Integrand size = 42, antiderivative size = 146 \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=-\frac {x \sqrt [3]{x-x^5+x^7}}{2 \left (1+x^2-x^4+x^6\right )}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{x-x^5+x^7}}{-2 x+\sqrt [3]{x-x^5+x^7}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (x+\sqrt [3]{x-x^5+x^7}\right )-\frac {1}{12} \log \left (x^2-x \sqrt [3]{x-x^5+x^7}+\left (x-x^5+x^7\right )^{2/3}\right ) \]
-x*(x^7-x^5+x)^(1/3)/(2*x^6-2*x^4+2*x^2+2)-1/6*arctan(3^(1/2)*(x^7-x^5+x)^ (1/3)/(-2*x+(x^7-x^5+x)^(1/3)))*3^(1/2)+1/6*ln(x+(x^7-x^5+x)^(1/3))-1/12*l n(x^2-x*(x^7-x^5+x)^(1/3)+(x^7-x^5+x)^(2/3))
Time = 3.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\frac {\sqrt [3]{x-x^5+x^7} \left (-\frac {6 x^{4/3}}{1+x^2-x^4+x^6}+\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{1-x^4+x^6}}\right )}{\sqrt [3]{1-x^4+x^6}}+\frac {2 \log \left (x^{2/3}+\sqrt [3]{1-x^4+x^6}\right )}{\sqrt [3]{1-x^4+x^6}}-\frac {\log \left (x^{4/3}-x^{2/3} \sqrt [3]{1-x^4+x^6}+\left (1-x^4+x^6\right )^{2/3}\right )}{\sqrt [3]{1-x^4+x^6}}\right )}{12 \sqrt [3]{x}} \]
((x - x^5 + x^7)^(1/3)*((-6*x^(4/3))/(1 + x^2 - x^4 + x^6) + (2*Sqrt[3]*Ar cTan[(Sqrt[3]*x^(2/3))/(x^(2/3) - 2*(1 - x^4 + x^6)^(1/3))])/(1 - x^4 + x^ 6)^(1/3) + (2*Log[x^(2/3) + (1 - x^4 + x^6)^(1/3)])/(1 - x^4 + x^6)^(1/3) - Log[x^(4/3) - x^(2/3)*(1 - x^4 + x^6)^(1/3) + (1 - x^4 + x^6)^(2/3)]/(1 - x^4 + x^6)^(1/3)))/(12*x^(1/3))
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.
| \(\displaystyle \int \frac {\left (2 x^6-x^4-1\right ) \sqrt [3]{x^7-x^5+x}}{\left (x^6-x^4+x^2+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x^7-x^5+x} \int -\frac {\sqrt [3]{x} \left (-2 x^6+x^4+1\right ) \sqrt [3]{x^6-x^4+1}}{\left (x^6-x^4+x^2+1\right )^2}dx}{\sqrt [3]{x} \sqrt [3]{x^6-x^4+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x^7-x^5+x} \int \frac {\sqrt [3]{x} \left (-2 x^6+x^4+1\right ) \sqrt [3]{x^6-x^4+1}}{\left (x^6-x^4+x^2+1\right )^2}dx}{\sqrt [3]{x} \sqrt [3]{x^6-x^4+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^7-x^5+x} \int \frac {x \left (-2 x^6+x^4+1\right ) \sqrt [3]{x^6-x^4+1}}{\left (x^6-x^4+x^2+1\right )^2}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^6-x^4+1}}\) |
| \(\Big \downarrow \) 7283 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^7-x^5+x} \int \frac {x^{2/3} \left (-2 x^3+x^2+1\right ) \sqrt [3]{x^3-x^2+1}}{\left (x^3-x^2+x+1\right )^2}dx^{2/3}}{2 \sqrt [3]{x} \sqrt [3]{x^6-x^4+1}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^7-x^5+x} \int \left (\frac {x^{2/3} \left (-x^2+2 x+3\right ) \sqrt [3]{x^3-x^2+1}}{\left (x^3-x^2+x+1\right )^2}-\frac {2 x^{2/3} \sqrt [3]{x^3-x^2+1}}{x^3-x^2+x+1}\right )dx^{2/3}}{2 \sqrt [3]{x} \sqrt [3]{x^6-x^4+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^7-x^5+x} \left (3 \int \frac {x^{2/3} \sqrt [3]{x^3-x^2+1}}{\left (x^3-x^2+x+1\right )^2}dx^{2/3}+2 \int \frac {x^{4/3} \sqrt [3]{x^3-x^2+1}}{\left (x^3-x^2+x+1\right )^2}dx^{2/3}-\int \frac {x^{7/3} \sqrt [3]{x^3-x^2+1}}{\left (x^3-x^2+x+1\right )^2}dx^{2/3}-2 \int \frac {x^{2/3} \sqrt [3]{x^3-x^2+1}}{x^3-x^2+x+1}dx^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{x^6-x^4+1}}\) |
3.21.43.3.1 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] :> With[{lst = PowerVariableExpn[u, m + 1, x ]}, Simp[1/lst[[2]] Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x] , x], x, (lst[[3]]*x)^lst[[2]]], x] /; !FalseQ[lst] && NeQ[lst[[2]], m + 1 ]] /; IntegerQ[m] && NeQ[m, -1] && NonsumQ[u] && (GtQ[m, 0] || !AlgebraicF unctionQ[u, x])
Time = 12.06 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.39
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\left (-2 x^{6}+2 x^{4}-2 x^{2}-2\right ) \ln \left (\frac {x +{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}}}{x}\right )+6 {\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}} x +\left (x^{6}-x^{4}+x^{2}+1\right ) \left (2 \arctan \left (\frac {\left (-2 {\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right ) \sqrt {3}+\ln \left (\frac {{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {2}{3}}-{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )\right )\right ) x}{12 \left (x +{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}}\right ) \left ({\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {2}{3}}+x \left (x -{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}}\right )\right )}\) | \(203\) |
| trager | \(\text {Expression too large to display}\) | \(739\) |
| risch | \(\text {Expression too large to display}\) | \(1528\) |
-1/12*((-2*x^6+2*x^4-2*x^2-2)*ln((x+(x*(x^6-x^4+1))^(1/3))/x)+6*(x*(x^6-x^ 4+1))^(1/3)*x+(x^6-x^4+x^2+1)*(2*arctan(1/3*(-2*(x*(x^6-x^4+1))^(1/3)+x)*3 ^(1/2)/x)*3^(1/2)+ln(((x*(x^6-x^4+1))^(2/3)-(x*(x^6-x^4+1))^(1/3)*x+x^2)/x ^2)))*x/(x+(x*(x^6-x^4+1))^(1/3))/((x*(x^6-x^4+1))^(2/3)+x*(x-(x*(x^6-x^4+ 1))^(1/3)))
Time = 1.80 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=-\frac {2 \, \sqrt {3} {\left (x^{6} - x^{4} + x^{2} + 1\right )} \arctan \left (-\frac {2 \, \sqrt {3} {\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{6} - x^{4} - x^{2} + 1\right )} - 2 \, \sqrt {3} {\left (x^{7} - x^{5} + x\right )}^{\frac {2}{3}}}{x^{6} - x^{4} + x^{2} + 1}\right ) - {\left (x^{6} - x^{4} + x^{2} + 1\right )} \log \left (\frac {x^{6} - x^{4} + x^{2} + 3 \, {\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{7} - x^{5} + x\right )}^{\frac {2}{3}} + 1}{x^{6} - x^{4} + x^{2} + 1}\right ) + 6 \, {\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} x}{12 \, {\left (x^{6} - x^{4} + x^{2} + 1\right )}} \]
-1/12*(2*sqrt(3)*(x^6 - x^4 + x^2 + 1)*arctan(-(2*sqrt(3)*(x^7 - x^5 + x)^ (1/3)*x + sqrt(3)*(x^6 - x^4 - x^2 + 1) - 2*sqrt(3)*(x^7 - x^5 + x)^(2/3)) /(x^6 - x^4 + x^2 + 1)) - (x^6 - x^4 + x^2 + 1)*log((x^6 - x^4 + x^2 + 3*( x^7 - x^5 + x)^(1/3)*x + 3*(x^7 - x^5 + x)^(2/3) + 1)/(x^6 - x^4 + x^2 + 1 )) + 6*(x^7 - x^5 + x)^(1/3)*x)/(x^6 - x^4 + x^2 + 1)
Timed out. \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\int { \frac {{\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} - x^{4} - 1\right )}}{{\left (x^{6} - x^{4} + x^{2} + 1\right )}^{2}} \,d x } \]
\[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\int { \frac {{\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} - x^{4} - 1\right )}}{{\left (x^{6} - x^{4} + x^{2} + 1\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\int -\frac {\left (-2\,x^6+x^4+1\right )\,{\left (x^7-x^5+x\right )}^{1/3}}{{\left (x^6-x^4+x^2+1\right )}^2} \,d x \]