Integrand size = 22, antiderivative size = 144 \[ \int \frac {1}{x^6 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {1}{2 x^5 \sqrt [3]{1-x^3}}-\frac {7 \left (1-x^3\right )^{2/3}}{10 x^5}-\frac {4 \left (1-x^3\right )^{2/3}}{5 x^2}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}} \]
1/2/x^5/(-x^3+1)^(1/3)-7/10*(-x^3+1)^(2/3)/x^5-4/5*(-x^3+1)^(2/3)/x^2-1/24 *ln(x^3+1)*2^(2/3)+1/8*ln(-2^(1/3)*x-(-x^3+1)^(1/3))*2^(2/3)-1/12*arctan(1 /3*(1-2*2^(1/3)*x/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)
Time = 0.58 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^6 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {1}{120} \left (-\frac {12 \left (2+x^3-8 x^6\right )}{x^5 \sqrt [3]{1-x^3}}-10\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )+10\ 2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )-5\ 2^{2/3} \log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \]
((-12*(2 + x^3 - 8*x^6))/(x^5*(1 - x^3)^(1/3)) - 10*2^(2/3)*Sqrt[3]*ArcTan [(Sqrt[3]*x)/(x - 2^(2/3)*(1 - x^3)^(1/3))] + 10*2^(2/3)*Log[2*x + 2^(2/3) *(1 - x^3)^(1/3)] - 5*2^(2/3)*Log[-2*x^2 + 2^(2/3)*x*(1 - x^3)^(1/3) - 2^( 1/3)*(1 - x^3)^(2/3)])/120
Time = 0.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {972, 1053, 25, 1053, 27, 901}
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.
| \(\displaystyle \int \frac {1}{x^6 \left (1-x^3\right )^{4/3} \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle \frac {1}{2} \int \frac {6 x^3+7}{x^6 \sqrt [3]{1-x^3} \left (x^3+1\right )}dx+\frac {1}{2 x^5 \sqrt [3]{1-x^3}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {1}{2} \left (-\frac {1}{5} \int -\frac {21 x^3+16}{x^3 \sqrt [3]{1-x^3} \left (x^3+1\right )}dx-\frac {7 \left (1-x^3\right )^{2/3}}{5 x^5}\right )+\frac {1}{2 x^5 \sqrt [3]{1-x^3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \int \frac {21 x^3+16}{x^3 \sqrt [3]{1-x^3} \left (x^3+1\right )}dx-\frac {7 \left (1-x^3\right )^{2/3}}{5 x^5}\right )+\frac {1}{2 x^5 \sqrt [3]{1-x^3}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (-\frac {1}{2} \int -\frac {10}{\sqrt [3]{1-x^3} \left (x^3+1\right )}dx-\frac {8 \left (1-x^3\right )^{2/3}}{x^2}\right )-\frac {7 \left (1-x^3\right )^{2/3}}{5 x^5}\right )+\frac {1}{2 x^5 \sqrt [3]{1-x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (5 \int \frac {1}{\sqrt [3]{1-x^3} \left (x^3+1\right )}dx-\frac {8 \left (1-x^3\right )^{2/3}}{x^2}\right )-\frac {7 \left (1-x^3\right )^{2/3}}{5 x^5}\right )+\frac {1}{2 x^5 \sqrt [3]{1-x^3}}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (5 \left (-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}}\right )-\frac {8 \left (1-x^3\right )^{2/3}}{x^2}\right )-\frac {7 \left (1-x^3\right )^{2/3}}{5 x^5}\right )+\frac {1}{2 x^5 \sqrt [3]{1-x^3}}\) |
1/(2*x^5*(1 - x^3)^(1/3)) + ((-7*(1 - x^3)^(2/3))/(5*x^5) + ((-8*(1 - x^3) ^(2/3))/x^2 + 5*(-(ArcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(2^ (1/3)*Sqrt[3])) - Log[1 + x^3]/(6*2^(1/3)) + Log[-(2^(1/3)*x) - (1 - x^3)^ (1/3)]/(2*2^(1/3))))/5)/2
3.7.50.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x ^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] & & IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Time = 22.17 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.12
| method | result | size |
| pseudoelliptic | \(\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) x^{5} \left (-x^{3}+1\right )^{\frac {1}{3}}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5} \left (-x^{3}+1\right )^{\frac {1}{3}}}{2}+2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5} \left (-x^{3}+1\right )^{\frac {1}{3}}+\frac {48 x^{6}}{5}-\frac {6 x^{3}}{5}-\frac {12}{5}}{12 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{5}}\) | \(161\) |
| trager | \(\text {Expression too large to display}\) | \(790\) |
| risch | \(\text {Expression too large to display}\) | \(958\) |
1/12/(-x^3+1)^(1/3)*(3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(-2^(2/3)*(-x^3+1) ^(1/3)+x)/x)*x^5*(-x^3+1)^(1/3)-1/2*2^(2/3)*ln((2^(2/3)*x^2-2^(1/3)*(-x^3+ 1)^(1/3)*x+(-x^3+1)^(2/3))/x^2)*x^5*(-x^3+1)^(1/3)+2^(2/3)*ln((2^(1/3)*x+( -x^3+1)^(1/3))/x)*x^5*(-x^3+1)^(1/3)+48/5*x^6-6/5*x^3-12/5)/x^5
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (109) = 218\).
Time = 1.69 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.19 \[ \int \frac {1}{x^6 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=-\frac {10 \, \sqrt {6} 2^{\frac {1}{6}} {\left (x^{8} - x^{5}\right )} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )} + 12 \, \sqrt {6} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 10 \cdot 2^{\frac {2}{3}} {\left (x^{8} - x^{5}\right )} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{3} + 1\right )} + 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 5 \cdot 2^{\frac {2}{3}} {\left (x^{8} - x^{5}\right )} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 12 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 36 \, {\left (8 \, x^{6} - x^{3} - 2\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{360 \, {\left (x^{8} - x^{5}\right )}} \]
-1/360*(10*sqrt(6)*2^(1/6)*(x^8 - x^5)*arctan(1/6*2^(1/6)*(6*sqrt(6)*2^(2/ 3)*(5*x^7 + 4*x^4 - x)*(-x^3 + 1)^(2/3) - sqrt(6)*2^(1/3)*(71*x^9 - 111*x^ 6 + 33*x^3 - 1) + 12*sqrt(6)*(19*x^8 - 16*x^5 + x^2)*(-x^3 + 1)^(1/3))/(10 9*x^9 - 105*x^6 + 3*x^3 + 1)) - 10*2^(2/3)*(x^8 - x^5)*log((6*2^(1/3)*(-x^ 3 + 1)^(1/3)*x^2 + 2^(2/3)*(x^3 + 1) + 6*(-x^3 + 1)^(2/3)*x)/(x^3 + 1)) + 5*2^(2/3)*(x^8 - x^5)*log((3*2^(2/3)*(5*x^4 - x)*(-x^3 + 1)^(2/3) + 2^(1/3 )*(19*x^6 - 16*x^3 + 1) - 12*(2*x^5 - x^2)*(-x^3 + 1)^(1/3))/(x^6 + 2*x^3 + 1)) + 36*(8*x^6 - x^3 - 2)*(-x^3 + 1)^(2/3))/(x^8 - x^5)
\[ \int \frac {1}{x^6 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\int \frac {1}{x^{6} \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {4}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
\[ \int \frac {1}{x^6 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {4}{3}} x^{6}} \,d x } \]
\[ \int \frac {1}{x^6 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {4}{3}} x^{6}} \,d x } \]
Timed out. \[ \int \frac {1}{x^6 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\int \frac {1}{x^6\,{\left (1-x^3\right )}^{4/3}\,\left (x^3+1\right )} \,d x \]