Integrand size = 29, antiderivative size = 129 \[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=-\sqrt {\frac {1}{3} \left (1+i \sqrt {2}\right )} \arctan \left (\frac {\sqrt {1-i \sqrt {2}} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\sqrt {\frac {1}{3} \left (1-i \sqrt {2}\right )} \arctan \left (\frac {\sqrt {1+i \sqrt {2}} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \] Output:
-1/3*(3+3*I*2^(1/2))^(1/2)*arctan((1-I*2^(1/2))^(1/2)*(x^3-x^2-x)^(1/2)/(x ^2-x-1))-1/3*(3-3*I*2^(1/2))^(1/2)*arctan((1+I*2^(1/2))^(1/2)*(x^3-x^2-x)^ (1/2)/(x^2-x-1))
Time = 0.69 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.08 \[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=-\frac {\sqrt {x} \sqrt {-1-x+x^2} \left (\sqrt {1+i \sqrt {2}} \arctan \left (\frac {\sqrt {1-i \sqrt {2}} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )+\sqrt {1-i \sqrt {2}} \arctan \left (\frac {\sqrt {1+i \sqrt {2}} \sqrt {x}}{\sqrt {-1-x+x^2}}\right )\right )}{\sqrt {3} \sqrt {x \left (-1-x+x^2\right )}} \] Input:
Integrate[(-1 + x^4)/(Sqrt[-x - x^2 + x^3]*(1 + x^4)),x]
Output:
-((Sqrt[x]*Sqrt[-1 - x + x^2]*(Sqrt[1 + I*Sqrt[2]]*ArcTan[(Sqrt[1 - I*Sqrt [2]]*Sqrt[x])/Sqrt[-1 - x + x^2]] + Sqrt[1 - I*Sqrt[2]]*ArcTan[(Sqrt[1 + I *Sqrt[2]]*Sqrt[x])/Sqrt[-1 - x + x^2]]))/(Sqrt[3]*Sqrt[x*(-1 - x + x^2)]))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.63 (sec) , antiderivative size = 1696, normalized size of antiderivative = 13.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2467, 25, 2035, 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.
| \(\displaystyle \int \frac {x^4-1}{\sqrt {x^3-x^2-x} \left (x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^2-x-1} \int -\frac {1-x^4}{\sqrt {x} \sqrt {x^2-x-1} \left (x^4+1\right )}dx}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^2-x-1} \int \frac {1-x^4}{\sqrt {x} \sqrt {x^2-x-1} \left (x^4+1\right )}dx}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \int \frac {1-x^4}{\sqrt {x^2-x-1} \left (x^4+1\right )}d\sqrt {x}}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \int \left (\frac {2}{\sqrt {x^2-x-1} \left (x^4+1\right )}-\frac {1}{\sqrt {x^2-x-1}}\right )d\sqrt {x}}{\sqrt {x^3-x^2-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2-x-1} \left (\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (1+\sqrt {5}\right ) \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\left (1+2 (-1)^{3/4}+\sqrt {5}\right ) \sqrt {x^2-x-1}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (1+\sqrt {5}\right ) \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\left (1-2 (-1)^{3/4}+\sqrt {5}\right ) \sqrt {x^2-x-1}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (1+\sqrt {5}\right ) \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\left (1+2 \sqrt [4]{-1}+\sqrt {5}\right ) \sqrt {x^2-x-1}}+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (1+\sqrt {5}\right ) \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\left (1-2 \sqrt [4]{-1}+\sqrt {5}\right ) \sqrt {x^2-x-1}}+\frac {(-1)^{3/4} \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}} \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+2 (-1)^{3/4}+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}-\frac {(-1)^{3/4} \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}} \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 (1-2 (-1)^{3/4}+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}+\frac {\sqrt [4]{-\frac {1}{5}} \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}} \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 \left (1+2 \sqrt [4]{-1}+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}-\frac {\sqrt [4]{-\frac {1}{5}} \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}} \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 \left (1-2 \sqrt [4]{-1}+\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^2-x-1}}-\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}} \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} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \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}}} \operatorname {EllipticPi}\left (-\frac {1}{2} \sqrt [4]{-1} \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 )}{4 \sqrt {2} \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}}} \operatorname {EllipticPi}\left (\frac {1}{2} \sqrt [4]{-1} \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 )}{4 \sqrt {2} \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}}} \operatorname {EllipticPi}\left (-\frac {1}{2} (-1)^{3/4} \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 )}{4 \sqrt {2} \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}}} \operatorname {EllipticPi}\left (\frac {1}{2} (-1)^{3/4} \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 )}{4 \sqrt {2} \sqrt {x^2-x-1}}\right )}{\sqrt {x^3-x^2-x}}\) |
Input:
Int[(-1 + x^4)/(Sqrt[-x - x^2 + x^3]*(1 + x^4)),x]
Output:
(-2*Sqrt[x]*Sqrt[-1 - x + x^2]*(((1/4 + I/4)*(1 + Sqrt[5])*Sqrt[-1 + Sqrt[ 5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*EllipticF[ArcSin[Sqrt[2/(1 + Sqrt[ 5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/((1 - 2*(-1)^(1/4) + Sqrt[5])*Sqrt[-1 - x + x^2]) - ((1/4 + I/4)*(1 + Sqrt[5])*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - ( 2*x)/(1 + Sqrt[5])]*EllipticF[ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/((1 + 2*(-1)^(1/4) + Sqrt[5])*Sqrt[-1 - x + x^2]) - ((1/4 - I/4)*(1 + Sqrt[5])*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]* EllipticF[ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/((1 - 2*(-1)^(3/4) + Sqrt[5])*Sqrt[-1 - x + x^2]) + ((1/4 - I/4)*(1 + Sqrt[5])*S qrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*EllipticF[ArcSin[Sqr t[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/((1 + 2*(-1)^(3/4) + Sqrt[ 5])*Sqrt[-1 - x + x^2]) - (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])/S qrt[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(2*5^(1/4)*Sqrt[(2 + (1 - S qrt[5])*x)^(-1)]*Sqrt[-1 - x + x^2]) - ((-1/5)^(1/4)*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])/( 2*(1 - 2*(-1)^(1/4) + Sqrt[5])*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*Sqrt[-1 - x + x^2]) + ((-1/5)^(1/4)*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])...
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]
Leaf count of result is larger than twice the leaf count of optimal. \(208\) vs. \(2(97)=194\).
Time = 3.98 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.62
| method | result | size |
| default | \(\frac {\sqrt {3}\, \left (\ln \left (\frac {x \sqrt {3}-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {2 \sqrt {3}-2}+x^{2}-x -1}{x}\right )-\ln \left (\frac {x \sqrt {3}+\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {2 \sqrt {3}-2}+x^{2}-x -1}{x}\right )\right ) \sqrt {2 \sqrt {3}+2}\, \sqrt {2 \sqrt {3}-2}+4 \left (3+\sqrt {3}\right ) \left (\arctan \left (\frac {\sqrt {2 \sqrt {3}-2}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2 \sqrt {3}+2}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {3}-2}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2 \sqrt {3}+2}}\right )\right )}{12 \sqrt {2 \sqrt {3}+2}}\) | \(209\) |
| pseudoelliptic | \(\frac {\sqrt {3}\, \left (\ln \left (\frac {x \sqrt {3}-\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {2 \sqrt {3}-2}+x^{2}-x -1}{x}\right )-\ln \left (\frac {x \sqrt {3}+\sqrt {x \left (x^{2}-x -1\right )}\, \sqrt {2 \sqrt {3}-2}+x^{2}-x -1}{x}\right )\right ) \sqrt {2 \sqrt {3}+2}\, \sqrt {2 \sqrt {3}-2}+4 \left (3+\sqrt {3}\right ) \left (\arctan \left (\frac {\sqrt {2 \sqrt {3}-2}\, x +2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2 \sqrt {3}+2}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {3}-2}\, x -2 \sqrt {x \left (x^{2}-x -1\right )}}{x \sqrt {2 \sqrt {3}+2}}\right )\right )}{12 \sqrt {2 \sqrt {3}+2}}\) | \(209\) |
| elliptic | \(\frac {2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}+\frac {5^{\frac {3}{4}} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\sqrt {5}-1\right ) \sqrt {\frac {-1+2 x +\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {1-2 x +\sqrt {5}}\, \sqrt {-\frac {x}{\sqrt {5}-1}}\, \left (3 \underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha +1+\sqrt {5}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-1\right )\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, -\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {5}}{6}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{3}}{6}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha }{6}-\frac {\sqrt {5}}{6}+\frac {1}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {5}}{6}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{\sqrt {x \left (x^{2}-x -1\right )}}\right )}{60}\) | \(288\) |
| trager | \(-\operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-4464 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{5} x^{2}+4464 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{5}-408 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3} x^{2}-2232 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3} x +336 \sqrt {x^{3}-x^{2}-x}\, \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+408 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3}+65 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x^{2}-390 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x +158 \sqrt {x^{3}-x^{2}-x}-65 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )}{12 x^{2} \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-12 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+x^{2}-2 x -1}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) \ln \left (-\frac {496 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{4} x^{2}-496 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{4}+120 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x^{2}-248 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x -120 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right )-\operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x^{2}-224 \sqrt {x^{3}-x^{2}-x}\, \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {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} \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-12 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+x^{2}+2 x -1}\right )}{6}\) | \(663\) |
Input:
int((x^4-1)/(x^3-x^2-x)^(1/2)/(x^4+1),x,method=_RETURNVERBOSE)
Output:
1/12*(3^(1/2)*(ln((x*3^(1/2)-(x*(x^2-x-1))^(1/2)*(2*3^(1/2)-2)^(1/2)+x^2-x -1)/x)-ln((x*3^(1/2)+(x*(x^2-x-1))^(1/2)*(2*3^(1/2)-2)^(1/2)+x^2-x-1)/x))* (2*3^(1/2)+2)^(1/2)*(2*3^(1/2)-2)^(1/2)+4*(3+3^(1/2))*(arctan(((2*3^(1/2)- 2)^(1/2)*x+2*(x*(x^2-x-1))^(1/2))/x/(2*3^(1/2)+2)^(1/2))-arctan(((2*3^(1/2 )-2)^(1/2)*x-2*(x*(x^2-x-1))^(1/2))/x/(2*3^(1/2)+2)^(1/2))))/(2*3^(1/2)+2) ^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (93) = 186\).
Time = 0.11 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.94 \[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=-\sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{6}} \arctan \left (-\frac {{\left (3 \, x^{2} - \sqrt {3} {\left (x^{2} + 2 \, x - 1\right )} - 3\right )} \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{6}}}{2 \, \sqrt {x^{3} - x^{2} - x}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{6} \, \sqrt {3} - \frac {1}{6}} \log \left (\frac {x^{4} - 4 \, x^{3} + 4 \, x^{2} + 4 \, \sqrt {x^{3} - x^{2} - x} {\left (\sqrt {3} {\left (x^{2} - x - 1\right )} + 3 \, x\right )} \sqrt {\frac {1}{6} \, \sqrt {3} - \frac {1}{6}} + 4 \, \sqrt {3} {\left (x^{3} - x^{2} - x\right )} + 4 \, x + 1}{x^{4} + 1}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{6} \, \sqrt {3} - \frac {1}{6}} \log \left (\frac {x^{4} - 4 \, x^{3} + 4 \, x^{2} - 4 \, \sqrt {x^{3} - x^{2} - x} {\left (\sqrt {3} {\left (x^{2} - x - 1\right )} + 3 \, x\right )} \sqrt {\frac {1}{6} \, \sqrt {3} - \frac {1}{6}} + 4 \, \sqrt {3} {\left (x^{3} - x^{2} - x\right )} + 4 \, x + 1}{x^{4} + 1}\right ) \] Input:
integrate((x^4-1)/(x^3-x^2-x)^(1/2)/(x^4+1),x, algorithm="fricas")
Output:
-sqrt(1/6*sqrt(3) + 1/6)*arctan(-1/2*(3*x^2 - sqrt(3)*(x^2 + 2*x - 1) - 3) *sqrt(1/6*sqrt(3) + 1/6)/sqrt(x^3 - x^2 - x)) - 1/4*sqrt(1/6*sqrt(3) - 1/6 )*log((x^4 - 4*x^3 + 4*x^2 + 4*sqrt(x^3 - x^2 - x)*(sqrt(3)*(x^2 - x - 1) + 3*x)*sqrt(1/6*sqrt(3) - 1/6) + 4*sqrt(3)*(x^3 - x^2 - x) + 4*x + 1)/(x^4 + 1)) + 1/4*sqrt(1/6*sqrt(3) - 1/6)*log((x^4 - 4*x^3 + 4*x^2 - 4*sqrt(x^3 - x^2 - x)*(sqrt(3)*(x^2 - x - 1) + 3*x)*sqrt(1/6*sqrt(3) - 1/6) + 4*sqrt (3)*(x^3 - x^2 - x) + 4*x + 1)/(x^4 + 1))
\[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt {x \left (x^{2} - x - 1\right )} \left (x^{4} + 1\right )}\, dx \] Input:
integrate((x**4-1)/(x**3-x**2-x)**(1/2)/(x**4+1),x)
Output:
Integral((x - 1)*(x + 1)*(x**2 + 1)/(sqrt(x*(x**2 - x - 1))*(x**4 + 1)), x )
\[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{4} + 1\right )} \sqrt {x^{3} - x^{2} - x}} \,d x } \] Input:
integrate((x^4-1)/(x^3-x^2-x)^(1/2)/(x^4+1),x, algorithm="maxima")
Output:
integrate((x^4 - 1)/((x^4 + 1)*sqrt(x^3 - x^2 - x)), x)
\[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{4} + 1\right )} \sqrt {x^{3} - x^{2} - x}} \,d x } \] Input:
integrate((x^4-1)/(x^3-x^2-x)^(1/2)/(x^4+1),x, algorithm="giac")
Output:
integrate((x^4 - 1)/((x^4 + 1)*sqrt(x^3 - x^2 - x)), x)
Time = 0.05 (sec) , antiderivative size = 682, normalized size of antiderivative = 5.29 \[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=\text {Too large to display} \] Input:
int((x^4 - 1)/((x^4 + 1)*(x^3 - x^2 - x)^(1/2)),x)
Output:
(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(asin((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)*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), a sin((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((...
\[ \int \frac {-1+x^4}{\sqrt {-x-x^2+x^3} \left (1+x^4\right )} \, dx=\int \frac {\sqrt {x}\, \sqrt {x^{2}-x -1}\, x^{3}}{x^{6}-x^{5}-x^{4}+x^{2}-x -1}d x -\left (\int \frac {\sqrt {x}\, \sqrt {x^{2}-x -1}}{x^{7}-x^{6}-x^{5}+x^{3}-x^{2}-x}d x \right ) \] Input:
int((x^4-1)/(x^3-x^2-x)^(1/2)/(x^4+1),x)
Output:
int((sqrt(x)*sqrt(x**2 - x - 1)*x**3)/(x**6 - x**5 - x**4 + x**2 - x - 1), x) - int((sqrt(x)*sqrt(x**2 - x - 1))/(x**7 - x**6 - x**5 + x**3 - x**2 - x),x)