^2+1)*x^2+2*(x^4+1)^(1/2)*RootOf(_Z
^2+RootOf(_Z^2+1))*RootOf(_Z^2+1)*x+RootOf(_Z^2+1)*(x^4+1)^(1/4)*x^3+(x^4+1)^(3/4)*x-(x^4+1)^(1/2)*RootOf(_Z^2
+RootOf(_Z^2+1))*RootOf(_Z^2+1)-3*RootOf(_Z^2+1)*(x^4+1)^(1/4)*x^2-2*RootOf(_Z^2+RootOf(_Z^2+1))*x^3-(x^4+1)^(
3/4)+3*RootOf(_Z^2+1)*(x^4+1)^(1/4)*x+3*RootOf(_Z^2+RootOf(_Z^2+1))*x^2-RootOf(_Z^2+1)*(x^4+1)^(1/4)-2*RootOf(
_Z^2+RootOf(_Z^2+1))*x)/(x^2-x+1)^2)-1/2*RootOf(_Z^2+1)*RootOf(_Z^2+RootOf(_Z^2+1))*ln(((x^4+1)^(1/2)*RootOf(_
Z^2+RootOf(_Z^2+1))*x^2-RootOf(_Z^2+1)*(x^4+1)^(1/4)*x^3+2*RootOf(_Z^2+RootOf(_Z^2+1))*RootOf(_Z^2+1)*x^3+(x^4
+1)^(3/4)*x-2*(x^4+1)^(1/2)*RootOf(_Z^2+RootOf(_Z^2+1))*x+3*RootOf(_Z^2+1)*(x^4+1)^(1/4)*x^2-3*RootOf(_Z^2+Roo
tOf(_Z^2+1))*RootOf(_Z^2+1)*x^2-(x^4+1)^(3/4)+(x^4+1)^(1/2)*RootOf(_Z^2+RootOf(_Z^2+1))-3*RootOf(_Z^2+1)*(x^4+
1)^(1/4)*x+2*RootOf(_Z^2+RootOf(_Z^2+1))*RootOf(_Z^2+1)*x+RootOf(_Z^2+1)*(x^4+1)^(1/4))/(x^2-x+1)^2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.26 (sec) , antiderivative size = 483, normalized size of antiderivative = 3.47 \[ \int \frac {-2 x+x^2}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} \sqrt {x^{4} + 1} {\left (\left (i + 1\right ) \, x^{2} - \left (2 i + 2\right ) \, x + i + 1\right )} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )} + \sqrt {2} {\left (-\left (2 i - 2\right ) \, x^{3} + \left (3 i - 3\right ) \, x^{2} - \left (2 i - 2\right ) \, x\right )} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (-i \, x^{3} + 3 i \, x^{2} - 3 i \, x + i\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) - \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} \sqrt {x^{4} + 1} {\left (-\left (i - 1\right ) \, x^{2} + \left (2 i - 2\right ) \, x - i + 1\right )} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )} + \sqrt {2} {\left (\left (2 i + 2\right ) \, x^{3} - \left (3 i + 3\right ) \, x^{2} + \left (2 i + 2\right ) \, x\right )} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (i \, x^{3} - 3 i \, x^{2} + 3 i \, x - i\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} \sqrt {x^{4} + 1} {\left (\left (i - 1\right ) \, x^{2} - \left (2 i - 2\right ) \, x + i - 1\right )} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )} + \sqrt {2} {\left (-\left (2 i + 2\right ) \, x^{3} + \left (3 i + 3\right ) \, x^{2} - \left (2 i + 2\right ) \, x\right )} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (i \, x^{3} - 3 i \, x^{2} + 3 i \, x - i\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) - \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} \sqrt {x^{4} + 1} {\left (-\left (i + 1\right ) \, x^{2} + \left (2 i + 2\right ) \, x - i - 1\right )} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )} + \sqrt {2} {\left (\left (2 i - 2\right ) \, x^{3} - \left (3 i - 3\right ) \, x^{2} + \left (2 i - 2\right ) \, x\right )} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (-i \, x^{3} + 3 i \, x^{2} - 3 i \, x + i\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \frac {1}{4} \, \arctan \left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x\right ) + \frac {1}{4} \, \log \left (2 \, x^{4} + 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x + 1\right ) \]

[In]

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

[Out]

(1/8*I - 1/8)*sqrt(2)*log((sqrt(2)*sqrt(x^4 + 1)*((I + 1)*x^2 - (2*I + 2)*x + I + 1) + 2*(x^4 + 1)^(3/4)*(x -
1) + sqrt(2)*(-(2*I - 2)*x^3 + (3*I - 3)*x^2 - (2*I - 2)*x) - 2*(x^4 + 1)^(1/4)*(-I*x^3 + 3*I*x^2 - 3*I*x + I)
)/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) - (1/8*I + 1/8)*sqrt(2)*log((sqrt(2)*sqrt(x^4 + 1)*(-(I - 1)*x^2 + (2*I - 2
)*x - I + 1) + 2*(x^4 + 1)^(3/4)*(x - 1) + sqrt(2)*((2*I + 2)*x^3 - (3*I + 3)*x^2 + (2*I + 2)*x) - 2*(x^4 + 1)
^(1/4)*(I*x^3 - 3*I*x^2 + 3*I*x - I))/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) + (1/8*I + 1/8)*sqrt(2)*log((sqrt(2)*sq
rt(x^4 + 1)*((I - 1)*x^2 - (2*I - 2)*x + I - 1) + 2*(x^4 + 1)^(3/4)*(x - 1) + sqrt(2)*(-(2*I + 2)*x^3 + (3*I +
3)*x^2 - (2*I + 2)*x) - 2*(x^4 + 1)^(1/4)*(I*x^3 - 3*I*x^2 + 3*I*x - I))/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) - (
1/8*I - 1/8)*sqrt(2)*log((sqrt(2)*sqrt(x^4 + 1)*(-(I + 1)*x^2 + (2*I + 2)*x - I - 1) + 2*(x^4 + 1)^(3/4)*(x -
1) + sqrt(2)*((2*I - 2)*x^3 - (3*I - 3)*x^2 + (2*I - 2)*x) - 2*(x^4 + 1)^(1/4)*(-I*x^3 + 3*I*x^2 - 3*I*x + I))
/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) + 1/4*arctan(2*(x^4 + 1)^(1/4)*x^3 + 2*(x^4 + 1)^(3/4)*x) + 1/4*log(2*x^4 +
2*(x^4 + 1)^(1/4)*x^3 + 2*sqrt(x^4 + 1)*x^2 + 2*(x^4 + 1)^(3/4)*x + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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