\(\int \frac {-1+x}{(1+x+x^2) \sqrt [4]{1+x^4}} \, dx\) [1700]

Optimal result

Integrand size = 21, antiderivative size = 114 \[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\frac {\arctan \left (\frac {-\frac {1}{\sqrt {2}}-\sqrt {2} x-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{(1+x) \sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\left (\sqrt {2}+\sqrt {2} x\right ) \sqrt [4]{1+x^4}}{1+2 x+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \]

[Out]

1/2*arctan((-1/2*2^(1/2)-x*2^(1/2)-1/2*2^(1/2)*x^2+1/2*(x^4+1)^(1/2)*2^(1/2))/(1+x)/(x^4+1)^(1/4))*2^(1/2)-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 = 0.95 (sec) , antiderivative size = 797, normalized size of antiderivative = 6.99, number of steps used = 46, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {6860, 2184, 1254, 385, 218, 214, 211, 524, 1262, 760, 408, 504, 1227, 551, 455, 65, 304, 209, 212} \[ \int \frac {-1+x}{\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 {\arctan \left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x^4+1}}\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]{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 {\text {arctanh}\left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x^4+1}}\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]{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[(-1 + x)/((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/((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4
)*(1 + x^4)^(1/4))]/(2*(-((I - Sqrt[3])/(I + Sqrt[3])))^(3/4)) - ((-((I - Sqrt[3])/(I + Sqrt[3])))^(3/4)*ArcTa
n[((-((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/((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*(1 + x^4)^(1/4))]/(2*(-((I - Sqrt[3])
/(I + Sqrt[3])))^(3/4)) - ((-((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 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 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 \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 \\ & = \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 \\ & = \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 \\ & = 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 \\ & = 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 ) \\ & = -\left (2 \int \frac {x^2}{\sqrt [4]{1+x^4} \left (1-i \sqrt {3}+2 x^4\right )} \, dx\right )-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 )}+\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 )}-\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 {\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 {\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 {\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 {\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 = 1.65 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.76 \[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\frac {\arctan \left (\frac {-1-2 x-x^2+\sqrt {1+x^4}}{\sqrt {2} (1+x) \sqrt [4]{1+x^4}}\right )-\text {arctanh}\left (\frac {\sqrt {2} (1+x) \sqrt [4]{1+x^4}}{1+2 x+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 3.28 (sec) , antiderivative size = 390, normalized size of antiderivative = 3.42

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x -\left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-3 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-3 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}+\left (x^{4}+1\right )^{\frac {3}{4}} x -\left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+\left (x^{4}+1\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x}{\left (x^{2}+x +1\right )^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+\left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+3 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +3 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x +2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x +\left (x^{4}+1\right )^{\frac {3}{4}} x +\left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+\sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+\left (x^{4}+1\right )^{\frac {3}{4}}}{\left (x^{2}+x +1\right )^{2}}\right )}{2}\)	\(390\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.55 (sec) , antiderivative size = 413, normalized size of antiderivative = 3.62 \[ \int \frac {-1+x}{\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 ) \]

[In]

integrate((-1+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))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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