Integrand size = 29, antiderivative size = 88 \[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^5}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-x+x^5}\right )-\frac {1}{4} \log \left (x^2+x \sqrt [3]{-x+x^5}+\left (-x+x^5\right )^{2/3}\right ) \]
-1/2*3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^5-x)^(1/3)))+1/2*ln(-x+(x^5-x)^(1/3) )-1/4*ln(x^2+x*(x^5-x)^(1/3)+(x^5-x)^(2/3))
Time = 15.46 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^5}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-x+x^5}\right )-\frac {1}{4} \log \left (x^2+x \sqrt [3]{-x+x^5}+\left (-x+x^5\right )^{2/3}\right ) \]
-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x + x^5)^(1/3))]) + Log[-x + (-x + x^5)^(1/3)]/2 - Log[x^2 + x*(-x + x^5)^(1/3) + (-x + x^5)^(2/3)]/4
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.26 (sec) , antiderivative size = 332, normalized size of antiderivative = 3.77, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2467, 25, 2035, 7266, 7279, 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.
| \(\displaystyle \int \frac {x^4+1}{\left (x^4-x^2-1\right ) \sqrt [3]{x^5-x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^4-1} \int -\frac {x^4+1}{\sqrt [3]{x} \left (-x^4+x^2+1\right ) \sqrt [3]{x^4-1}}dx}{\sqrt [3]{x^5-x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{x^4-1} \int \frac {x^4+1}{\sqrt [3]{x} \left (-x^4+x^2+1\right ) \sqrt [3]{x^4-1}}dx}{\sqrt [3]{x^5-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4-1} \int \frac {\sqrt [3]{x} \left (x^4+1\right )}{\left (-x^4+x^2+1\right ) \sqrt [3]{x^4-1}}d\sqrt [3]{x}}{\sqrt [3]{x^5-x}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4-1} \int \frac {x^2+1}{\left (-x^2+x+1\right ) \sqrt [3]{x^2-1}}dx^{2/3}}{2 \sqrt [3]{x^5-x}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4-1} \int \left (\frac {x+2}{\left (-x^2+x+1\right ) \sqrt [3]{x^2-1}}-\frac {1}{\sqrt [3]{x^2-1}}\right )dx^{2/3}}{2 \sqrt [3]{x^5-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4-1} \left (\frac {\left (1-\sqrt {5}\right ) \sqrt [3]{1-x^2} x^{4/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^2,\frac {2 x^2}{3-\sqrt {5}}\right )}{4 \left (3-\sqrt {5}\right ) \sqrt [3]{x^2-1}}+\frac {\left (1+\sqrt {5}\right ) \sqrt [3]{1-x^2} x^{4/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^2,\frac {2 x^2}{3+\sqrt {5}}\right )}{4 \left (3+\sqrt {5}\right ) \sqrt [3]{x^2-1}}+\frac {\sqrt [3]{1-x^2} x^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},1,\frac {7}{6},x^2,\frac {2 x^2}{3-\sqrt {5}}\right )}{\sqrt [3]{x^2-1}}+\frac {\sqrt [3]{1-x^2} x^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},1,\frac {1}{3},\frac {7}{6},\frac {2 x^2}{3+\sqrt {5}},x^2\right )}{\sqrt [3]{x^2-1}}-\frac {\sqrt [3]{1-x^2} x^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^2\right )}{\sqrt [3]{x^2-1}}\right )}{2 \sqrt [3]{x^5-x}}\) |
(-3*x^(1/3)*(-1 + x^4)^(1/3)*((x^(2/3)*(1 - x^2)^(1/3)*AppellF1[1/6, 1/3, 1, 7/6, x^2, (2*x^2)/(3 - Sqrt[5])])/(-1 + x^2)^(1/3) + (x^(2/3)*(1 - x^2) ^(1/3)*AppellF1[1/6, 1, 1/3, 7/6, (2*x^2)/(3 + Sqrt[5]), x^2])/(-1 + x^2)^ (1/3) + ((1 - Sqrt[5])*x^(4/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, (2*x^2)/(3 - Sqrt[5])])/(4*(3 - Sqrt[5])*(-1 + x^2)^(1/3)) + ((1 + Sq rt[5])*x^(4/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, (2*x^2)/(3 + Sqrt[5])])/(4*(3 + Sqrt[5])*(-1 + x^2)^(1/3)) - (x^(2/3)*(1 - x^2)^(1/3) *Hypergeometric2F1[1/6, 1/3, 7/6, x^2])/(-1 + x^2)^(1/3)))/(2*(-x + x^5)^( 1/3))
3.13.2.3.1 Defintions of rubi rules used
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_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, x]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 7.35 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.91
| method | result | size |
| pseudoelliptic | \(\frac {\ln \left (\frac {-x +\left (x^{5}-x \right )^{\frac {1}{3}}}{x}\right )}{2}-\frac {\ln \left (\frac {x^{2}+x \left (x^{5}-x \right )^{\frac {1}{3}}+\left (x^{5}-x \right )^{\frac {2}{3}}}{x^{2}}\right )}{4}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{5}-x \right )^{\frac {1}{3}}\right )}{3 x}\right )}{2}\) | \(80\) |
| trager | \(\frac {\ln \left (\frac {11545253597330080 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}-20271516034318916 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-43294700989987800 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}-2579689613765595 x^{4}+41156279639771682 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {2}{3}}-4180130780591496 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}} x -32579356521190130 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+18488074429590093 \left (x^{5}-x \right )^{\frac {2}{3}}-20578139819885841 x \left (x^{5}-x \right )^{\frac {1}{3}}-11545253597330080 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-3267606844103087 x^{2}+20271516034318916 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+2579689613765595}{x^{4}-x^{2}-1}\right )}{2}+\operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {2751668921349968 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}+12307840486871214 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-10318758455062380 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+13709988646829470 x^{4}+41156279639771682 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {2}{3}}-36976148859180186 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}} x -20271516034318916 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+2090065390295748 \left (x^{5}-x \right )^{\frac {2}{3}}-20578139819885841 x \left (x^{5}-x \right )^{\frac {1}{3}}-2751668921349968 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+2886313399332520 x^{2}-12307840486871214 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-13709988646829470}{x^{4}-x^{2}-1}\right )\) | \(397\) |
1/2*ln((-x+(x^5-x)^(1/3))/x)-1/4*ln((x^2+x*(x^5-x)^(1/3)+(x^5-x)^(2/3))/x^ 2)+1/2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^5-x)^(1/3)))
Time = 0.71 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.24 \[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=-\frac {1}{2} \, \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{4} - 1\right )} - 2 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {2}{3}}}{x^{4} + 8 \, x^{2} - 1}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} - x^{2} + 3 \, {\left (x^{5} - x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{5} - x\right )}^{\frac {2}{3}} - 1}{x^{4} - x^{2} - 1}\right ) \]
-1/2*sqrt(3)*arctan(-(4*sqrt(3)*(x^5 - x)^(1/3)*x + sqrt(3)*(x^4 - 1) - 2* sqrt(3)*(x^5 - x)^(2/3))/(x^4 + 8*x^2 - 1)) + 1/4*log((x^4 - x^2 + 3*(x^5 - x)^(1/3)*x - 3*(x^5 - x)^(2/3) - 1)/(x^4 - x^2 - 1))
\[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\int \frac {x^{4} + 1}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} - x^{2} - 1\right )}\, dx \]
\[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\int { \frac {x^{4} + 1}{{\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{4} - x^{2} - 1\right )}} \,d x } \]
\[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\int { \frac {x^{4} + 1}{{\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{4} - x^{2} - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {1+x^4}{\left (-1-x^2+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\int -\frac {x^4+1}{{\left (x^5-x\right )}^{1/3}\,\left (-x^4+x^2+1\right )} \,d x \]