Integrand size = 62, antiderivative size = 124 \[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+2 x^3-x^4-x^7}}\right )-\log \left (-x+\sqrt [3]{x+2 x^3-x^4-x^7}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x+2 x^3-x^4-x^7}+\left (x+2 x^3-x^4-x^7\right )^{2/3}\right ) \]
-3^(1/2)*arctan(3^(1/2)*x/(x+2*(-x^7-x^4+2*x^3+x)^(1/3)))-ln(-x+(-x^7-x^4+ 2*x^3+x)^(1/3))+1/2*ln(x^2+x*(-x^7-x^4+2*x^3+x)^(1/3)+(-x^7-x^4+2*x^3+x)^( 2/3))
Time = 1.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.36 \[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=-\frac {x^{2/3} \left (-1-2 x^2+x^3+x^6\right )^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{-1-2 x^2+x^3+x^6}}\right )+2 \log \left (x^{2/3}+\sqrt [3]{-1-2 x^2+x^3+x^6}\right )-\log \left (x^{4/3}-x^{2/3} \sqrt [3]{-1-2 x^2+x^3+x^6}+\left (-1-2 x^2+x^3+x^6\right )^{2/3}\right )\right )}{2 \left (-x \left (-1-2 x^2+x^3+x^6\right )\right )^{2/3}} \]
Integrate[((2 + x^3 + 4*x^6)*(x + 2*x^3 - x^4 - x^7)^(1/3))/((-1 - 2*x^2 + x^3 + x^6)*(-1 - x^2 + x^3 + x^6)),x]
-1/2*(x^(2/3)*(-1 - 2*x^2 + x^3 + x^6)^(2/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^ (2/3))/(x^(2/3) - 2*(-1 - 2*x^2 + x^3 + x^6)^(1/3))] + 2*Log[x^(2/3) + (-1 - 2*x^2 + x^3 + x^6)^(1/3)] - Log[x^(4/3) - x^(2/3)*(-1 - 2*x^2 + x^3 + x ^6)^(1/3) + (-1 - 2*x^2 + x^3 + x^6)^(2/3)]))/(-(x*(-1 - 2*x^2 + x^3 + x^6 )))^(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.
| \(\displaystyle \int \frac {\left (4 x^6+x^3+2\right ) \sqrt [3]{-x^7-x^4+2 x^3+x}}{\left (x^6+x^3-2 x^2-1\right ) \left (x^6+x^3-x^2-1\right )} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\sqrt [3]{-x^7-x^4+2 x^3+x} \left (4 x^6+x^3+2\right )}{7 (x-1) \left (x^6+x^3-2 x^2-1\right )}+\frac {\left (-x^4-2 x^3-3 x^2-5 x-6\right ) \sqrt [3]{-x^7-x^4+2 x^3+x} \left (4 x^6+x^3+2\right )}{7 \left (x^5+x^4+x^3+2 x^2+x+1\right ) \left (x^6+x^3-2 x^2-1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [3]{-x^7-x^4+2 x^3+x} \text {Subst}\left (\int \frac {1}{(1-x) \left (-x^{18}-x^9+2 x^6+1\right )^{2/3}}dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-x^6-x^3+2 x^2+1}}-\frac {12 \sqrt [3]{-x^7-x^4+2 x^3+x} \text {Subst}\left (\int \frac {x^3}{\left (-x^{18}-x^9+2 x^6+1\right )^{2/3}}dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-x^6-x^3+2 x^2+1}}+\frac {\sqrt [3]{-x^7-x^4+2 x^3+x} \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-1} x+1\right ) \left (-x^{18}-x^9+2 x^6+1\right )^{2/3}}dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-x^6-x^3+2 x^2+1}}+\frac {\sqrt [3]{-x^7-x^4+2 x^3+x} \text {Subst}\left (\int \frac {1}{\left (1-(-1)^{2/3} x\right ) \left (-x^{18}-x^9+2 x^6+1\right )^{2/3}}dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-x^6-x^3+2 x^2+1}}-\frac {3 \sqrt [3]{-x^7-x^4+2 x^3+x} \text {Subst}\left (\int \frac {1}{\left (x^{15}+x^{12}+x^9+2 x^6+x^3+1\right ) \left (-x^{18}-x^9+2 x^6+1\right )^{2/3}}dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-x^6-x^3+2 x^2+1}}+\frac {12 \sqrt [3]{-x^7-x^4+2 x^3+x} \text {Subst}\left (\int \frac {x^3}{\left (x^{15}+x^{12}+x^9+2 x^6+x^3+1\right ) \left (-x^{18}-x^9+2 x^6+1\right )^{2/3}}dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-x^6-x^3+2 x^2+1}}+\frac {6 \sqrt [3]{-x^7-x^4+2 x^3+x} \text {Subst}\left (\int \frac {x^6}{\left (x^{15}+x^{12}+x^9+2 x^6+x^3+1\right ) \left (-x^{18}-x^9+2 x^6+1\right )^{2/3}}dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-x^6-x^3+2 x^2+1}}+\frac {15 \sqrt [3]{-x^7-x^4+2 x^3+x} \text {Subst}\left (\int \frac {x^9}{\left (x^{15}+x^{12}+x^9+2 x^6+x^3+1\right ) \left (-x^{18}-x^9+2 x^6+1\right )^{2/3}}dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-x^6-x^3+2 x^2+1}}+\frac {3 \sqrt [3]{-x^7-x^4+2 x^3+x} \text {Subst}\left (\int \frac {x^{12}}{\left (x^{15}+x^{12}+x^9+2 x^6+x^3+1\right ) \left (-x^{18}-x^9+2 x^6+1\right )^{2/3}}dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-x^6-x^3+2 x^2+1}}\) |
Int[((2 + x^3 + 4*x^6)*(x + 2*x^3 - x^4 - x^7)^(1/3))/((-1 - 2*x^2 + x^3 + x^6)*(-1 - x^2 + x^3 + x^6)),x]
3.19.27.3.1 Defintions of rubi rules used
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]
Time = 15.78 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93
| method | result | size |
| pseudoelliptic | \(\frac {\ln \left (\frac {{\left (-x \left (x^{6}+x^{3}-2 x^{2}-1\right )\right )}^{\frac {2}{3}}+{\left (-x \left (x^{6}+x^{3}-2 x^{2}-1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (-x \left (x^{6}+x^{3}-2 x^{2}-1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )-\ln \left (\frac {{\left (-x \left (x^{6}+x^{3}-2 x^{2}-1\right )\right )}^{\frac {1}{3}}-x}{x}\right )\) | \(115\) |
| trager | \(\text {Expression too large to display}\) | \(536\) |
int((4*x^6+x^3+2)*(-x^7-x^4+2*x^3+x)^(1/3)/(x^6+x^3-2*x^2-1)/(x^6+x^3-x^2- 1),x,method=_RETURNVERBOSE)
1/2*ln(((-x*(x^6+x^3-2*x^2-1))^(2/3)+(-x*(x^6+x^3-2*x^2-1))^(1/3)*x+x^2)/x ^2)+3^(1/2)*arctan(1/3*(2*(-x*(x^6+x^3-2*x^2-1))^(1/3)+x)*3^(1/2)/x)-ln((( -x*(x^6+x^3-2*x^2-1))^(1/3)-x)/x)
Time = 1.65 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.40 \[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=\sqrt {3} \arctan \left (-\frac {70 \, \sqrt {3} {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (32 \, x^{6} + 32 \, x^{3} - 39 \, x^{2} - 32\right )} - 56 \, \sqrt {3} {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {2}{3}}}{64 \, x^{6} + 64 \, x^{3} - 253 \, x^{2} - 64}\right ) - \frac {1}{2} \, \log \left (\frac {x^{6} + x^{3} - x^{2} - 3 \, {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {2}{3}} - 1}{x^{6} + x^{3} - x^{2} - 1}\right ) \]
integrate((4*x^6+x^3+2)*(-x^7-x^4+2*x^3+x)^(1/3)/(x^6+x^3-2*x^2-1)/(x^6+x^ 3-x^2-1),x, algorithm="fricas")
sqrt(3)*arctan(-(70*sqrt(3)*(-x^7 - x^4 + 2*x^3 + x)^(1/3)*x - sqrt(3)*(32 *x^6 + 32*x^3 - 39*x^2 - 32) - 56*sqrt(3)*(-x^7 - x^4 + 2*x^3 + x)^(2/3))/ (64*x^6 + 64*x^3 - 253*x^2 - 64)) - 1/2*log((x^6 + x^3 - x^2 - 3*(-x^7 - x ^4 + 2*x^3 + x)^(1/3)*x + 3*(-x^7 - x^4 + 2*x^3 + x)^(2/3) - 1)/(x^6 + x^3 - x^2 - 1))
\[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=\int \frac {\sqrt [3]{- x \left (x^{6} + x^{3} - 2 x^{2} - 1\right )} \left (4 x^{6} + x^{3} + 2\right )}{\left (x - 1\right ) \left (x^{6} + x^{3} - 2 x^{2} - 1\right ) \left (x^{5} + x^{4} + x^{3} + 2 x^{2} + x + 1\right )}\, dx \]
integrate((4*x**6+x**3+2)*(-x**7-x**4+2*x**3+x)**(1/3)/(x**6+x**3-2*x**2-1 )/(x**6+x**3-x**2-1),x)
Integral((-x*(x**6 + x**3 - 2*x**2 - 1))**(1/3)*(4*x**6 + x**3 + 2)/((x - 1)*(x**6 + x**3 - 2*x**2 - 1)*(x**5 + x**4 + x**3 + 2*x**2 + x + 1)), x)
\[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=\int { \frac {{\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} {\left (4 \, x^{6} + x^{3} + 2\right )}}{{\left (x^{6} + x^{3} - x^{2} - 1\right )} {\left (x^{6} + x^{3} - 2 \, x^{2} - 1\right )}} \,d x } \]
integrate((4*x^6+x^3+2)*(-x^7-x^4+2*x^3+x)^(1/3)/(x^6+x^3-2*x^2-1)/(x^6+x^ 3-x^2-1),x, algorithm="maxima")
integrate((-x^7 - x^4 + 2*x^3 + x)^(1/3)*(4*x^6 + x^3 + 2)/((x^6 + x^3 - x ^2 - 1)*(x^6 + x^3 - 2*x^2 - 1)), x)
\[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=\int { \frac {{\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} {\left (4 \, x^{6} + x^{3} + 2\right )}}{{\left (x^{6} + x^{3} - x^{2} - 1\right )} {\left (x^{6} + x^{3} - 2 \, x^{2} - 1\right )}} \,d x } \]
integrate((4*x^6+x^3+2)*(-x^7-x^4+2*x^3+x)^(1/3)/(x^6+x^3-2*x^2-1)/(x^6+x^ 3-x^2-1),x, algorithm="giac")
integrate((-x^7 - x^4 + 2*x^3 + x)^(1/3)*(4*x^6 + x^3 + 2)/((x^6 + x^3 - x ^2 - 1)*(x^6 + x^3 - 2*x^2 - 1)), x)
Timed out. \[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=\int \frac {\left (4\,x^6+x^3+2\right )\,{\left (-x^7-x^4+2\,x^3+x\right )}^{1/3}}{\left (-x^6-x^3+2\,x^2+1\right )\,\left (-x^6-x^3+x^2+1\right )} \,d x \]
int(((x^3 + 4*x^6 + 2)*(x + 2*x^3 - x^4 - x^7)^(1/3))/((2*x^2 - x^3 - x^6 + 1)*(x^2 - x^3 - x^6 + 1)),x)