Integrand size = 24, antiderivative size = 85 \[ \int \frac {-1+x^4}{\sqrt {-x+x^3} \left (1+x^4\right )} \, dx=\frac {\arctan \left (\frac {2^{3/4} \sqrt {-x+x^3}}{1+\sqrt {2} x-x^2}\right )}{2^{3/4}}-\frac {\text {arctanh}\left (\frac {-\frac {1}{2^{3/4}}+\frac {x}{\sqrt [4]{2}}+\frac {x^2}{2^{3/4}}}{\sqrt {-x+x^3}}\right )}{2^{3/4}} \]
1/2*arctan(2^(3/4)*(x^3-x)^(1/2)/(1+x*2^(1/2)-x^2))*2^(1/4)-1/2*arctanh((- 1/2*2^(1/4)+1/2*x*2^(3/4)+1/2*x^2*2^(1/4))/(x^3-x)^(1/2))*2^(1/4)
Time = 0.43 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.25 \[ \int \frac {-1+x^4}{\sqrt {-x+x^3} \left (1+x^4\right )} \, dx=\frac {\sqrt {x} \sqrt {-1+x^2} \left (\arctan \left (\frac {2^{3/4} \sqrt {x} \sqrt {-1+x^2}}{1+\sqrt {2} x-x^2}\right )-\text {arctanh}\left (\frac {2^{3/4} \sqrt {x} \sqrt {-1+x^2}}{-1+\sqrt {2} x+x^2}\right )\right )}{2^{3/4} \sqrt {x \left (-1+x^2\right )}} \]
(Sqrt[x]*Sqrt[-1 + x^2]*(ArcTan[(2^(3/4)*Sqrt[x]*Sqrt[-1 + x^2])/(1 + Sqrt [2]*x - x^2)] - ArcTanh[(2^(3/4)*Sqrt[x]*Sqrt[-1 + x^2])/(-1 + Sqrt[2]*x + x^2)]))/(2^(3/4)*Sqrt[x*(-1 + x^2)])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.27 (sec) , antiderivative size = 697, normalized size of antiderivative = 8.20, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2467, 25, 1388, 2035, 25, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^4-1}{\sqrt {x^3-x} \left (x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^2-1} \int -\frac {1-x^4}{\sqrt {x} \sqrt {x^2-1} \left (x^4+1\right )}dx}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^2-1} \int \frac {1-x^4}{\sqrt {x} \sqrt {x^2-1} \left (x^4+1\right )}dx}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^2-1} \int \frac {\left (-x^2-1\right ) \sqrt {x^2-1}}{\sqrt {x} \left (x^4+1\right )}dx}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-1} \int -\frac {\sqrt {x^2-1} \left (x^2+1\right )}{x^4+1}d\sqrt {x}}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \int \frac {\sqrt {x^2-1} \left (x^2+1\right )}{x^4+1}d\sqrt {x}}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {x^2-1} \int \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x^2-1}}{x^2+i}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {x^2-1}}{i-x^2}\right )d\sqrt {x}}{\sqrt {x^3-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-1} \left (-\frac {(1+i) \sqrt {1-x} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-1\right )}{\left (2 \sqrt {2}+(2+2 i)\right ) \sqrt {x^2-1}}-\frac {(1-i) \sqrt {1-x} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-1\right )}{\left (2 \sqrt {2}+(2-2 i)\right ) \sqrt {x^2-1}}-\frac {(1+i) \sqrt {1-x} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-1\right )}{\left (-2 \sqrt {2}+(2+2 i)\right ) \sqrt {x^2-1}}-\frac {(1-i) \sqrt {1-x} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-1\right )}{\left (-2 \sqrt {2}+(2-2 i)\right ) \sqrt {x^2-1}}+\frac {\sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{2 \left (\sqrt {2}+(1+i)\right ) \sqrt {x^2-1}}+\frac {\sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{2 \left (\sqrt {2}+(1-i)\right ) \sqrt {x^2-1}}+\frac {\sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{2 \left (\sqrt {2}+(-1+i)\right ) \sqrt {x^2-1}}+\frac {\sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{2 \left (\sqrt {2}+(-1-i)\right ) \sqrt {x^2-1}}-\frac {\sqrt {x-1} \sqrt {x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2-1}}+\frac {\sqrt {1-x} \sqrt {x+1} \operatorname {EllipticPi}\left (-\sqrt [4]{-1},\arcsin \left (\sqrt {x}\right ),-1\right )}{2 \sqrt {x^2-1}}+\frac {\sqrt {1-x} \sqrt {x+1} \operatorname {EllipticPi}\left (\sqrt [4]{-1},\arcsin \left (\sqrt {x}\right ),-1\right )}{2 \sqrt {x^2-1}}+\frac {\sqrt {1-x} \sqrt {x+1} \operatorname {EllipticPi}\left (-(-1)^{3/4},\arcsin \left (\sqrt {x}\right ),-1\right )}{2 \sqrt {x^2-1}}+\frac {\sqrt {1-x} \sqrt {x+1} \operatorname {EllipticPi}\left ((-1)^{3/4},\arcsin \left (\sqrt {x}\right ),-1\right )}{2 \sqrt {x^2-1}}\right )}{\sqrt {x^3-x}}\) |
(-2*Sqrt[x]*Sqrt[-1 + x^2]*(((-1 + I)*Sqrt[1 - x]*Sqrt[1 + x]*EllipticF[Ar cSin[Sqrt[x]], -1])/(((2 - 2*I) - 2*Sqrt[2])*Sqrt[-1 + x^2]) - ((1 + I)*Sq rt[1 - x]*Sqrt[1 + x]*EllipticF[ArcSin[Sqrt[x]], -1])/(((2 + 2*I) - 2*Sqrt [2])*Sqrt[-1 + x^2]) - ((1 - I)*Sqrt[1 - x]*Sqrt[1 + x]*EllipticF[ArcSin[S qrt[x]], -1])/(((2 - 2*I) + 2*Sqrt[2])*Sqrt[-1 + x^2]) - ((1 + I)*Sqrt[1 - x]*Sqrt[1 + x]*EllipticF[ArcSin[Sqrt[x]], -1])/(((2 + 2*I) + 2*Sqrt[2])*S qrt[-1 + x^2]) - (Sqrt[-1 + x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[ x])/Sqrt[-1 + x]], 1/2])/(Sqrt[2]*Sqrt[-1 + x^2]) + (Sqrt[-1 + x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2])/(2*((-1 - I) + Sqrt[2])*Sqrt[-1 + x^2]) + (Sqrt[-1 + x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sq rt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2])/(2*((-1 + I) + Sqrt[2])*Sqrt[-1 + x^2] ) + (Sqrt[-1 + x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2])/(2*((1 - I) + Sqrt[2])*Sqrt[-1 + x^2]) + (Sqrt[-1 + x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2])/(2*((1 + I) + Sqrt[2])*Sqrt[-1 + x^2]) + (Sqrt[1 - x]*Sqrt[1 + x]*EllipticPi[-(-1)^(1/4 ), ArcSin[Sqrt[x]], -1])/(2*Sqrt[-1 + x^2]) + (Sqrt[1 - x]*Sqrt[1 + x]*Ell ipticPi[(-1)^(1/4), ArcSin[Sqrt[x]], -1])/(2*Sqrt[-1 + x^2]) + (Sqrt[1 - x ]*Sqrt[1 + x]*EllipticPi[-(-1)^(3/4), ArcSin[Sqrt[x]], -1])/(2*Sqrt[-1 + x ^2]) + (Sqrt[1 - x]*Sqrt[1 + x]*EllipticPi[(-1)^(3/4), ArcSin[Sqrt[x]], -1 ])/(2*Sqrt[-1 + x^2])))/Sqrt[-x + x^3]
3.12.38.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 8.57 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.22
| method | result | size |
| default | \(\frac {2^{\frac {1}{4}} \left (\ln \left (\frac {-2^{\frac {3}{4}} \sqrt {x^{3}-x}+x \sqrt {2}+x^{2}-1}{2^{\frac {3}{4}} \sqrt {x^{3}-x}+x \sqrt {2}+x^{2}-1}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {x^{3}-x}+x}{x}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {x^{3}-x}-x}{x}\right )\right )}{4}\) | \(104\) |
| pseudoelliptic | \(\frac {2^{\frac {1}{4}} \left (\ln \left (\frac {-2^{\frac {3}{4}} \sqrt {x^{3}-x}+x \sqrt {2}+x^{2}-1}{2^{\frac {3}{4}} \sqrt {x^{3}-x}+x \sqrt {2}+x^{2}-1}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {x^{3}-x}+x}{x}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {x^{3}-x}-x}{x}\right )\right )}{4}\) | \(104\) |
| elliptic | \(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha -1\right ) \sqrt {1+x}\, \sqrt {1-x}\, \sqrt {-x}\, \operatorname {EllipticPi}\left (\sqrt {1+x}, -\frac {1}{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+\frac {1}{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-\frac {1}{2} \underline {\hspace {1.25 ex}}\alpha +\frac {1}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x \left (x^{2}-1\right )}}\right )}{4}\) | \(119\) |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{4} x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right )-12 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{4} x \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right )-8 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right )-50 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) x +50 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right )+78 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}+200 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {x^{3}-x}+104 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) x -78 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right )-56 \sqrt {x^{3}-x}}{2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}+6 x^{2}+8 x -6}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{5} x^{2}-12 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{5} x -8 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{5}+50 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3} x^{2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3} x -50 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3}-200 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {x^{3}-x}+78 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) x^{2}+104 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) x -78 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )-56 \sqrt {x^{3}-x}}{2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}-6 x^{2}-8 x +6}\right )}{4}\) | \(476\) |
1/4*2^(1/4)*(ln((-2^(3/4)*(x^3-x)^(1/2)+x*2^(1/2)+x^2-1)/(2^(3/4)*(x^3-x)^ (1/2)+x*2^(1/2)+x^2-1))+2*arctan((2^(1/4)*(x^3-x)^(1/2)+x)/x)+2*arctan((2^ (1/4)*(x^3-x)^(1/2)-x)/x))
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 265, normalized size of antiderivative = 3.12 \[ \int \frac {-1+x^4}{\sqrt {-x+x^3} \left (1+x^4\right )} \, dx=-\left (\frac {1}{8} i + \frac {1}{8}\right ) \cdot 2^{\frac {1}{4}} \log \left (\frac {x^{4} - 4 \, x^{2} - 2 \, \sqrt {2} {\left (i \, x^{3} - i \, x\right )} + \sqrt {x^{3} - x} {\left (2^{\frac {3}{4}} {\left (-\left (i - 1\right ) \, x^{2} + i - 1\right )} - \left (2 i + 2\right ) \cdot 2^{\frac {1}{4}} x\right )} + 1}{x^{4} + 1}\right ) + \left (\frac {1}{8} i + \frac {1}{8}\right ) \cdot 2^{\frac {1}{4}} \log \left (\frac {x^{4} - 4 \, x^{2} - 2 \, \sqrt {2} {\left (i \, x^{3} - i \, x\right )} + \sqrt {x^{3} - x} {\left (2^{\frac {3}{4}} {\left (\left (i - 1\right ) \, x^{2} - i + 1\right )} + \left (2 i + 2\right ) \cdot 2^{\frac {1}{4}} x\right )} + 1}{x^{4} + 1}\right ) + \left (\frac {1}{8} i - \frac {1}{8}\right ) \cdot 2^{\frac {1}{4}} \log \left (\frac {x^{4} - 4 \, x^{2} - 2 \, \sqrt {2} {\left (-i \, x^{3} + i \, x\right )} + \sqrt {x^{3} - x} {\left (2^{\frac {3}{4}} {\left (\left (i + 1\right ) \, x^{2} - i - 1\right )} + \left (2 i - 2\right ) \cdot 2^{\frac {1}{4}} x\right )} + 1}{x^{4} + 1}\right ) - \left (\frac {1}{8} i - \frac {1}{8}\right ) \cdot 2^{\frac {1}{4}} \log \left (\frac {x^{4} - 4 \, x^{2} - 2 \, \sqrt {2} {\left (-i \, x^{3} + i \, x\right )} + \sqrt {x^{3} - x} {\left (2^{\frac {3}{4}} {\left (-\left (i + 1\right ) \, x^{2} + i + 1\right )} - \left (2 i - 2\right ) \cdot 2^{\frac {1}{4}} x\right )} + 1}{x^{4} + 1}\right ) \]
-(1/8*I + 1/8)*2^(1/4)*log((x^4 - 4*x^2 - 2*sqrt(2)*(I*x^3 - I*x) + sqrt(x ^3 - x)*(2^(3/4)*(-(I - 1)*x^2 + I - 1) - (2*I + 2)*2^(1/4)*x) + 1)/(x^4 + 1)) + (1/8*I + 1/8)*2^(1/4)*log((x^4 - 4*x^2 - 2*sqrt(2)*(I*x^3 - I*x) + sqrt(x^3 - x)*(2^(3/4)*((I - 1)*x^2 - I + 1) + (2*I + 2)*2^(1/4)*x) + 1)/( x^4 + 1)) + (1/8*I - 1/8)*2^(1/4)*log((x^4 - 4*x^2 - 2*sqrt(2)*(-I*x^3 + I *x) + sqrt(x^3 - x)*(2^(3/4)*((I + 1)*x^2 - I - 1) + (2*I - 2)*2^(1/4)*x) + 1)/(x^4 + 1)) - (1/8*I - 1/8)*2^(1/4)*log((x^4 - 4*x^2 - 2*sqrt(2)*(-I*x ^3 + I*x) + sqrt(x^3 - x)*(2^(3/4)*(-(I + 1)*x^2 + I + 1) - (2*I - 2)*2^(1 /4)*x) + 1)/(x^4 + 1))
\[ \int \frac {-1+x^4}{\sqrt {-x+x^3} \left (1+x^4\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x^{4} + 1\right )}\, dx \]
\[ \int \frac {-1+x^4}{\sqrt {-x+x^3} \left (1+x^4\right )} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{4} + 1\right )} \sqrt {x^{3} - x}} \,d x } \]
\[ \int \frac {-1+x^4}{\sqrt {-x+x^3} \left (1+x^4\right )} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{4} + 1\right )} \sqrt {x^{3} - x}} \,d x } \]
Time = 0.03 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.41 \[ \int \frac {-1+x^4}{\sqrt {-x+x^3} \left (1+x^4\right )} \, dx=\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\sqrt {2}\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right );\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\sqrt {2}\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right );\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\sqrt {2}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right );\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\sqrt {2}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right );\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}-\frac {2\,\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}} \]
((-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi(2^(1/2)*(- 1/2 - 1i/2), asin((-x)^(1/2)), -1))/(x^3 - x)^(1/2) + ((-x)^(1/2)*(1 - x)^(1/2)*(x + 1 )^(1/2)*ellipticPi(2^(1/2)*(- 1/2 + 1i/2), asin((-x)^(1/2)), -1))/(x^3 - x )^(1/2) + ((-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi(2^(1/2)*(1/2 - 1i/2), asin((-x)^(1/2)), -1))/(x^3 - x)^(1/2) + ((-x)^(1/2)*(1 - x)^(1/2 )*(x + 1)^(1/2)*ellipticPi(2^(1/2)*(1/2 + 1i/2), asin((-x)^(1/2)), -1))/(x ^3 - x)^(1/2) - (2*(-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticF(asin(( -x)^(1/2)), -1))/(x^3 - x)^(1/2)