Integrand size = 28, antiderivative size = 138 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3+x^6\right )}{x^6 \left (-1+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (2+3 x^3\right )}{10 x^5}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}} \]
1/10*(x^3-1)^(2/3)*(3*x^3+2)/x^5+1/6*arctan(3^(1/2)*x/(x+2^(2/3)*(x^3-1)^( 1/3)))*2^(2/3)*3^(1/2)-1/6*ln(-2*x+2^(2/3)*(x^3-1)^(1/3))*2^(2/3)+1/12*ln( 2*x^2+2^(2/3)*x*(x^3-1)^(1/3)+2^(1/3)*(x^3-1)^(2/3))*2^(2/3)
Time = 0.34 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3+x^6\right )}{x^6 \left (-1+x^6\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (2+3 x^3\right )}{10 x^5}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}} \]
((-1 + x^3)^(2/3)*(2 + 3*x^3))/(10*x^5) + ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)* (-1 + x^3)^(1/3))]/(2^(1/3)*Sqrt[3]) - Log[-2*x + 2^(2/3)*(-1 + x^3)^(1/3) ]/(3*2^(1/3)) + Log[2*x^2 + 2^(2/3)*x*(-1 + x^3)^(1/3) + 2^(1/3)*(-1 + x^3 )^(2/3)]/(6*2^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(407\) vs. \(2(138)=276\).
Time = 0.99 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1388, 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.
\(\displaystyle \int \frac {\left (x^3-1\right )^{2/3} \left (x^6+x^3+1\right )}{x^6 \left (x^6-1\right )} \, dx\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle \int \frac {x^6+x^3+1}{x^6 \sqrt [3]{x^3-1} \left (x^3+1\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {1}{3 (x+1) \sqrt [3]{x^3-1}}+\frac {1}{x^6 \sqrt [3]{x^3-1}}+\frac {2-x}{3 \left (x^2-x+1\right ) \sqrt [3]{x^3-1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\arctan \left (\frac {1-2^{2/3} \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{18 \sqrt [3]{2}}+\frac {\log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}\right )}{9 \sqrt [3]{2}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (x^3-1\right )^{2/3}}+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}+1\right )}{18 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{x^3-1}+\sqrt [3]{2}\right )}{6 \sqrt [3]{2}}-\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1}-x+1\right )}{6 \sqrt [3]{2}}+\frac {\left (x^3-1\right )^{2/3}}{5 x^5}+\frac {3 \left (x^3-1\right )^{2/3}}{10 x^2}+\frac {\log \left ((1-x) (x+1)^2\right )}{18 \sqrt [3]{2}}\) |
(-1 + x^3)^(2/3)/(5*x^5) + (3*(-1 + x^3)^(2/3))/(10*x^2) + ArcTan[(1 - (2^ (1/3)*(1 - x))/(-1 + x^3)^(1/3))/Sqrt[3]]/(3*2^(1/3)*Sqrt[3]) - ArcTan[(1 + (2*2^(1/3)*(1 - x))/(-1 + x^3)^(1/3))/Sqrt[3]]/(3*2^(1/3)*Sqrt[3]) + (2^ (2/3)*ArcTan[(1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + ArcTan[(1 - 2^(2/3)*(-1 + x^3)^(1/3))/Sqrt[3]]/(3*2^(1/3)*Sqrt[3]) + Log[( 1 - x)*(1 + x)^2]/(18*2^(1/3)) + Log[1 + x^3]/(18*2^(1/3)) + Log[1 - (2^(1 /3)*(1 - x))/(-1 + x^3)^(1/3)]/(9*2^(1/3)) - Log[1 + (2^(2/3)*(1 - x)^2)/( -1 + x^3)^(2/3) + (2^(1/3)*(1 - x))/(-1 + x^3)^(1/3)]/(18*2^(1/3)) - Log[2 ^(1/3)*x - (-1 + x^3)^(1/3)]/(3*2^(1/3)) + Log[2^(1/3) + (-1 + x^3)^(1/3)] /(6*2^(1/3)) - Log[1 - x + 2^(2/3)*(-1 + x^3)^(1/3)]/(6*2^(1/3))
3.20.60.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(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, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 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 = 14.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {-10 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}+5 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}-10 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+18 x^{3} \left (x^{3}-1\right )^{\frac {2}{3}}+12 \left (x^{3}-1\right )^{\frac {2}{3}}}{60 x^{5}}\) | \(128\) |
trager | \(\text {Expression too large to display}\) | \(760\) |
risch | \(\text {Expression too large to display}\) | \(930\) |
1/60*(-10*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)/x*(x+2^(2/3)*(x^3-1)^(1/3)))* x^5+5*2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*x*(x^3-1)^(1/3)+(x^3-1)^(2/3))/x^2)* x^5-10*2^(2/3)*ln((-2^(1/3)*x+(x^3-1)^(1/3))/x)*x^5+18*x^3*(x^3-1)^(2/3)+1 2*(x^3-1)^(2/3))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (105) = 210\).
Time = 1.66 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.18 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3+x^6\right )}{x^6 \left (-1+x^6\right )} \, dx=-\frac {10 \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 12 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{6 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 10 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 5 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 12 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 18 \, {\left (3 \, x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{180 \, x^{5}} \]
-1/180*(10*sqrt(6)*2^(1/6)*(-1)^(1/3)*x^5*arctan(1/6*2^(1/6)*(6*sqrt(6)*2^ (2/3)*(-1)^(2/3)*(5*x^7 + 4*x^4 - x)*(x^3 - 1)^(2/3) + 12*sqrt(6)*(-1)^(1/ 3)*(19*x^8 - 16*x^5 + x^2)*(x^3 - 1)^(1/3) - sqrt(6)*2^(1/3)*(71*x^9 - 111 *x^6 + 33*x^3 - 1))/(109*x^9 - 105*x^6 + 3*x^3 + 1)) - 10*2^(2/3)*(-1)^(1/ 3)*x^5*log(-(6*2^(1/3)*(-1)^(2/3)*(x^3 - 1)^(1/3)*x^2 + 2^(2/3)*(-1)^(1/3) *(x^3 + 1) - 6*(x^3 - 1)^(2/3)*x)/(x^3 + 1)) + 5*2^(2/3)*(-1)^(1/3)*x^5*lo g(-(3*2^(2/3)*(-1)^(1/3)*(5*x^4 - x)*(x^3 - 1)^(2/3) - 2^(1/3)*(-1)^(2/3)* (19*x^6 - 16*x^3 + 1) - 12*(2*x^5 - x^2)*(x^3 - 1)^(1/3))/(x^6 + 2*x^3 + 1 )) - 18*(3*x^3 + 2)*(x^3 - 1)^(2/3))/x^5
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3+x^6\right )}{x^6 \left (-1+x^6\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + x^{3} + 1\right )}{x^{6} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
Integral(((x - 1)*(x**2 + x + 1))**(2/3)*(x**6 + x**3 + 1)/(x**6*(x - 1)*( x + 1)*(x**2 - x + 1)*(x**2 + x + 1)), x)
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3+x^6\right )}{x^6 \left (-1+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 1\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3+x^6\right )}{x^6 \left (-1+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 1\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3+x^6\right )}{x^6 \left (-1+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+x^3+1\right )}{x^6\,\left (x^6-1\right )} \,d x \]