Integrand size = 32, antiderivative size = 144 \[ \int \frac {\left (-4+x^3\right ) \left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^9 \left (-2+3 x^3\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (5+4 x^3+16 x^6\right )}{10 x^8}+\frac {5 \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{2^{2/3} \sqrt {3}}+\frac {5 \log \left (x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )}{3\ 2^{2/3}}-\frac {5 \log \left (x^2-\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )}{6\ 2^{2/3}} \]
1/10*(x^3-1)^(2/3)*(16*x^6+4*x^3+5)/x^8+5/6*arctan(3^(1/2)*x/(-x+2*2^(1/3) *(x^3-1)^(1/3)))*2^(1/3)*3^(1/2)+5/6*ln(x+2^(1/3)*(x^3-1)^(1/3))*2^(1/3)-5 /12*ln(x^2-2^(1/3)*x*(x^3-1)^(1/3)+2^(2/3)*(x^3-1)^(2/3))*2^(1/3)
Time = 0.35 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-4+x^3\right ) \left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^9 \left (-2+3 x^3\right )} \, dx=\frac {1}{60} \left (\frac {6 \left (-1+x^3\right )^{2/3} \left (5+4 x^3+16 x^6\right )}{x^8}-50 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )+50 \sqrt [3]{2} \log \left (x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )-25 \sqrt [3]{2} \log \left (x^2-\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )\right ) \]
((6*(-1 + x^3)^(2/3)*(5 + 4*x^3 + 16*x^6))/x^8 - 50*2^(1/3)*Sqrt[3]*ArcTan [(Sqrt[3]*x)/(x - 2*2^(1/3)*(-1 + x^3)^(1/3))] + 50*2^(1/3)*Log[x + 2^(1/3 )*(-1 + x^3)^(1/3)] - 25*2^(1/3)*Log[x^2 - 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^ (2/3)*(-1 + x^3)^(2/3)])/60
Time = 0.73 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {7276, 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.
\(\displaystyle \int \frac {\left (x^3-4\right ) \left (x^3-2\right ) \left (x^3-1\right )^{2/3}}{x^9 \left (3 x^3-2\right )} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {15 \left (x^3-1\right )^{2/3}}{3 x^3-2}-\frac {5 \left (x^3-1\right )^{2/3}}{x^3}-\frac {4 \left (x^3-1\right )^{2/3}}{x^9}-\frac {3 \left (x^3-1\right )^{2/3}}{x^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5 \arctan \left (\frac {1-\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {5 \log \left (3 x^3-2\right )}{6\ 2^{2/3}}+\frac {5 \log \left (-\sqrt [3]{x^3-1}-\frac {x}{\sqrt [3]{2}}\right )}{2\ 2^{2/3}}-\frac {\left (x^3-1\right )^{5/3}}{2 x^8}-\frac {9 \left (x^3-1\right )^{5/3}}{10 x^5}+\frac {5 \left (x^3-1\right )^{2/3}}{2 x^2}\) |
(5*(-1 + x^3)^(2/3))/(2*x^2) - (-1 + x^3)^(5/3)/(2*x^8) - (9*(-1 + x^3)^(5 /3))/(10*x^5) - (5*ArcTan[(1 - (2^(2/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(2^ (2/3)*Sqrt[3]) - (5*Log[-2 + 3*x^3])/(6*2^(2/3)) + (5*Log[-(x/2^(1/3)) - ( -1 + x^3)^(1/3)])/(2*2^(2/3))
3.21.23.3.1 Defintions of rubi rules used
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 = 14.00 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(\frac {\left (96 x^{6}+24 x^{3}+30\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+50 \,2^{\frac {1}{3}} x^{8} \left (\arctan \left (\frac {\sqrt {3}\, \left (x -2 \,2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {2^{\frac {2}{3}} x +2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {-2^{\frac {2}{3}} x {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} x^{2}+2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}-\frac {\ln \left (2\right )}{2}\right )}{60 x^{8}}\) | \(139\) |
risch | \(\text {Expression too large to display}\) | \(563\) |
trager | \(\text {Expression too large to display}\) | \(753\) |
1/60*((96*x^6+24*x^3+30)*(x^3-1)^(2/3)+50*2^(1/3)*x^8*(arctan(1/3*3^(1/2)* (x-2*2^(1/3)*(x^3-1)^(1/3))/x)*3^(1/2)+ln((2^(2/3)*x+2*((-1+x)*(x^2+x+1))^ (1/3))/x)-1/2*ln((-2^(2/3)*x*((-1+x)*(x^2+x+1))^(1/3)+2^(1/3)*x^2+2*((-1+x )*(x^2+x+1))^(2/3))/x^2)-1/2*ln(2)))/x^8
Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (109) = 218\).
Time = 4.61 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.85 \[ \int \frac {\left (-4+x^3\right ) \left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^9 \left (-2+3 x^3\right )} \, dx=\frac {100 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{8} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (3 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \sqrt {3} {\left (27 \, x^{9} - 72 \, x^{6} + 36 \, x^{3} + 8\right )} - 12 \, \sqrt {3} {\left (9 \, x^{8} - 6 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (27 \, x^{9} - 36 \, x^{3} + 8\right )}}\right ) + 50 \cdot 4^{\frac {2}{3}} x^{8} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (3 \, x^{3} - 2\right )} + 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{3 \, x^{3} - 2}\right ) - 25 \cdot 4^{\frac {2}{3}} x^{8} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x - 4^{\frac {1}{3}} {\left (9 \, x^{6} - 6 \, x^{3} - 4\right )} + 6 \, {\left (3 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{9 \, x^{6} - 12 \, x^{3} + 4}\right ) + 36 \, {\left (16 \, x^{6} + 4 \, x^{3} + 5\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{360 \, x^{8}} \]
1/360*(100*4^(1/6)*sqrt(3)*x^8*arctan(1/6*4^(1/6)*(12*4^(2/3)*sqrt(3)*(3*x ^4 - 2*x)*(x^3 - 1)^(2/3) - 4^(1/3)*sqrt(3)*(27*x^9 - 72*x^6 + 36*x^3 + 8) - 12*sqrt(3)*(9*x^8 - 6*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(27*x^9 - 36*x^3 + 8)) + 50*4^(2/3)*x^8*log((6*4^(1/3)*(x^3 - 1)^(1/3)*x^2 + 4^(2/3)*(3*x^3 - 2) + 12*(x^3 - 1)^(2/3)*x)/(3*x^3 - 2)) - 25*4^(2/3)*x^8*log((6*4^(2/3)*( x^3 - 1)^(2/3)*x - 4^(1/3)*(9*x^6 - 6*x^3 - 4) + 6*(3*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(9*x^6 - 12*x^3 + 4)) + 36*(16*x^6 + 4*x^3 + 5)*(x^3 - 1)^(2/3)) /x^8
\[ \int \frac {\left (-4+x^3\right ) \left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^9 \left (-2+3 x^3\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} - 4\right ) \left (x^{3} - 2\right )}{x^{9} \cdot \left (3 x^{3} - 2\right )}\, dx \]
\[ \int \frac {\left (-4+x^3\right ) \left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^9 \left (-2+3 x^3\right )} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )} {\left (x^{3} - 4\right )}}{{\left (3 \, x^{3} - 2\right )} x^{9}} \,d x } \]
\[ \int \frac {\left (-4+x^3\right ) \left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^9 \left (-2+3 x^3\right )} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )} {\left (x^{3} - 4\right )}}{{\left (3 \, x^{3} - 2\right )} x^{9}} \,d x } \]
Timed out. \[ \int \frac {\left (-4+x^3\right ) \left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^9 \left (-2+3 x^3\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3-2\right )\,\left (x^3-4\right )}{x^9\,\left (3\,x^3-2\right )} \,d x \]