Integrand size = 38, antiderivative size = 145 \[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+2 x+2 x^3+x^6}}\right )}{\sqrt [3]{2}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{1+2 x+2 x^3+x^6}\right )}{\sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+2 x+2 x^3+x^6}+\sqrt [3]{2} \left (1+2 x+2 x^3+x^6\right )^{2/3}\right )}{2 \sqrt [3]{2}} \]
-1/2*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(x^6+2*x^3+2*x+1)^(1/3)))*2^(2/3) +1/2*ln(-2*x+2^(2/3)*(x^6+2*x^3+2*x+1)^(1/3))*2^(2/3)-1/4*ln(2*x^2+2^(2/3) *x*(x^6+2*x^3+2*x+1)^(1/3)+2^(1/3)*(x^6+2*x^3+2*x+1)^(2/3))*2^(2/3)
Time = 0.37 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.94 \[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+2 x+2 x^3+x^6}}\right )-2 \log \left (-2 x+2^{2/3} \sqrt [3]{1+2 x+2 x^3+x^6}\right )+\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+2 x+2 x^3+x^6}+\sqrt [3]{2} \left (1+2 x+2 x^3+x^6\right )^{2/3}\right )}{2 \sqrt [3]{2}} \]
-1/2*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + 2*x + 2*x^3 + x^6)^(1 /3))] - 2*Log[-2*x + 2^(2/3)*(1 + 2*x + 2*x^3 + x^6)^(1/3)] + Log[2*x^2 + 2^(2/3)*x*(1 + 2*x + 2*x^3 + x^6)^(1/3) + 2^(1/3)*(1 + 2*x + 2*x^3 + x^6)^ (2/3)])/2^(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.
| \(\displaystyle \int \frac {3 x^6-4 x-3}{\left (x^6+2 x+1\right ) \sqrt [3]{x^6+2 x^3+2 x+1}} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (x^4-2 x^3+3 x^2-4 x+5\right ) \left (3 x^6-4 x-3\right )}{4 \left (x^5-x^4+x^3-x^2+x+1\right ) \sqrt [3]{x^6+2 x^3+2 x+1}}-\frac {3 x^6-4 x-3}{4 (x+1) \sqrt [3]{x^6+2 x^3+2 x+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \int \frac {1}{\sqrt [3]{x^6+2 x^3+2 x+1}}dx-\int \frac {1}{(x+1) \sqrt [3]{x^6+2 x^3+2 x+1}}dx-5 \int \frac {1}{\left (x^5-x^4+x^3-x^2+x+1\right ) \sqrt [3]{x^6+2 x^3+2 x+1}}dx-4 \int \frac {x}{\left (x^5-x^4+x^3-x^2+x+1\right ) \sqrt [3]{x^6+2 x^3+2 x+1}}dx+3 \int \frac {x^2}{\left (x^5-x^4+x^3-x^2+x+1\right ) \sqrt [3]{x^6+2 x^3+2 x+1}}dx-2 \int \frac {x^3}{\left (x^5-x^4+x^3-x^2+x+1\right ) \sqrt [3]{x^6+2 x^3+2 x+1}}dx+\int \frac {x^4}{\left (x^5-x^4+x^3-x^2+x+1\right ) \sqrt [3]{x^6+2 x^3+2 x+1}}dx\) |
3.21.36.3.1 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 = 24.71 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.83
| method | result | size |
| pseudoelliptic | \(\frac {2^{\frac {2}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {1}{3}}\right )}{3 x}\right )+2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {1}{3}} x +\left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{4}\) | \(121\) |
| trager | \(\text {Expression too large to display}\) | \(606\) |
1/4*2^(2/3)*(2*3^(1/2)*arctan(1/3*3^(1/2)*(x+2^(2/3)*(x^6+2*x^3+2*x+1)^(1/ 3))/x)+2*ln((-2^(1/3)*x+(x^6+2*x^3+2*x+1)^(1/3))/x)-ln((2^(2/3)*x^2+2^(1/3 )*(x^6+2*x^3+2*x+1)^(1/3)*x+(x^6+2*x^3+2*x+1)^(2/3))/x^2))
Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (118) = 236\).
Time = 27.36 (sec) , antiderivative size = 478, normalized size of antiderivative = 3.30 \[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=-\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{18} + 36 \, x^{15} + 6 \, x^{13} + 183 \, x^{12} + 144 \, x^{10} + 288 \, x^{9} + 12 \, x^{8} + 372 \, x^{7} + 183 \, x^{6} + 144 \, x^{5} + 144 \, x^{4} + 44 \, x^{3} + 12 \, x^{2} + 6 \, x + 1\right )} + 12 \, \sqrt {2} {\left (x^{14} + 18 \, x^{11} + 4 \, x^{9} + 38 \, x^{8} + 36 \, x^{6} + 18 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} + x^{2}\right )} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}} + 12 \cdot 2^{\frac {1}{6}} {\left (x^{13} + 6 \, x^{10} + 4 \, x^{8} + 2 \, x^{7} + 12 \, x^{5} + 6 \, x^{4} + 4 \, x^{3} + 4 \, x^{2} + x\right )} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (x^{18} + 6 \, x^{13} - 105 \, x^{12} - 216 \, x^{9} + 12 \, x^{8} - 204 \, x^{7} - 105 \, x^{6} + 8 \, x^{3} + 12 \, x^{2} + 6 \, x + 1\right )}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{6} + 2 \, x + 1\right )} - 6 \, {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {2}{3}} x}{x^{6} + 2 \, x + 1}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{7} + 6 \, x^{4} + 2 \, x^{2} + x\right )} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{12} + 18 \, x^{9} + 4 \, x^{7} + 38 \, x^{6} + 36 \, x^{4} + 18 \, x^{3} + 4 \, x^{2} + 4 \, x + 1\right )} + 12 \, {\left (x^{8} + 3 \, x^{5} + 2 \, x^{3} + x^{2}\right )} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{7} + 2 \, x^{6} + 4 \, x^{2} + 4 \, x + 1}\right ) \]
-1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(x^18 + 36*x^15 + 6*x^13 + 183*x^12 + 144*x^10 + 288*x^9 + 12*x^8 + 372*x^7 + 183*x^6 + 144 *x^5 + 144*x^4 + 44*x^3 + 12*x^2 + 6*x + 1) + 12*sqrt(2)*(x^14 + 18*x^11 + 4*x^9 + 38*x^8 + 36*x^6 + 18*x^5 + 4*x^4 + 4*x^3 + x^2)*(x^6 + 2*x^3 + 2* x + 1)^(1/3) + 12*2^(1/6)*(x^13 + 6*x^10 + 4*x^8 + 2*x^7 + 12*x^5 + 6*x^4 + 4*x^3 + 4*x^2 + x)*(x^6 + 2*x^3 + 2*x + 1)^(2/3))/(x^18 + 6*x^13 - 105*x ^12 - 216*x^9 + 12*x^8 - 204*x^7 - 105*x^6 + 8*x^3 + 12*x^2 + 6*x + 1)) + 1/6*2^(2/3)*log((6*2^(1/3)*(x^6 + 2*x^3 + 2*x + 1)^(1/3)*x^2 + 2^(2/3)*(x^ 6 + 2*x + 1) - 6*(x^6 + 2*x^3 + 2*x + 1)^(2/3)*x)/(x^6 + 2*x + 1)) - 1/12* 2^(2/3)*log((3*2^(2/3)*(x^7 + 6*x^4 + 2*x^2 + x)*(x^6 + 2*x^3 + 2*x + 1)^( 2/3) + 2^(1/3)*(x^12 + 18*x^9 + 4*x^7 + 38*x^6 + 36*x^4 + 18*x^3 + 4*x^2 + 4*x + 1) + 12*(x^8 + 3*x^5 + 2*x^3 + x^2)*(x^6 + 2*x^3 + 2*x + 1)^(1/3))/ (x^12 + 4*x^7 + 2*x^6 + 4*x^2 + 4*x + 1))
\[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=\int \frac {3 x^{6} - 4 x - 3}{\sqrt [3]{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 2 x + 1\right )} \left (x + 1\right ) \left (x^{5} - x^{4} + x^{3} - x^{2} + x + 1\right )}\, dx \]
Integral((3*x**6 - 4*x - 3)/(((x**2 + 1)*(x**4 - x**2 + 2*x + 1))**(1/3)*( x + 1)*(x**5 - x**4 + x**3 - x**2 + x + 1)), x)
\[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=\int { \frac {3 \, x^{6} - 4 \, x - 3}{{\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}} {\left (x^{6} + 2 \, x + 1\right )}} \,d x } \]
\[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=\int { \frac {3 \, x^{6} - 4 \, x - 3}{{\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}} {\left (x^{6} + 2 \, x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=\int -\frac {-3\,x^6+4\,x+3}{\left (x^6+2\,x+1\right )\,{\left (x^6+2\,x^3+2\,x+1\right )}^{1/3}} \,d x \]