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

Optimal. Leaf size=129 \[ -\sqrt {\frac {1}{3} \left (1+i \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {1-i \sqrt {2}} \sqrt {x^3-x^2-x}}{x^2-x-1}\right )-\sqrt {\frac {1}{3} \left (1-i \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {1+i \sqrt {2}} \sqrt {x^3-x^2-x}}{x^2-x-1}\right ) \]

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Rubi [C]  time = 1.98, antiderivative size = 625, normalized size of antiderivative = 4.84, number of steps used = 27, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2056, 6725, 716, 1098, 934, 168, 538, 537} \begin {gather*} \frac {\sqrt {x} \sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (-\frac {1}{2} \sqrt [4]{-1} \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} \sqrt [4]{-1} \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (-\frac {1}{2} (-1)^{3/4} \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} (-1)^{3/4} \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {x^3-x^2-x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[x]*Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2 + (1 + Sqrt[5])*x)/(2 + (1 - Sqrt[5])*x)]*EllipticF[ArcSin[(Sqrt[2
]*5^(1/4)*Sqrt[x])/Sqrt[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(5^(1/4)*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*S
qrt[-x - x^2 + x^3]) - (Sqrt[3 + Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*Ellip
ticPi[-1/2*((-1)^(1/4)*(1 + Sqrt[5])), ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(2*Sqrt[-x -
x^2 + x^3]) - (Sqrt[3 + Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*EllipticPi[((-
1)^(1/4)*(1 + Sqrt[5]))/2, ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(2*Sqrt[-x - x^2 + x^3])
- (Sqrt[3 + Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*EllipticPi[-1/2*((-1)^(3/4
)*(1 + Sqrt[5])), ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(2*Sqrt[-x - x^2 + x^3]) - (Sqrt[3
+ Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*EllipticPi[((-1)^(3/4)*(1 + Sqrt[5]
))/2, ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(2*Sqrt[-x - x^2 + x^3])

Rule 168

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] && GtQ[(d*e - c
*f)/d, 0]

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 1098

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

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^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^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}{\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 (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1-x+x^2} \left (1+x^4\right )} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=-\frac {\left (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {i}{2 \sqrt {x} \left (i-x^2\right ) \sqrt {-1-x+x^2}}+\frac {i}{2 \sqrt {x} \left (i+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} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (i-x^2\right ) \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (i+x^2\right ) \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-x\right ) \sqrt {x} \sqrt {-1-x+x^2}}-\frac {(-1)^{3/4}}{2 \sqrt {x} \left (\sqrt [4]{-1}+x\right ) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-x\right ) \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt [4]{-1}}{2 \sqrt {x} \left (-(-1)^{3/4}+x\right ) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt [4]{-1}+x\right ) \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}+\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\left (-(-1)^{3/4}-x\right ) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}+\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-(-1)^{3/4}+x\right ) \sqrt {-1-x+x^2}} \, dx}{2 \sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}}-\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt [4]{-1}+x\right ) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}}+\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\left (-(-1)^{3/4}-x\right ) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}}+\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\sqrt {x} \left (-(-1)^{3/4}+x\right ) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{2 \sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [4]{-1}-x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [4]{-1}+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}-\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left ((-1)^{3/4}-x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}-\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left ((-1)^{3/4}+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [4]{-1}-x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [4]{-1}+x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}-\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left ((-1)^{3/4}-x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}-\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left ((-1)^{3/4}+x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (-\frac {1}{2} \sqrt [4]{-1} \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} \sqrt [4]{-1} \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (-\frac {1}{2} (-1)^{3/4} \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} (-1)^{3/4} \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {-x-x^2+x^3}}\\ \end {align*}

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Mathematica [C]  time = 1.43, size = 329, normalized size = 2.55 \begin {gather*} -\frac {i \sqrt {\frac {2}{\sqrt {5}-1}} \sqrt {-\frac {1}{x^2}-\frac {1}{x}+1} x^{3/2} \left (2 F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{1+\sqrt {5}}}}{\sqrt {x}}\right )|-\frac {3}{2}-\frac {\sqrt {5}}{2}\right )-\Pi \left (-\frac {1}{2} \sqrt [4]{-1} \left (1+\sqrt {5}\right );i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{1+\sqrt {5}}}}{\sqrt {x}}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )-\Pi \left (\frac {1}{2} \sqrt [4]{-1} \left (1+\sqrt {5}\right );i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{1+\sqrt {5}}}}{\sqrt {x}}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )-\Pi \left (-\frac {1}{2} (-1)^{3/4} \left (1+\sqrt {5}\right );i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{1+\sqrt {5}}}}{\sqrt {x}}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )-\Pi \left (\frac {1}{2} (-1)^{3/4} \left (1+\sqrt {5}\right );i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{1+\sqrt {5}}}}{\sqrt {x}}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )\right )}{\sqrt {x \left (x^2-x-1\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*Sqrt[2/(-1 + Sqrt[5])]*Sqrt[1 - x^(-2) - x^(-1)]*x^(3/2)*(2*EllipticF[I*ArcSinh[Sqrt[2/(1 + Sqrt[5])]/Sq
rt[x]], -3/2 - Sqrt[5]/2] - EllipticPi[-1/2*((-1)^(1/4)*(1 + Sqrt[5])), I*ArcSinh[Sqrt[2/(1 + Sqrt[5])]/Sqrt[x
]], (-3 - Sqrt[5])/2] - EllipticPi[((-1)^(1/4)*(1 + Sqrt[5]))/2, I*ArcSinh[Sqrt[2/(1 + Sqrt[5])]/Sqrt[x]], (-3
- Sqrt[5])/2] - EllipticPi[-1/2*((-1)^(3/4)*(1 + Sqrt[5])), I*ArcSinh[Sqrt[2/(1 + Sqrt[5])]/Sqrt[x]], (-3 - S
qrt[5])/2] - EllipticPi[((-1)^(3/4)*(1 + Sqrt[5]))/2, I*ArcSinh[Sqrt[2/(1 + Sqrt[5])]/Sqrt[x]], (-3 - Sqrt[5])
/2]))/Sqrt[x*(-1 - x + x^2)]

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IntegrateAlgebraic [A]  time = 0.41, size = 129, normalized size = 1.00 \begin {gather*} -\sqrt {\frac {1}{3} \left (1+i \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {1-i \sqrt {2}} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\sqrt {\frac {1}{3} \left (1-i \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {1+i \sqrt {2}} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[(1 + I*Sqrt[2])/3]*ArcTan[(Sqrt[1 - I*Sqrt[2]]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)]) - Sqrt[(1 - I*Sqr
t[2])/3]*ArcTan[(Sqrt[1 + I*Sqrt[2]]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)]

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fricas [B]  time = 0.99, size = 2368, normalized size = 18.36

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*3^(1/4)*sqrt(sqrt(3) + 3)*(sqrt(3) - 1)*log(3*(3*x^4 - 12*x^3 + 2*3^(1/4)*sqrt(x^3 - x^2 - x)*(3*x^2 - s
qrt(3)*(x^2 - 4*x - 1) - 6*x - 3)*sqrt(sqrt(3) + 3) + 12*x^2 + 12*sqrt(3)*(x^3 - x^2 - x) + 12*x + 3)/(x^4 + 1
)) + 1/24*3^(1/4)*sqrt(sqrt(3) + 3)*(sqrt(3) - 1)*log(3*(3*x^4 - 12*x^3 - 2*3^(1/4)*sqrt(x^3 - x^2 - x)*(3*x^2
- sqrt(3)*(x^2 - 4*x - 1) - 6*x - 3)*sqrt(sqrt(3) + 3) + 12*x^2 + 12*sqrt(3)*(x^3 - x^2 - x) + 12*x + 3)/(x^4
+ 1)) - 1/6*3^(1/4)*sqrt(2)*sqrt(sqrt(3) + 3)*arctan(-1/36*(18*sqrt(3)*sqrt(2)*(x^11 - 3*x^10 - 3*x^9 + 8*x^8
+ 4*x^7 - 6*x^6 - 4*x^5 + 8*x^4 + 3*x^3 - 3*x^2 - x) + 3*sqrt(x^3 - x^2 - x)*(3^(3/4)*(sqrt(3)*sqrt(2)*(x^10
- 4*x^9 - 9*x^8 + 36*x^7 + 26*x^6 - 72*x^5 - 26*x^4 + 36*x^3 + 9*x^2 - 4*x - 1) - sqrt(2)*(x^10 + 2*x^9 - 29*x
^8 + 12*x^7 + 110*x^6 - 28*x^5 - 110*x^4 + 12*x^3 + 29*x^2 + 2*x - 1)) + 4*3^(1/4)*(sqrt(3)*sqrt(2)*(2*x^9 + 3
*x^8 - 5*x^7 - 23*x^6 + 12*x^5 + 23*x^4 - 5*x^3 - 3*x^2 + 2*x) - 3*sqrt(2)*(x^9 + 4*x^8 - 11*x^7 - 8*x^6 + 18*
x^5 + 8*x^4 - 11*x^3 - 4*x^2 + x)))*sqrt(sqrt(3) + 3) - sqrt(3)*(24*sqrt(3)*sqrt(2)*(x^10 + 2*x^9 - 13*x^8 + 2
*x^7 + 22*x^6 - 2*x^5 - 13*x^4 - 2*x^3 + x^2) + sqrt(x^3 - x^2 - x)*(3^(3/4)*(sqrt(3)*sqrt(2)*(x^10 - 6*x^9 -
9*x^8 + 68*x^7 - 38*x^6 - 108*x^5 + 38*x^4 + 68*x^3 + 9*x^2 - 6*x - 1) - sqrt(2)*(x^10 - 4*x^9 - 5*x^8 + 84*x^
7 - 106*x^6 - 136*x^5 + 106*x^4 + 84*x^3 + 5*x^2 - 4*x - 1)) + 4*3^(1/4)*(sqrt(3)*sqrt(2)*(2*x^9 - 3*x^8 - 23*
x^7 + 31*x^6 + 36*x^5 - 31*x^4 - 23*x^3 + 3*x^2 + 2*x) - 3*sqrt(2)*(x^9 - 15*x^7 + 12*x^6 + 26*x^5 - 12*x^4 -
15*x^3 + x)))*sqrt(sqrt(3) + 3) - 48*sqrt(2)*(x^10 - 7*x^8 + 2*x^7 + 12*x^6 - 2*x^5 - 7*x^4 + x^2) + 2*sqrt(3)
*(sqrt(3)*sqrt(2)*(x^11 - x^10 - 17*x^9 + 48*x^8 + 2*x^7 - 82*x^6 - 2*x^5 + 48*x^4 + 17*x^3 - x^2 - x) - 3*sqr
t(2)*(x^11 - 5*x^10 - 9*x^9 + 52*x^8 - 6*x^7 - 90*x^6 + 6*x^5 + 52*x^4 + 9*x^3 - 5*x^2 - x)))*sqrt((3*x^4 - 12
*x^3 + 2*3^(1/4)*sqrt(x^3 - x^2 - x)*(3*x^2 - sqrt(3)*(x^2 - 4*x - 1) - 6*x - 3)*sqrt(sqrt(3) + 3) + 12*x^2 +
12*sqrt(3)*(x^3 - x^2 - x) + 12*x + 3)/(x^4 + 1)) - 18*sqrt(2)*(x^11 + 5*x^10 - 35*x^9 + 116*x^7 - 22*x^6 - 11
6*x^5 + 35*x^3 + 5*x^2 - x) + 6*sqrt(3)*(sqrt(3)*sqrt(2)*(x^11 + 5*x^10 - 35*x^9 + 116*x^7 - 22*x^6 - 116*x^5
+ 35*x^3 + 5*x^2 - x) - 3*sqrt(2)*(x^11 - 3*x^10 - 3*x^9 + 8*x^8 + 4*x^7 - 6*x^6 - 4*x^5 + 8*x^4 + 3*x^3 - 3*x
^2 - x)))/(x^11 - 9*x^10 - 17*x^9 + 104*x^8 - 14*x^7 - 178*x^6 + 14*x^5 + 104*x^4 + 17*x^3 - 9*x^2 - x)) - 1/6
*3^(1/4)*sqrt(2)*sqrt(sqrt(3) + 3)*arctan(1/36*(18*sqrt(3)*sqrt(2)*(x^11 - 3*x^10 - 3*x^9 + 8*x^8 + 4*x^7 - 6*
x^6 - 4*x^5 + 8*x^4 + 3*x^3 - 3*x^2 - x) - 3*sqrt(x^3 - x^2 - x)*(3^(3/4)*(sqrt(3)*sqrt(2)*(x^10 - 4*x^9 - 9*x
^8 + 36*x^7 + 26*x^6 - 72*x^5 - 26*x^4 + 36*x^3 + 9*x^2 - 4*x - 1) - sqrt(2)*(x^10 + 2*x^9 - 29*x^8 + 12*x^7 +
110*x^6 - 28*x^5 - 110*x^4 + 12*x^3 + 29*x^2 + 2*x - 1)) + 4*3^(1/4)*(sqrt(3)*sqrt(2)*(2*x^9 + 3*x^8 - 5*x^7
- 23*x^6 + 12*x^5 + 23*x^4 - 5*x^3 - 3*x^2 + 2*x) - 3*sqrt(2)*(x^9 + 4*x^8 - 11*x^7 - 8*x^6 + 18*x^5 + 8*x^4 -
11*x^3 - 4*x^2 + x)))*sqrt(sqrt(3) + 3) - sqrt(3)*(24*sqrt(3)*sqrt(2)*(x^10 + 2*x^9 - 13*x^8 + 2*x^7 + 22*x^6
- 2*x^5 - 13*x^4 - 2*x^3 + x^2) - sqrt(x^3 - x^2 - x)*(3^(3/4)*(sqrt(3)*sqrt(2)*(x^10 - 6*x^9 - 9*x^8 + 68*x^
7 - 38*x^6 - 108*x^5 + 38*x^4 + 68*x^3 + 9*x^2 - 6*x - 1) - sqrt(2)*(x^10 - 4*x^9 - 5*x^8 + 84*x^7 - 106*x^6 -
136*x^5 + 106*x^4 + 84*x^3 + 5*x^2 - 4*x - 1)) + 4*3^(1/4)*(sqrt(3)*sqrt(2)*(2*x^9 - 3*x^8 - 23*x^7 + 31*x^6
+ 36*x^5 - 31*x^4 - 23*x^3 + 3*x^2 + 2*x) - 3*sqrt(2)*(x^9 - 15*x^7 + 12*x^6 + 26*x^5 - 12*x^4 - 15*x^3 + x)))
*sqrt(sqrt(3) + 3) - 48*sqrt(2)*(x^10 - 7*x^8 + 2*x^7 + 12*x^6 - 2*x^5 - 7*x^4 + x^2) + 2*sqrt(3)*(sqrt(3)*sqr
t(2)*(x^11 - x^10 - 17*x^9 + 48*x^8 + 2*x^7 - 82*x^6 - 2*x^5 + 48*x^4 + 17*x^3 - x^2 - x) - 3*sqrt(2)*(x^11 -
5*x^10 - 9*x^9 + 52*x^8 - 6*x^7 - 90*x^6 + 6*x^5 + 52*x^4 + 9*x^3 - 5*x^2 - x)))*sqrt((3*x^4 - 12*x^3 - 2*3^(1
/4)*sqrt(x^3 - x^2 - x)*(3*x^2 - sqrt(3)*(x^2 - 4*x - 1) - 6*x - 3)*sqrt(sqrt(3) + 3) + 12*x^2 + 12*sqrt(3)*(x
^3 - x^2 - x) + 12*x + 3)/(x^4 + 1)) - 18*sqrt(2)*(x^11 + 5*x^10 - 35*x^9 + 116*x^7 - 22*x^6 - 116*x^5 + 35*x^
3 + 5*x^2 - x) + 6*sqrt(3)*(sqrt(3)*sqrt(2)*(x^11 + 5*x^10 - 35*x^9 + 116*x^7 - 22*x^6 - 116*x^5 + 35*x^3 + 5*
x^2 - x) - 3*sqrt(2)*(x^11 - 3*x^10 - 3*x^9 + 8*x^8 + 4*x^7 - 6*x^6 - 4*x^5 + 8*x^4 + 3*x^3 - 3*x^2 - x)))/(x^
11 - 9*x^10 - 17*x^9 + 104*x^8 - 14*x^7 - 178*x^6 + 14*x^5 + 104*x^4 + 17*x^3 - 9*x^2 - x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} - 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^4-1)/(x^3-x^2-x)^(1/2)/(x^4+1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 2.90, size = 288, normalized size = 2.23

method	result	size
default	\(\frac {2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticF \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}+\frac {5^{\frac {3}{4}} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\sqrt {5}-1\right ) \sqrt {\frac {2 x -1+\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-2 x +1+\sqrt {5}}\, \sqrt {-\frac {x}{\sqrt {5}-1}}\, \left (3 \underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha +1+\sqrt {5}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-1\right )\right ) \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, -\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {5}}{6}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{3}}{6}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha }{6}-\frac {\sqrt {5}}{6}+\frac {1}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {5}}{6}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{\sqrt {x \left (x^{2}-x -1\right )}}\right )}{60}\)	\(288\)
elliptic	\(\frac {2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticF \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}+\frac {5^{\frac {3}{4}} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\sqrt {5}-1\right ) \sqrt {\frac {2 x -1+\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-2 x +1+\sqrt {5}}\, \sqrt {-\frac {x}{\sqrt {5}-1}}\, \left (3 \underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha +1+\sqrt {5}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-1\right )\right ) \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, -\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {5}}{6}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{3}}{6}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha }{6}-\frac {\sqrt {5}}{6}+\frac {1}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {5}}{6}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{\sqrt {x \left (x^{2}-x -1\right )}}\right )}{60}\)	\(288\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) \ln \left (\frac {496 \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{4} x^{2}-496 \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{4}+120 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x^{2}-248 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x -120 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right )-\RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x^{2}+224 \sqrt {x^{3}-x^{2}-x}\, \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+2 \RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x +\RootOf \left (\textit {\_Z}^{2}+36 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right )-68 \sqrt {x^{3}-x^{2}-x}}{12 x^{2} \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-12 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+x^{2}+2 x -1}\right )}{6}+\RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {4464 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{5} x^{2}-4464 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{5}+408 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3} x^{2}+2232 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3} x -408 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3}+336 \sqrt {x^{3}-x^{2}-x}\, \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-65 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x^{2}+390 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x +65 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )+158 \sqrt {x^{3}-x^{2}-x}}{12 x^{2} \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-12 \RootOf \left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+x^{2}-2 x -1}\right )\)	\(661\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(1/2*5^(1/2)-1/2)*((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2)*(-5*(x-1/2-1/2*5^(1/2))*5^(1/2))^(1/2)*(-x
/(1/2*5^(1/2)-1/2))^(1/2)/(x^3-x^2-x)^(1/2)*EllipticF(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),1/5*5^(1/2
)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2))+1/60*5^(3/4)*sum(_alpha*(5^(1/2)-1)*((2*x-1+5^(1/2))/(5^(1/2)-1))^(1/2)*(
-2*x+1+5^(1/2))^(1/2)*(-x/(5^(1/2)-1))^(1/2)/(x*(x^2-x-1))^(1/2)*(3*_alpha^3-_alpha^2+2*_alpha+1+5^(1/2)*(_alp
ha^3-_alpha^2-1))*EllipticPi(((x-1/2+1/2*5^(1/2))/(1/2*5^(1/2)-1/2))^(1/2),-1/6*_alpha^3*5^(1/2)-1/6*_alpha^3+
1/3*_alpha^2+1/6*_alpha-1/6*5^(1/2)+1/2-1/6*_alpha*5^(1/2),1/5*5^(1/2)*((1/2*5^(1/2)-1/2)*5^(1/2))^(1/2)),_alp
ha=RootOf(_Z^4+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} - 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^4-1)/(x^3-x^2-x)^(1/2)/(x^4+1),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 682, normalized size = 5.29

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(5^(1/2)/2 + 1/2)*(x/(5^(1/2)/2 + 1/2))^(1/2)*((x + 5^(1/2)/2 - 1/2)/(5^(1/2)/2 - 1/2))^(1/2)*ellipticF(asi
n((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/2))*((5^(1/2)/2 - x + 1/2)/(5^(1/2)/2 + 1/2)
)^(1/2))/(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(1/2) - ((5^(1/2)/2 + 1/2)*(x/(5^(1/2)/2 + 1/2))^
(1/2)*((x + 5^(1/2)/2 - 1/2)/(5^(1/2)/2 - 1/2))^(1/2)*((5^(1/2)/2 - x + 1/2)/(5^(1/2)/2 + 1/2))^(1/2)*elliptic
Pi(2^(1/2)*(5^(1/2)/2 + 1/2)*(- 1/2 + 1i/2), asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2
- 1/2)))/(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(1/2) - ((5^(1/2)/2 + 1/2)*(x/(5^(1/2)/2 + 1/2))^
(1/2)*((x + 5^(1/2)/2 - 1/2)/(5^(1/2)/2 - 1/2))^(1/2)*((5^(1/2)/2 - x + 1/2)/(5^(1/2)/2 + 1/2))^(1/2)*elliptic
Pi(2^(1/2)*(5^(1/2)/2 + 1/2)*(1/2 - 1i/2), asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 -
1/2)))/(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(1/2) - ((5^(1/2)/2 + 1/2)*(x/(5^(1/2)/2 + 1/2))^(1
/2)*((x + 5^(1/2)/2 - 1/2)/(5^(1/2)/2 - 1/2))^(1/2)*((5^(1/2)/2 - x + 1/2)/(5^(1/2)/2 + 1/2))^(1/2)*ellipticPi
(2^(1/2)*(5^(1/2)/2 + 1/2)*(1/2 + 1i/2), asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/
2)))/(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(1/2) - ((5^(1/2)/2 + 1/2)*(x/(5^(1/2)/2 + 1/2))^(1/2
)*((x + 5^(1/2)/2 - 1/2)/(5^(1/2)/2 - 1/2))^(1/2)*((5^(1/2)/2 - x + 1/2)/(5^(1/2)/2 + 1/2))^(1/2)*ellipticPi(2
^(1/2)*(5^(1/2)/2 + 1/2)*(- 1/2 - 1i/2), asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/
2)))/(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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