3.13.20 \(\int \frac {1-x^2+x^4}{\sqrt {x+x^2+x^3} (-1+x^4)} \, dx\)

Optimal. Leaf size=89 \[ -\frac {1}{4} \tan ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right )-\frac {3}{2} \tanh ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x^3+x^2+x}}{x^2+x+1}\right )}{4 \sqrt {3}} \]

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Rubi [C]  time = 2.23, antiderivative size = 592, normalized size of antiderivative = 6.65, number of steps used = 31, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2056, 6725, 716, 1103, 934, 169, 538, 537} \begin {gather*} \frac {\sqrt {x} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x^3+x^2+x}}-\frac {3 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (-i-\sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}-\frac {\sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (1-i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}-\frac {\sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (-1+i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}-\frac {3 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (i+\sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[x]*(1 + x)*Sqrt[(1 + x + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt[x]], 1/4])/Sqrt[x + x^2 + x^3] - (3*Sqr
t[x]*Sqrt[1 + (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(-I - Sqrt[3])/2, ArcSin[((1 -
I*Sqrt[3])*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/((1 - I*Sqrt[3])*Sqrt[x + x^2 + x^3]) - (Sqrt[x]*Sqrt[1
+ (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(1 - I*Sqrt[3])/2, ArcSin[((1 - I*Sqrt[3]
)*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/((1 - I*Sqrt[3])*Sqrt[x + x^2 + x^3]) - (Sqrt[x]*Sqrt[1 + (2*x)/(
1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(-1 + I*Sqrt[3])/2, ArcSin[((1 - I*Sqrt[3])*Sqrt[x]
)/2], (I + Sqrt[3])/(I - Sqrt[3])])/((1 - I*Sqrt[3])*Sqrt[x + x^2 + x^3]) - (3*Sqrt[x]*Sqrt[1 + (2*x)/(1 - I*S
qrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(I + Sqrt[3])/2, ArcSin[((1 - I*Sqrt[3])*Sqrt[x])/2], (I +
Sqrt[3])/(I - Sqrt[3])])/((1 - I*Sqrt[3])*Sqrt[x + x^2 + x^3])

Rule 169

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] &&  !SimplerQ[e
+ f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 538

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

Rule 716

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

Rule 934

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x])/Sqrt[a + b*x + c*x^2], Int[1/((d +
e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1103

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6725

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

Rubi steps

\begin {align*} \int \frac {1-x^2+x^4}{\sqrt {x+x^2+x^3} \left (-1+x^4\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1-x^2+x^4}{\sqrt {x} \sqrt {1+x+x^2} \left (-1+x^4\right )} \, dx}{\sqrt {x+x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {1+x+x^2}}+\frac {2-x^2}{\sqrt {x} \sqrt {1+x+x^2} \left (-1+x^4\right )}\right ) \, dx}{\sqrt {x+x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x+x^2}} \, dx}{\sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {2-x^2}{\sqrt {x} \sqrt {1+x+x^2} \left (-1+x^4\right )} \, dx}{\sqrt {x+x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (-\frac {1}{2 \sqrt {x} \left (1-x^2\right ) \sqrt {1+x+x^2}}-\frac {3}{2 \sqrt {x} \left (1+x^2\right ) \sqrt {1+x+x^2}}\right ) \, dx}{\sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^2+x^3}}\\ &=\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1-x^2\right ) \sqrt {1+x+x^2}} \, dx}{2 \sqrt {x+x^2+x^3}}-\frac {\left (3 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1+x^2\right ) \sqrt {1+x+x^2}} \, dx}{2 \sqrt {x+x^2+x^3}}\\ &=\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (\frac {1}{2 (1-x) \sqrt {x} \sqrt {1+x+x^2}}+\frac {1}{2 \sqrt {x} (1+x) \sqrt {1+x+x^2}}\right ) \, dx}{2 \sqrt {x+x^2+x^3}}-\frac {\left (3 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (\frac {i}{2 (i-x) \sqrt {x} \sqrt {1+x+x^2}}+\frac {i}{2 \sqrt {x} (i+x) \sqrt {1+x+x^2}}\right ) \, dx}{2 \sqrt {x+x^2+x^3}}\\ &=\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (3 i \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{(i-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{4 \sqrt {x+x^2+x^3}}-\frac {\left (3 i \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} (i+x) \sqrt {1+x+x^2}} \, dx}{4 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{4 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} (1+x) \sqrt {1+x+x^2}} \, dx}{4 \sqrt {x+x^2+x^3}}\\ &=\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (3 i \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{(i-x) \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{4 \sqrt {x+x^2+x^3}}-\frac {\left (3 i \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{\sqrt {x} (i+x) \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{4 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{4 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{\sqrt {x} (1+x) \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{4 \sqrt {x+x^2+x^3}}\\ &=\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (3 i \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i-x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (3 i \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1-x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^2+x^3}}\\ &=\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (3 i \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i-x^2\right ) \sqrt {1+i \sqrt {3}+2 x^2} \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (3 i \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+x^2\right ) \sqrt {1+i \sqrt {3}+2 x^2} \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1-x^2\right ) \sqrt {1+i \sqrt {3}+2 x^2} \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1+i \sqrt {3}+2 x^2} \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^2+x^3}}\\ &=\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (3 i \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i-x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (3 i \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1-x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^2+x^3}}\\ &=\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {3 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (-i-\sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (1-i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (-1+i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}-\frac {3 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (i+\sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}\\ \end {align*}

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Mathematica [C]  time = 3.07, size = 774, normalized size = 8.70 \begin {gather*} -\frac {(-1)^{2/3} x^{3/2} \left (-4 \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} F\left (i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )+\sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left (\sqrt [3]{-1};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )+3 \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left (-(-1)^{5/6};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )+3 \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left ((-1)^{5/6};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )-\frac {\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}} \sqrt {\frac {(-1)^{2/3} \sqrt {x}-1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}} \sqrt {-\frac {(-1)^{2/3} \sqrt {x}+1}{\sqrt {x}+\sqrt [3]{-1}-1}} \left (\sqrt [3]{-1} \sqrt {x}-1\right )^2 \left (\left (1+\sqrt [3]{-1}\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}}\right )\right |-3\right )-2 \sqrt [3]{-1} \Pi \left (-1;\left .\sin ^{-1}\left (\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}}\right )\right |-3\right )\right )}{\left (\sqrt [3]{-1}-1\right )^2 x}+\frac {\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}} \sqrt {\frac {(-1)^{2/3} \sqrt {x}-1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}} \sqrt {-\frac {(-1)^{2/3} \sqrt {x}+1}{\sqrt {x}+\sqrt [3]{-1}-1}} \left (\sqrt [3]{-1} \sqrt {x}-1\right )^2 \left (\left (\sqrt [3]{-1}-1\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}}\right )\right |-3\right )-2 \sqrt [3]{-1} \Pi \left (3;\left .\sin ^{-1}\left (\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}}\right )\right |-3\right )\right )}{\left (\sqrt [3]{-1}-1\right )^2 x}\right )}{2 \sqrt {x \left (x^2+x+1\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*((-1)^(2/3)*x^(3/2)*(-4*Sqrt[1 - (-1)^(2/3)/x]*Sqrt[((-1)^(1/3) + x)/x]*EllipticF[I*ArcSinh[(-1)^(5/6)/Sq
rt[x]], (-1)^(2/3)] - (Sqrt[(1 - (-1)^(1/3) + Sqrt[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[x]))]*(-1 + (-1
)^(1/3)*Sqrt[x])^2*Sqrt[(-1 + (-1)^(2/3)*Sqrt[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[x]))]*Sqrt[-((1 + (-
1)^(2/3)*Sqrt[x])/(-1 + (-1)^(1/3) + Sqrt[x]))]*((1 + (-1)^(1/3))*EllipticF[ArcSin[Sqrt[(1 - (-1)^(1/3) + Sqrt
[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[x]))]], -3] - 2*(-1)^(1/3)*EllipticPi[-1, ArcSin[Sqrt[(1 - (-1)^(
1/3) + Sqrt[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[x]))]], -3]))/((-1 + (-1)^(1/3))^2*x) + (Sqrt[(1 - (-1
)^(1/3) + Sqrt[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[x]))]*(-1 + (-1)^(1/3)*Sqrt[x])^2*Sqrt[(-1 + (-1)^(
2/3)*Sqrt[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[x]))]*Sqrt[-((1 + (-1)^(2/3)*Sqrt[x])/(-1 + (-1)^(1/3) +
Sqrt[x]))]*((-1 + (-1)^(1/3))*EllipticF[ArcSin[Sqrt[(1 - (-1)^(1/3) + Sqrt[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(
1/3)*Sqrt[x]))]], -3] - 2*(-1)^(1/3)*EllipticPi[3, ArcSin[Sqrt[(1 - (-1)^(1/3) + Sqrt[x])/((1 + (-1)^(1/3))*(-
1 + (-1)^(1/3)*Sqrt[x]))]], -3]))/((-1 + (-1)^(1/3))^2*x) + Sqrt[1 - (-1)^(2/3)/x]*Sqrt[((-1)^(1/3) + x)/x]*El
lipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)/Sqrt[x]], (-1)^(2/3)] + 3*Sqrt[1 - (-1)^(2/3)/x]*Sqrt[((-1)^(1/3) +
x)/x]*EllipticPi[-(-1)^(5/6), I*ArcSinh[(-1)^(5/6)/Sqrt[x]], (-1)^(2/3)] + 3*Sqrt[1 - (-1)^(2/3)/x]*Sqrt[((-1)
^(1/3) + x)/x]*EllipticPi[(-1)^(5/6), I*ArcSinh[(-1)^(5/6)/Sqrt[x]], (-1)^(2/3)]))/Sqrt[x*(1 + x + x^2)]

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IntegrateAlgebraic [A]  time = 0.16, size = 89, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \tan ^{-1}\left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )-\frac {3}{2} \tanh ^{-1}\left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x+x^2+x^3}}{1+x+x^2}\right )}{4 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - x^2 + x^4)/(Sqrt[x + x^2 + x^3]*(-1 + x^4)),x]

[Out]

-1/4*ArcTan[Sqrt[x + x^2 + x^3]/(1 + x + x^2)] - (3*ArcTanh[Sqrt[x + x^2 + x^3]/(1 + x + x^2)])/2 - ArcTanh[(S
qrt[3]*Sqrt[x + x^2 + x^3])/(1 + x + x^2)]/(4*Sqrt[3])

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fricas [A]  time = 1.15, size = 120, normalized size = 1.35 \begin {gather*} \frac {1}{48} \, \sqrt {3} \log \left (\frac {x^{4} + 20 \, x^{3} - 4 \, \sqrt {3} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 4 \, x + 1\right )} + 30 \, x^{2} + 20 \, x + 1}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) + \frac {1}{8} \, \arctan \left (\frac {x^{2} + 1}{2 \, \sqrt {x^{3} + x^{2} + x}}\right ) + \frac {3}{4} \, \log \left (\frac {x^{2} + 2 \, x - 2 \, \sqrt {x^{3} + x^{2} + x} + 1}{x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*sqrt(3)*log((x^4 + 20*x^3 - 4*sqrt(3)*sqrt(x^3 + x^2 + x)*(x^2 + 4*x + 1) + 30*x^2 + 20*x + 1)/(x^4 - 4*x
^3 + 6*x^2 - 4*x + 1)) + 1/8*arctan(1/2*(x^2 + 1)/sqrt(x^3 + x^2 + x)) + 3/4*log((x^2 + 2*x - 2*sqrt(x^3 + x^2
+ x) + 1)/(x^2 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} - x^{2} + 1}{{\left (x^{4} - 1\right )} \sqrt {x^{3} + x^{2} + x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.84, size = 136, normalized size = 1.53

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+4 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +\RootOf \left (\textit {\_Z}^{2}-3\right )-6 \sqrt {x^{3}+x^{2}+x}}{\left (-1+x \right )^{2}}\right )}{24}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{3}+x^{2}+x}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right )^{2}}\right )}{8}+\frac {3 \ln \left (\frac {-x^{2}+2 \sqrt {x^{3}+x^{2}+x}-2 x -1}{x^{2}+1}\right )}{4}\)	\(136\)
default	\(\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}\right )}-\frac {3 \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}\right )}+\frac {3 \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}\right )}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{6 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}+\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{6 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}\)	\(1080\)
elliptic	\(\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{\sqrt {x^{3}+x^{2}+x}}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}}+\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{2 \sqrt {x^{3}+x^{2}+x}}-\frac {3 \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}}-\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}+i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}}+\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}+i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{2 \sqrt {x^{3}+x^{2}+x}}+\frac {3 \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}+i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}}-\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{12 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}+\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{12 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}\)	\(1764\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-x^2+1)/(x^3+x^2+x)^(1/2)/(x^4-1),x,method=_RETURNVERBOSE)

[Out]

1/24*RootOf(_Z^2-3)*ln(-(RootOf(_Z^2-3)*x^2+4*RootOf(_Z^2-3)*x+RootOf(_Z^2-3)-6*(x^3+x^2+x)^(1/2))/(-1+x)^2)+1
/8*RootOf(_Z^2+1)*ln((-RootOf(_Z^2+1)*x^2+2*(x^3+x^2+x)^(1/2)-RootOf(_Z^2+1))/(1+x)^2)+3/4*ln((-x^2+2*(x^3+x^2
+x)^(1/2)-2*x-1)/(x^2+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} - x^{2} + 1}{{\left (x^{4} - 1\right )} \sqrt {x^{3} + x^{2} + x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 698, normalized size = 7.84

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*((3^(1/2)*1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2)
)^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticF(asin((x/((3^(1/2)*1i)/2 - 1/2))^(1
/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 +
1/2))^(1/2) - (3*((3^(1/2)*1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/
2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi(- 3^(1/2)/2 - 1i/2
, asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(2*(x^2 + x^3 - x*(
(3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) - (3*((3^(1/2)*1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1
/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1
/2))^(1/2)*ellipticPi(3^(1/2)/2 + 1i/2, asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1
/2)*1i)/2 + 1/2)))/(2*(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) - (((3^(1/2)*1i)/2
- 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1
/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi(1/2 - (3^(1/2)*1i)/2, asin((x/((3^(1/2)*1i)/2 - 1/2)
)^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(2*(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*
1i)/2 + 1/2))^(1/2)) - (((3^(1/2)*1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/(
(3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi((3^(1/2)*1i)
/2 - 1/2, asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(2*(x^2 + x
^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} - x^{2} + 1}{\sqrt {x \left (x^{2} + x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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