\(\int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx\) [1838]

Optimal result

Integrand size = 24, antiderivative size = 125 \[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\frac {1}{2} \arctan \left (\frac {\sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\frac {1}{4} \sqrt {1-2 i} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\frac {1}{4} \sqrt {1+2 i} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \]

[Out]

1/2*arctan((x^3-x^2-x)^(1/2)/(x^2-x-1))-1/4*(1-2*I)^(1/2)*arctan((1-2*I)^(1/2)*(x^3-x^2-x)^(1/2)/(x^2-x-1))-1/
4*(1+2*I)^(1/2)*arctan((1+2*I)^(1/2)*(x^3-x^2-x)^(1/2)/(x^2-x-1))

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 3.34 (sec) , antiderivative size = 1650, normalized size of antiderivative = 13.20, number of steps used = 55, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.792, Rules used = {2081, 6857, 932, 6865, 1730, 1201, 1112, 1198, 1228, 1470, 554, 432, 430, 552, 551, 1726, 1714, 1712, 211} \[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=-\frac {\sqrt {1-2 i} \sqrt {x^3-x^2-x} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {x^2-x-1}}\right )}{4 \sqrt {x} \sqrt {x^2-x-1}}-\frac {\sqrt {1+2 i} \sqrt {x^3-x^2-x} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {x}}{\sqrt {x^2-x-1}}\right )}{4 \sqrt {x} \sqrt {x^2-x-1}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \sqrt {x^3-x^2-x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {x} \left (-x^2+x+1\right )}+\frac {\left (1+\sqrt {5}\right ) \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \sqrt {x^3-x^2-x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{2 \sqrt {2} \left (1-\sqrt {5}\right ) \sqrt {x} \left (-x^2+x+1\right )}-\frac {\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}} \sqrt {x^3-x^2-x} \operatorname {EllipticF}\left (\arcsin \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 )}{2 \sqrt [4]{5} \left (3+\sqrt {5}\right ) \sqrt {x} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \left (-x^2+x+1\right )}+\frac {\left (\frac {1}{8}+\frac {i}{24}\right ) \left ((1+2 i)+\sqrt {5}\right ) \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}} \sqrt {x^3-x^2-x} \operatorname {EllipticF}\left (\arcsin \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 {x} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \left (-x^2+x+1\right )}+\frac {\left (\frac {1}{8}-\frac {i}{24}\right ) \left ((1-2 i)+\sqrt {5}\right ) \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}} \sqrt {x^3-x^2-x} \operatorname {EllipticF}\left (\arcsin \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 {x} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \left (-x^2+x+1\right )}+\frac {\left (1-\sqrt {5}\right ) \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}} \sqrt {x^3-x^2-x} \operatorname {EllipticF}\left (\arcsin \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 )}{4 \sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \left (-x^2+x+1\right )}-\frac {\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}} \sqrt {x^3-x^2-x} \operatorname {EllipticF}\left (\arcsin \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 )}{2 \sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {x} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \left (-x^2+x+1\right )}-\frac {\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}} \sqrt {x^3-x^2-x} \operatorname {EllipticF}\left (\arcsin \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 )}{12 \sqrt [4]{5} \sqrt {x} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \left (-x^2+x+1\right )}-\frac {\left (2+\sqrt {5}\right ) \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \sqrt {x^3-x^2-x} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-1-\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {x} \left (-x^2+x+1\right )}+\frac {\sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \sqrt {x^3-x^2-x} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {2} \left (1-\sqrt {5}\right ) \sqrt {x} \left (-x^2+x+1\right )} \]

[In]

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

[Out]

-1/4*(Sqrt[1 - 2*I]*Sqrt[-x - x^2 + x^3]*ArcTan[(Sqrt[1 - 2*I]*Sqrt[x])/Sqrt[-1 - x + x^2]])/(Sqrt[x]*Sqrt[-1
- x + x^2]) - (Sqrt[1 + 2*I]*Sqrt[-x - x^2 + x^3]*ArcTan[(Sqrt[1 + 2*I]*Sqrt[x])/Sqrt[-1 - x + x^2]])/(4*Sqrt[
x]*Sqrt[-1 - x + x^2]) + ((1 + Sqrt[5])*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*Sqrt[-x - x^2 +
x^3]*EllipticF[ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(2*Sqrt[2]*(1 - Sqrt[5])*Sqrt[x]*(1
+ x - x^2)) + ((1 + Sqrt[5])*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*Sqrt[-x - x^2 + x^3]*Ellip
ticF[ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(2*Sqrt[2]*(3 + Sqrt[5])*Sqrt[x]*(1 + x - x^2))
- (Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2 + (1 + Sqrt[5])*x)/(2 + (1 - Sqrt[5])*x)]*Sqrt[-x - x^2 + x^3]*Elliptic
F[ArcSin[(Sqrt[2]*5^(1/4)*Sqrt[x])/Sqrt[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(12*5^(1/4)*Sqrt[x]*Sqrt[(2
+ (1 - Sqrt[5])*x)^(-1)]*(1 + x - x^2)) - (Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2 + (1 + Sqrt[5])*x)/(2 + (1 - Sq
rt[5])*x)]*Sqrt[-x - x^2 + x^3]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*Sqrt[x])/Sqrt[-2 - (1 - Sqrt[5])*x]], (5 - S
qrt[5])/10])/(2*5^(1/4)*(1 - Sqrt[5])*Sqrt[x]*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*(1 + x - x^2)) + ((1 - Sqrt[5])
*Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2 + (1 + Sqrt[5])*x)/(2 + (1 - Sqrt[5])*x)]*Sqrt[-x - x^2 + x^3]*EllipticF[A
rcSin[(Sqrt[2]*5^(1/4)*Sqrt[x])/Sqrt[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(4*5^(1/4)*Sqrt[x]*Sqrt[(2 + (
1 - Sqrt[5])*x)^(-1)]*(1 + x - x^2)) + ((1/8 - I/24)*((1 - 2*I) + Sqrt[5])*Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2
+ (1 + Sqrt[5])*x)/(2 + (1 - Sqrt[5])*x)]*Sqrt[-x - x^2 + x^3]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*Sqrt[x])/Sqrt
[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(5^(1/4)*Sqrt[x]*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*(1 + x - x^2)) +
((1/8 + I/24)*((1 + 2*I) + Sqrt[5])*Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2 + (1 + Sqrt[5])*x)/(2 + (1 - Sqrt[5])*
x)]*Sqrt[-x - x^2 + x^3]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*Sqrt[x])/Sqrt[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])
/10])/(5^(1/4)*Sqrt[x]*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*(1 + x - x^2)) - (Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2 +
(1 + Sqrt[5])*x)/(2 + (1 - Sqrt[5])*x)]*Sqrt[-x - x^2 + x^3]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*Sqrt[x])/Sqrt[
-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(2*5^(1/4)*(3 + Sqrt[5])*Sqrt[x]*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*(
1 + x - x^2)) - ((2 + Sqrt[5])*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*Sqrt[-x - x^2 + x^3]*Ell
ipticPi[(-1 - Sqrt[5])/2, ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(Sqrt[2]*(3 + Sqrt[5])*Sqr
t[x]*(1 + x - x^2)) + (Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*Sqrt[-x - x^2 + x^3]*EllipticPi[
(1 + Sqrt[5])/2, ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/(Sqrt[2]*(1 - Sqrt[5])*Sqrt[x]*(1 +
x - x^2))

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 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !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 552

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

Rule 554

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

Rule 932

Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :>
Simp[2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + b*x + c*x^2]/(e*(2*m + 5))), x] - Dist[1/(e*(2*m + 5)), Int[(
(d + e*x)^m/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[b*d*f - 3*a*e*f + a*d*g + 2*(c*d*f - b*e*f + b*d*g - a
*e*g)*x - (c*e*f - 3*c*d*g + b*e*g)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[e*f - d*g, 0]
&& NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] &&  !LtQ[m, -1]

Rule 1112

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]/(2*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)
*x^2)]))*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)], x]] /; FreeQ[{a, b, c}, x] && Gt
Q[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]

Rule 1198

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

Rule 1201

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

Rule 1228

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

Rule 1470

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^
(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPar
t[p]), Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p,
q, r}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p]

Rule 1712

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

Rule 1714

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

Rule 1726

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[P4x,
x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Dist[-C/e^2, Int[(d - e*x^2)/Sqrt[a + b*x^2 + c*x^4], x],
x] + Dist[1/e^2, Int[(C*d^2 + A*e^2 + B*e^2*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a,
b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a
*e^2, 0]

Rule 1730

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[P4x,
x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Dist[-(e^2)^(-1), Int[(C*d - B*e - C*e*x^2)/Sqrt[a + b*x^
2 + c*x^4], x], x] + Dist[(C*d^2 - B*d*e + A*e^2)/e^2, Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /;
FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Ne
Q[c*d^2 - a*e^2, 0]

Rule 2081

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 6857

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]

Rule 6865

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-x-x^2+x^3} \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{-1+x^4} \, dx}{\sqrt {x} \sqrt {-1-x+x^2}} \\ & = \frac {\sqrt {-x-x^2+x^3} \int \left (-\frac {\sqrt {x} \sqrt {-1-x+x^2}}{2 \left (1-x^2\right )}-\frac {\sqrt {x} \sqrt {-1-x+x^2}}{2 \left (1+x^2\right )}\right ) \, dx}{\sqrt {x} \sqrt {-1-x+x^2}} \\ & = -\frac {\sqrt {-x-x^2+x^3} \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{1-x^2} \, dx}{2 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{1+x^2} \, dx}{2 \sqrt {x} \sqrt {-1-x+x^2}} \\ & = -\frac {\sqrt {-x-x^2+x^3} \int \left (\frac {i \sqrt {x} \sqrt {-1-x+x^2}}{2 (i-x)}+\frac {i \sqrt {x} \sqrt {-1-x+x^2}}{2 (i+x)}\right ) \, dx}{2 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \int \left (\frac {\sqrt {x} \sqrt {-1-x+x^2}}{2 (1-x)}+\frac {\sqrt {x} \sqrt {-1-x+x^2}}{2 (1+x)}\right ) \, dx}{2 \sqrt {x} \sqrt {-1-x+x^2}} \\ & = -\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{i-x} \, dx}{4 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{i+x} \, dx}{4 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{1-x} \, dx}{4 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \int \frac {\sqrt {x} \sqrt {-1-x+x^2}}{1+x} \, dx}{4 \sqrt {x} \sqrt {-1-x+x^2}} \\ & = -\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \int \frac {-i-(2+2 i) x-(1-3 i) x^2}{(i-x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{12 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \int \frac {-i+(2-2 i) x+(1+3 i) x^2}{\sqrt {x} (i+x) \sqrt {-1-x+x^2}} \, dx}{12 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \int \frac {-1-4 x+2 x^2}{(1-x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{12 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \int \frac {-1+4 x^2}{\sqrt {x} (1+x) \sqrt {-1-x+x^2}} \, dx}{12 \sqrt {x} \sqrt {-1-x+x^2}} \\ & = -\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {-i-(2+2 i) x^2-(1-3 i) x^4}{\left (i-x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {-i+(2-2 i) x^2+(1+3 i) x^4}{\left (i+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \text {Subst}\left (\int \frac {-1-4 x^2+2 x^4}{\left (1-x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \text {Subst}\left (\int \frac {-1+4 x^4}{\left (1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}} \\ & = -\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {(1-4 i)-(2+2 i) x^2}{\left (i-x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\left (i \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {(-1-4 i)+(2-2 i) x^2}{\left (i+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \text {Subst}\left (\int \frac {4-4 x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \text {Subst}\left (\int \frac {-2+2 x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\left (\left (\frac {1}{2}-\frac {i}{6}\right ) \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {i-x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (\left (\frac {1}{2}+\frac {i}{6}\right ) \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {i+x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}} \\ & = -\frac {\left (\left (\frac {1}{4}-\frac {i}{12}\right ) \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {-1-\sqrt {5}+2 x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}+-\frac {\left (\left (\frac {1}{4}+\frac {i}{2}\right ) \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {i-x^2}{\left (i+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}--\frac {\left (\left (\frac {1}{12}+\frac {i}{6}\right ) \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}+\frac {\left (\left (\frac {1}{12}-\frac {i}{6}\right ) \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \text {Subst}\left (\int \frac {-1-\sqrt {5}+2 x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (\left (\frac {1}{4}+\frac {i}{12}\right ) \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {-1-\sqrt {5}+2 x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (\left (\frac {1}{4}-\frac {i}{2}\right ) \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {i+x^2}{\left (i-x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \text {Subst}\left (\int \frac {-1-\sqrt {5}+2 x^2}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\left (\left (\frac {1}{4}-\frac {i}{12}\right ) \left ((-1+2 i)-\sqrt {5}\right ) \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \text {Subst}\left (\int \frac {-1-\sqrt {5}+2 x^2}{\left (1-x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-\sqrt {5}\right ) \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1-\sqrt {5}\right ) \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {-1-x+x^2}}-\frac {\left (\left (\frac {1}{4}+\frac {i}{12}\right ) \left ((1+2 i)+\sqrt {5}\right ) \sqrt {-x-x^2+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1-x+x^2}}-\frac {\sqrt {-x-x^2+x^3} \text {Subst}\left (\int \frac {-1-\sqrt {5}+2 x^2}{\left (1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (3+\sqrt {5}\right ) \sqrt {x} \sqrt {-1-x+x^2}}+\frac {\sqrt {-x-x^2+x^3} \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\left (3+\sqrt {5}\right ) \sqrt {x} \sqrt {-1-x+x^2}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\frac {\sqrt {x} \sqrt {-1-x+x^2} \left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt {-1-x+x^2}}\right )-\sqrt {1-2 i} \arctan \left (\frac {\sqrt {1-2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )-\sqrt {1+2 i} \arctan \left (\frac {\sqrt {1+2 i} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )\right )}{4 \sqrt {x \left (-1-x+x^2\right )}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(217\) vs. \(2(105)=210\).

Time = 3.51 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.74

method	result	size
default	\(\frac {-\ln \left (\frac {-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\ln \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\left (-\sqrt {5}-1\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )+\left (\sqrt {5}+1\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )-2 \arctan \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}}{x}\right ) \sqrt {2+2 \sqrt {5}}}{4 \sqrt {2+2 \sqrt {5}}}\)	\(218\)
pseudoelliptic	\(\frac {-\ln \left (\frac {-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\ln \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {-2+2 \sqrt {5}}+x \sqrt {5}+x^{2}-x -1}{x}\right )+\left (-\sqrt {5}-1\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )+\left (\sqrt {5}+1\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2+2 \sqrt {5}}}\right )-2 \arctan \left (\frac {\sqrt {x \left (x^{2}-x -1\right )}}{x}\right ) \sqrt {2+2 \sqrt {5}}}{4 \sqrt {2+2 \sqrt {5}}}\)	\(218\)
trager	\(\text {Expression too large to display}\)	\(913\)
elliptic	\(\text {Expression too large to display}\)	\(1430\)

[In]

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

[Out]

1/4*(-ln((-(x*(x^2-x-1))^(1/2)*(-2+2*5^(1/2))^(1/2)+x*5^(1/2)+x^2-x-1)/x)+ln(((x*(x^2-x-1))^(1/2)*(-2+2*5^(1/2
))^(1/2)+x*5^(1/2)+x^2-x-1)/x)+(-5^(1/2)-1)*arctan(((-2+2*5^(1/2))^(1/2)*x-2*(x*(x^2-x-1))^(1/2))/x/(2+2*5^(1/
2))^(1/2))+(5^(1/2)+1)*arctan(((-2+2*5^(1/2))^(1/2)*x+2*(x*(x^2-x-1))^(1/2))/x/(2+2*5^(1/2))^(1/2))-2*arctan((
x*(x^2-x-1))^(1/2)/x)*(2+2*5^(1/2))^(1/2))/(2+2*5^(1/2))^(1/2)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (97) = 194\).

Time = 0.30 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.27 \[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\frac {1}{16} \, \sqrt {2 i - 1} \log \left (\frac {x^{4} + \left (4 i - 2\right ) \, x^{3} + 2 \, \sqrt {2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (x^{2} + 2 i \, x - 1\right )} - \left (4 i + 6\right ) \, x^{2} - \left (4 i - 2\right ) \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{16} \, \sqrt {2 i - 1} \log \left (\frac {x^{4} + \left (4 i - 2\right ) \, x^{3} - 2 \, \sqrt {2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (x^{2} + 2 i \, x - 1\right )} - \left (4 i + 6\right ) \, x^{2} - \left (4 i - 2\right ) \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{16} \, \sqrt {-2 i - 1} \log \left (\frac {x^{4} - \left (4 i + 2\right ) \, x^{3} + 2 \, \sqrt {-2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (x^{2} - 2 i \, x - 1\right )} + \left (4 i - 6\right ) \, x^{2} + \left (4 i + 2\right ) \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{16} \, \sqrt {-2 i - 1} \log \left (\frac {x^{4} - \left (4 i + 2\right ) \, x^{3} - 2 \, \sqrt {-2 i - 1} \sqrt {x^{3} - x^{2} - x} {\left (x^{2} - 2 i \, x - 1\right )} + \left (4 i - 6\right ) \, x^{2} + \left (4 i + 2\right ) \, x + 1}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{4} \, \arctan \left (\frac {x^{2} - 2 \, x - 1}{2 \, \sqrt {x^{3} - x^{2} - x}}\right ) \]

[In]

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

[Out]

1/16*sqrt(2*I - 1)*log((x^4 + (4*I - 2)*x^3 + 2*sqrt(2*I - 1)*sqrt(x^3 - x^2 - x)*(x^2 + 2*I*x - 1) - (4*I + 6
)*x^2 - (4*I - 2)*x + 1)/(x^4 + 2*x^2 + 1)) - 1/16*sqrt(2*I - 1)*log((x^4 + (4*I - 2)*x^3 - 2*sqrt(2*I - 1)*sq
rt(x^3 - x^2 - x)*(x^2 + 2*I*x - 1) - (4*I + 6)*x^2 - (4*I - 2)*x + 1)/(x^4 + 2*x^2 + 1)) + 1/16*sqrt(-2*I - 1
)*log((x^4 - (4*I + 2)*x^3 + 2*sqrt(-2*I - 1)*sqrt(x^3 - x^2 - x)*(x^2 - 2*I*x - 1) + (4*I - 6)*x^2 + (4*I + 2
)*x + 1)/(x^4 + 2*x^2 + 1)) - 1/16*sqrt(-2*I - 1)*log((x^4 - (4*I + 2)*x^3 - 2*sqrt(-2*I - 1)*sqrt(x^3 - x^2 -
x)*(x^2 - 2*I*x - 1) + (4*I - 6)*x^2 + (4*I + 2)*x + 1)/(x^4 + 2*x^2 + 1)) - 1/4*arctan(1/2*(x^2 - 2*x - 1)/s
qrt(x^3 - x^2 - x))

Sympy [F]

\[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\int \frac {\sqrt {x \left (x^{2} - x - 1\right )}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} - x}}{x^{4} - 1} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} - x}}{x^{4} - 1} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 537, normalized size of antiderivative = 4.30 \[ \int \frac {\sqrt {-x-x^2+x^3}}{-1+x^4} \, dx=\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (-\frac {\sqrt {5}}{2}-\frac {1}{2};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )}{2\,\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}+\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (\frac {\sqrt {5}}{2}+\frac {1}{2};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )}{2\,\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}+\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (-\frac {\sqrt {5}\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\,\left (-\frac {1}{2}+1{}\mathrm {i}\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}}+\frac {\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\Pi \left (\frac {\sqrt {5}\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\,\left (-\frac {1}{2}-\mathrm {i}\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}} \]

[In]

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

[Out]

((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(- 5^(1/2)/2 - 1/2, asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2
+ 1/2)/(5^(1/2)/2 - 1/2)))/(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(5^(1/2)/2 + 1/2, asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1
/2)))/(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)*ellipti
cPi(- (5^(1/2)*1i)/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))*(1/2 - 1
i))/(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((5
^(1/2)*1i)/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))*(1/2 + 1i))/(x^3
- x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(1/2)