Integrand size = 74, antiderivative size = 318 \[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x-x^3-x^4+x^6}}{-2 x+\sqrt [3]{-x-x^3-x^4+x^6}}\right )-\sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x-x^3-x^4+x^6}}{-2 \sqrt [3]{2} x+\sqrt [3]{-x-x^3-x^4+x^6}}\right )-\log \left (x+\sqrt [3]{-x-x^3-x^4+x^6}\right )+\sqrt [3]{2} \log \left (\sqrt [3]{2} x+\sqrt [3]{-x-x^3-x^4+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x-x^3-x^4+x^6}+\left (-x-x^3-x^4+x^6\right )^{2/3}\right )-\frac {\log \left (2^{2/3} x^2-\sqrt [3]{2} x \sqrt [3]{-x-x^3-x^4+x^6}+\left (-x-x^3-x^4+x^6\right )^{2/3}\right )}{2^{2/3}} \]
3^(1/2)*arctan(3^(1/2)*(x^6-x^4-x^3-x)^(1/3)/(-2*x+(x^6-x^4-x^3-x)^(1/3))) -2^(1/3)*3^(1/2)*arctan(3^(1/2)*(x^6-x^4-x^3-x)^(1/3)/(-2*2^(1/3)*x+(x^6-x ^4-x^3-x)^(1/3)))-ln(x+(x^6-x^4-x^3-x)^(1/3))+2^(1/3)*ln(2^(1/3)*x+(x^6-x^ 4-x^3-x)^(1/3))+1/2*ln(x^2-x*(x^6-x^4-x^3-x)^(1/3)+(x^6-x^4-x^3-x)^(2/3))- 1/2*ln(2^(2/3)*x^2-2^(1/3)*x*(x^6-x^4-x^3-x)^(1/3)+(x^6-x^4-x^3-x)^(2/3))* 2^(1/3)
Time = 1.16 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.10 \[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\frac {x^{2/3} \left (-1-x^2-x^3+x^5\right )^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{-1-x^2-x^3+x^5}}\right )+2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2^{2/3} \sqrt [3]{-1-x^2-x^3+x^5}}\right )-2 \log \left (x^{2/3}+\sqrt [3]{-1-x^2-x^3+x^5}\right )+2 \sqrt [3]{2} \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{-1-x^2-x^3+x^5}\right )+\log \left (x^{4/3}-x^{2/3} \sqrt [3]{-1-x^2-x^3+x^5}+\left (-1-x^2-x^3+x^5\right )^{2/3}\right )-\sqrt [3]{2} \log \left (-2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{-1-x^2-x^3+x^5}-\sqrt [3]{2} \left (-1-x^2-x^3+x^5\right )^{2/3}\right )\right )}{2 \left (x \left (-1-x^2-x^3+x^5\right )\right )^{2/3}} \]
Integrate[((2 - 2*x + 2*x^2 - 3*x^3 + 3*x^4)*(-x - x^3 - x^4 + x^6)^(1/3)) /((1 + x)*(-1 + 2*x - 2*x^2 + x^3)*(-1 - x^3 + x^5)),x]
(x^(2/3)*(-1 - x^2 - x^3 + x^5)^(2/3)*(-2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3)) /(x^(2/3) - 2*(-1 - x^2 - x^3 + x^5)^(1/3))] + 2*2^(1/3)*Sqrt[3]*ArcTan[(S qrt[3]*x^(2/3))/(x^(2/3) - 2^(2/3)*(-1 - x^2 - x^3 + x^5)^(1/3))] - 2*Log[ x^(2/3) + (-1 - x^2 - x^3 + x^5)^(1/3)] + 2*2^(1/3)*Log[2*x^(2/3) + 2^(2/3 )*(-1 - x^2 - x^3 + x^5)^(1/3)] + Log[x^(4/3) - x^(2/3)*(-1 - x^2 - x^3 + x^5)^(1/3) + (-1 - x^2 - x^3 + x^5)^(2/3)] - 2^(1/3)*Log[-2*x^(4/3) + 2^(2 /3)*x^(2/3)*(-1 - x^2 - x^3 + x^5)^(1/3) - 2^(1/3)*(-1 - x^2 - x^3 + x^5)^ (2/3)]))/(2*(x*(-1 - x^2 - x^3 + x^5))^(2/3))
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.
| \(\displaystyle \int \frac {\left (3 x^4-3 x^3+2 x^2-2 x+2\right ) \sqrt [3]{x^6-x^4-x^3-x}}{(x+1) \left (x^3-2 x^2+2 x-1\right ) \left (x^5-x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (3 x^4-3 x^3+2 x^2-2 x+2\right ) \sqrt [3]{x^6-x^4-x^3-x}}{(x-1) (x+1) \left (x^5-x^3-1\right )}-\frac {x \left (3 x^4-3 x^3+2 x^2-2 x+2\right ) \sqrt [3]{x^6-x^4-x^3-x}}{(x+1) \left (x^2-x+1\right ) \left (x^5-x^3-1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 x^4-3 x^3+2 x^2-2 x+2\right ) \sqrt [3]{x^6-x^4-x^3-x}}{(1-x) (x+1) \left (x^2-x+1\right ) \left (-x^5+x^3+1\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x^6-x^4-x^3-x} \int \frac {\sqrt [3]{x} \left (3 x^4-3 x^3+2 x^2-2 x+2\right ) \sqrt [3]{x^5-x^3-x^2-1}}{(1-x) (x+1) \left (x^2-x+1\right ) \left (-x^5+x^3+1\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^5-x^3-x^2-1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x^6-x^4-x^3-x} \int \frac {x \left (3 x^4-3 x^3+2 x^2-2 x+2\right ) \sqrt [3]{x^5-x^3-x^2-1}}{(1-x) (x+1) \left (x^2-x+1\right ) \left (-x^5+x^3+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^5-x^3-x^2-1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{x^6-x^4-x^3-x} \int \left (\frac {\left (5 x^2-3\right ) \sqrt [3]{x^5-x^3-x^2-1} x^2}{x^5-x^3-1}-\frac {\sqrt [3]{x^5-x^3-x^2-1}}{3 \left (\sqrt [3]{x}-1\right )}-\frac {2 \sqrt [3]{x^5-x^3-x^2-1}}{3 \left (\sqrt [3]{x}+1\right )}+\frac {2 \left (\sqrt [3]{x}-2\right ) \sqrt [3]{x^5-x^3-x^2-1}}{3 \left (x^{2/3}-\sqrt [3]{x}+1\right )}+\frac {\left (\sqrt [3]{x}+2\right ) \sqrt [3]{x^5-x^3-x^2-1}}{3 \left (x^{2/3}+\sqrt [3]{x}+1\right )}+\frac {(1-2 x) \sqrt [3]{x^5-x^3-x^2-1}}{x^2-x+1}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^5-x^3-x^2-1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x^6-x^4-x^3-x} \left (-\frac {1}{3} \int \frac {\sqrt [3]{x^5-x^3-x^2-1}}{\sqrt [3]{x}-1}d\sqrt [3]{x}-\frac {2}{3} \int \frac {\sqrt [3]{x^5-x^3-x^2-1}}{\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {2}{3} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^5-x^3-x^2-1}}{2 \sqrt [3]{x}-i \sqrt {3}-1}d\sqrt [3]{x}+\frac {1}{3} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^5-x^3-x^2-1}}{2 \sqrt [3]{x}-i \sqrt {3}+1}d\sqrt [3]{x}+\frac {2}{3} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^5-x^3-x^2-1}}{2 \sqrt [3]{x}+i \sqrt {3}-1}d\sqrt [3]{x}+\frac {1}{3} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^5-x^3-x^2-1}}{2 \sqrt [3]{x}+i \sqrt {3}+1}d\sqrt [3]{x}+\frac {2 \int \frac {\sqrt [3]{x^5-x^3-x^2-1}}{\sqrt [3]{-2} \sqrt [3]{x}+\sqrt [3]{1-i \sqrt {3}}}d\sqrt [3]{x}}{3 \left (1-i \sqrt {3}\right )^{2/3}}+\frac {2 \int \frac {\sqrt [3]{x^5-x^3-x^2-1}}{\sqrt [3]{-2} \sqrt [3]{x}+\sqrt [3]{1+i \sqrt {3}}}d\sqrt [3]{x}}{3 \left (1+i \sqrt {3}\right )^{2/3}}+\frac {2 \int \frac {\sqrt [3]{x^5-x^3-x^2-1}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{3 \left (1-i \sqrt {3}\right )^{2/3}}+\frac {2 \int \frac {\sqrt [3]{x^5-x^3-x^2-1}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{3 \left (1+i \sqrt {3}\right )^{2/3}}+\frac {2 \int \frac {\sqrt [3]{x^5-x^3-x^2-1}}{\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{3 \left (1-i \sqrt {3}\right )^{2/3}}+\frac {2 \int \frac {\sqrt [3]{x^5-x^3-x^2-1}}{\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{3 \left (1+i \sqrt {3}\right )^{2/3}}-3 \int \frac {x^2 \sqrt [3]{x^5-x^3-x^2-1}}{x^5-x^3-1}d\sqrt [3]{x}+5 \int \frac {x^4 \sqrt [3]{x^5-x^3-x^2-1}}{x^5-x^3-1}d\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{x^5-x^3-x^2-1}}\) |
Int[((2 - 2*x + 2*x^2 - 3*x^3 + 3*x^4)*(-x - x^3 - x^4 + x^6)^(1/3))/((1 + x)*(-1 + 2*x - 2*x^2 + x^3)*(-1 - x^3 + x^5)),x]
3.29.96.3.1 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[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
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_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, 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 = 75.37 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.80
| method | result | size |
| pseudoelliptic | \(-\frac {2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} {\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {1}{3}} x +{\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}+\frac {\ln \left (\frac {{\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {2}{3}}-{\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}+2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +{\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {{\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {1}{3}}+x}{x}\right )+\left (-\arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} {\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {1}{3}}+x \right )}{3 x}\right ) 2^{\frac {1}{3}}+\arctan \left (\frac {\left (x -2 {\left (x \left (x^{5}-x^{3}-x^{2}-1\right )\right )}^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right )\right ) \sqrt {3}\) | \(255\) |
| trager | \(\text {Expression too large to display}\) | \(3486\) |
int((3*x^4-3*x^3+2*x^2-2*x+2)*(x^6-x^4-x^3-x)^(1/3)/(1+x)/(x^3-2*x^2+2*x-1 )/(x^5-x^3-1),x,method=_RETURNVERBOSE)
-1/2*2^(1/3)*ln((2^(2/3)*x^2-2^(1/3)*(x*(x^5-x^3-x^2-1))^(1/3)*x+(x*(x^5-x ^3-x^2-1))^(2/3))/x^2)+1/2*ln(((x*(x^5-x^3-x^2-1))^(2/3)-(x*(x^5-x^3-x^2-1 ))^(1/3)*x+x^2)/x^2)+2^(1/3)*ln((2^(1/3)*x+(x*(x^5-x^3-x^2-1))^(1/3))/x)-l n(((x*(x^5-x^3-x^2-1))^(1/3)+x)/x)+(-arctan(1/3*3^(1/2)*(-2^(2/3)*(x*(x^5- x^3-x^2-1))^(1/3)+x)/x)*2^(1/3)+arctan(1/3*(x-2*(x*(x^5-x^3-x^2-1))^(1/3)) *3^(1/2)/x))*3^(1/2)
Timed out. \[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\text {Timed out} \]
integrate((3*x^4-3*x^3+2*x^2-2*x+2)*(x^6-x^4-x^3-x)^(1/3)/(1+x)/(x^3-2*x^2 +2*x-1)/(x^5-x^3-1),x, algorithm="fricas")
\[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\int \frac {\sqrt [3]{x \left (x^{5} - x^{3} - x^{2} - 1\right )} \left (3 x^{4} - 3 x^{3} + 2 x^{2} - 2 x + 2\right )}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{5} - x^{3} - 1\right )}\, dx \]
integrate((3*x**4-3*x**3+2*x**2-2*x+2)*(x**6-x**4-x**3-x)**(1/3)/(1+x)/(x* *3-2*x**2+2*x-1)/(x**5-x**3-1),x)
Integral((x*(x**5 - x**3 - x**2 - 1))**(1/3)*(3*x**4 - 3*x**3 + 2*x**2 - 2 *x + 2)/((x - 1)*(x + 1)*(x**2 - x + 1)*(x**5 - x**3 - 1)), x)
\[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\int { \frac {{\left (x^{6} - x^{4} - x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x^{4} - 3 \, x^{3} + 2 \, x^{2} - 2 \, x + 2\right )}}{{\left (x^{5} - x^{3} - 1\right )} {\left (x^{3} - 2 \, x^{2} + 2 \, x - 1\right )} {\left (x + 1\right )}} \,d x } \]
integrate((3*x^4-3*x^3+2*x^2-2*x+2)*(x^6-x^4-x^3-x)^(1/3)/(1+x)/(x^3-2*x^2 +2*x-1)/(x^5-x^3-1),x, algorithm="maxima")
integrate((x^6 - x^4 - x^3 - x)^(1/3)*(3*x^4 - 3*x^3 + 2*x^2 - 2*x + 2)/(( x^5 - x^3 - 1)*(x^3 - 2*x^2 + 2*x - 1)*(x + 1)), x)
\[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\int { \frac {{\left (x^{6} - x^{4} - x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x^{4} - 3 \, x^{3} + 2 \, x^{2} - 2 \, x + 2\right )}}{{\left (x^{5} - x^{3} - 1\right )} {\left (x^{3} - 2 \, x^{2} + 2 \, x - 1\right )} {\left (x + 1\right )}} \,d x } \]
integrate((3*x^4-3*x^3+2*x^2-2*x+2)*(x^6-x^4-x^3-x)^(1/3)/(1+x)/(x^3-2*x^2 +2*x-1)/(x^5-x^3-1),x, algorithm="giac")
integrate((x^6 - x^4 - x^3 - x)^(1/3)*(3*x^4 - 3*x^3 + 2*x^2 - 2*x + 2)/(( x^5 - x^3 - 1)*(x^3 - 2*x^2 + 2*x - 1)*(x + 1)), x)
Timed out. \[ \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx=\int -\frac {{\left (x^6-x^4-x^3-x\right )}^{1/3}\,\left (3\,x^4-3\,x^3+2\,x^2-2\,x+2\right )}{\left (x+1\right )\,\left (-x^5+x^3+1\right )\,\left (x^3-2\,x^2+2\,x-1\right )} \,d x \]
int(-((x^6 - x^3 - x^4 - x)^(1/3)*(2*x^2 - 2*x - 3*x^3 + 3*x^4 + 2))/((x + 1)*(x^3 - x^5 + 1)*(2*x - 2*x^2 + x^3 - 1)),x)