Integrand size = 37, antiderivative size = 131 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\frac {\left (-2-17 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+2 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-\frac {2 \log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{3}}+\frac {\log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{\sqrt [3]{3}} \]
1/10*(-17*x^3-2)*(x^3+1)^(2/3)/x^5+2*3^(1/6)*arctan(3^(5/6)*x/(3^(1/3)*x+2 *(x^3+1)^(1/3)))-2/3*ln(-3*x+3^(2/3)*(x^3+1)^(1/3))*3^(2/3)+1/3*ln(3*x^2+3 ^(2/3)*x*(x^3+1)^(1/3)+3^(1/3)*(x^3+1)^(2/3))*3^(2/3)
Time = 0.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\frac {\left (-2-17 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+2 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-\frac {2 \log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{3}}+\frac {\log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{\sqrt [3]{3}} \]
((-2 - 17*x^3)*(1 + x^3)^(2/3))/(10*x^5) + 2*3^(1/6)*ArcTan[(3^(5/6)*x)/(3 ^(1/3)*x + 2*(1 + x^3)^(1/3))] - (2*Log[-3*x + 3^(2/3)*(1 + x^3)^(1/3)])/3 ^(1/3) + Log[3*x^2 + 3^(2/3)*x*(1 + x^3)^(1/3) + 3^(1/3)*(1 + x^3)^(2/3)]/ 3^(1/3)
Time = 0.61 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {1387, 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.
\(\displaystyle \int \frac {\left (x^3+1\right )^{2/3} \left (2 x^6-2 x^3-1\right )}{x^6 \left (2 x^6+x^3-1\right )} \, dx\) |
\(\Big \downarrow \) 1387 |
\(\displaystyle \int \frac {2 x^6-2 x^3-1}{x^6 \sqrt [3]{x^3+1} \left (2 x^3-1\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (-\frac {6}{\left (2 x^3-1\right ) \sqrt [3]{x^3+1}}+\frac {4}{x^3 \sqrt [3]{x^3+1}}+\frac {1}{x^6 \sqrt [3]{x^3+1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \sqrt [6]{3} \arctan \left (\frac {\frac {2 \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\frac {\log \left (2 x^3-1\right )}{\sqrt [3]{3}}-3^{2/3} \log \left (\sqrt [3]{3} x-\sqrt [3]{x^3+1}\right )-\frac {\left (x^3+1\right )^{2/3}}{5 x^5}-\frac {17 \left (x^3+1\right )^{2/3}}{10 x^2}\) |
-1/5*(1 + x^3)^(2/3)/x^5 - (17*(1 + x^3)^(2/3))/(10*x^2) + 2*3^(1/6)*ArcTa n[(1 + (2*3^(1/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]] + Log[-1 + 2*x^3]/3^(1/3) - 3^(2/3)*Log[3^(1/3)*x - (1 + x^3)^(1/3)]
3.19.94.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)* (x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^ p, x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
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 = 12.96 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98
method | result | size |
pseudoelliptic | \(\frac {10 \,3^{\frac {2}{3}} \ln \left (\frac {3^{\frac {2}{3}} x^{2}+3^{\frac {1}{3}} \left (x^{3}+1\right )^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}-20 \,3^{\frac {2}{3}} \ln \left (\frac {-3^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}-60 \,3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (2 \,3^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) x^{5}-51 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}-6 \left (x^{3}+1\right )^{\frac {2}{3}}}{30 x^{5}}\) | \(128\) |
risch | \(\text {Expression too large to display}\) | \(893\) |
trager | \(\text {Expression too large to display}\) | \(1120\) |
1/30*(10*3^(2/3)*ln((3^(2/3)*x^2+3^(1/3)*(x^3+1)^(1/3)*x+(x^3+1)^(2/3))/x^ 2)*x^5-20*3^(2/3)*ln((-3^(1/3)*x+(x^3+1)^(1/3))/x)*x^5-60*3^(1/6)*arctan(1 /9*3^(1/2)*(2*3^(2/3)*(x^3+1)^(1/3)+3*x)/x)*x^5-51*x^3*(x^3+1)^(2/3)-6*(x^ 3+1)^(2/3))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (104) = 208\).
Time = 1.80 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.23 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\frac {20 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (\frac {9 \cdot 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )} - 9 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{3} - 1}\right ) - 10 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (31 \, x^{6} + 23 \, x^{3} + 1\right )} - 9 \, {\left (5 \, x^{5} + 2 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{4 \, x^{6} - 4 \, x^{3} + 1}\right ) - 60 \cdot 3^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3^{\frac {1}{6}} {\left (6 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (14 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 18 \, \left (-1\right )^{\frac {1}{3}} {\left (31 \, x^{8} + 23 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 3^{\frac {1}{3}} {\left (127 \, x^{9} + 201 \, x^{6} + 48 \, x^{3} + 1\right )}\right )}}{3 \, {\left (251 \, x^{9} + 231 \, x^{6} + 6 \, x^{3} - 1\right )}}\right ) - 9 \, {\left (17 \, x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{90 \, x^{5}} \]
1/90*(20*3^(2/3)*(-1)^(1/3)*x^5*log((9*3^(1/3)*(-1)^(2/3)*(x^3 + 1)^(1/3)* x^2 + 3^(2/3)*(-1)^(1/3)*(2*x^3 - 1) - 9*(x^3 + 1)^(2/3)*x)/(2*x^3 - 1)) - 10*3^(2/3)*(-1)^(1/3)*x^5*log(-(3*3^(2/3)*(-1)^(1/3)*(7*x^4 + x)*(x^3 + 1 )^(2/3) - 3^(1/3)*(-1)^(2/3)*(31*x^6 + 23*x^3 + 1) - 9*(5*x^5 + 2*x^2)*(x^ 3 + 1)^(1/3))/(4*x^6 - 4*x^3 + 1)) - 60*3^(1/6)*(-1)^(1/3)*x^5*arctan(1/3* 3^(1/6)*(6*3^(2/3)*(-1)^(2/3)*(14*x^7 - 5*x^4 - x)*(x^3 + 1)^(2/3) + 18*(- 1)^(1/3)*(31*x^8 + 23*x^5 + x^2)*(x^3 + 1)^(1/3) - 3^(1/3)*(127*x^9 + 201* x^6 + 48*x^3 + 1))/(251*x^9 + 231*x^6 + 6*x^3 - 1)) - 9*(17*x^3 + 2)*(x^3 + 1)^(2/3))/x^5
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \cdot \left (2 x^{6} - 2 x^{3} - 1\right )}{x^{6} \left (x + 1\right ) \left (2 x^{3} - 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Integral(((x + 1)*(x**2 - x + 1))**(2/3)*(2*x**6 - 2*x**3 - 1)/(x**6*(x + 1)*(2*x**3 - 1)*(x**2 - x + 1)), x)
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} + x^{3} - 1\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} + x^{3} - 1\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\int -\frac {{\left (x^3+1\right )}^{2/3}\,\left (-2\,x^6+2\,x^3+1\right )}{x^6\,\left (2\,x^6+x^3-1\right )} \,d x \]