Integrand size = 22, antiderivative size = 291 \[ \int \frac {x^6}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {1}{2} x \sqrt [3]{1-x^3}+\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}+\frac {\log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (2 \sqrt [3]{2}+\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}} \] Output:
-1/2*x*(-x^3+1)^(1/3)+1/6*arctan(1/3*(1-2*2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^ (1/2))*2^(1/3)*3^(1/2)+1/12*arctan(1/3*(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^ (1/2))*2^(1/3)*3^(1/2)+1/12*ln(2^(2/3)-(1-x)/(-x^3+1)^(1/3))*2^(1/3)-1/12* ln(1+2^(2/3)*(1-x)^2/(-x^3+1)^(2/3)-2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)+ 1/6*ln(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)-1/24*ln(2*2^(1/3)+(1-x)^2/( -x^3+1)^(2/3)+2^(2/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.40 \[ \int \frac {x^6}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{2} x \sqrt [3]{1-x^3} \left (-1-\frac {4 \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},1,\frac {4}{3},x^3,-x^3\right )}{\left (1+x^3\right ) \left (-4 \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},1,\frac {4}{3},x^3,-x^3\right )+x^3 \left (3 \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{3},2,\frac {7}{3},x^3,-x^3\right )+\operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},x^3,-x^3\right )\right )\right )}\right ) \] Input:
Integrate[x^6/((1 - x^3)^(2/3)*(1 + x^3)),x]
Output:
(x*(1 - x^3)^(1/3)*(-1 - (4*AppellF1[1/3, -1/3, 1, 4/3, x^3, -x^3])/((1 + x^3)*(-4*AppellF1[1/3, -1/3, 1, 4/3, x^3, -x^3] + x^3*(3*AppellF1[4/3, -1/ 3, 2, 7/3, x^3, -x^3] + AppellF1[4/3, 2/3, 1, 7/3, x^3, -x^3])))))/2
Time = 0.68 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {979, 927, 982, 821, 16, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {x^6}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 979 |
| \(\displaystyle \frac {1}{2} \int \frac {\sqrt [3]{1-x^3}}{x^3+1}dx-\frac {1}{2} x \sqrt [3]{1-x^3}\) |
| \(\Big \downarrow \) 927 |
| \(\displaystyle -\frac {9}{2} \int \frac {1-x}{\sqrt [3]{1-x^3} \left (4-\frac {(1-x)^3}{1-x^3}\right ) \left (\frac {2 (1-x)^3}{1-x^3}+1\right )}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {1}{2} \sqrt [3]{1-x^3} x\) |
| \(\Big \downarrow \) 982 |
| \(\displaystyle -\frac {9}{2} \left (\frac {1}{9} \int \frac {1-x}{\sqrt [3]{1-x^3} \left (4-\frac {(1-x)^3}{1-x^3}\right )}d\frac {1-x}{\sqrt [3]{1-x^3}}+\frac {2}{9} \int \frac {1-x}{\sqrt [3]{1-x^3} \left (\frac {2 (1-x)^3}{1-x^3}+1\right )}d\frac {1-x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{2} \sqrt [3]{1-x^3} x\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle -\frac {9}{2} \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\int \frac {1}{\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}\right )+\frac {1}{9} \left (\frac {\int \frac {1}{2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\int \frac {2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}\right )\right )-\frac {1}{2} \sqrt [3]{1-x^3} x\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {9}{2} \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\int \frac {2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {1}{2} \sqrt [3]{1-x^3} x\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}+\frac {\int -\frac {\sqrt [3]{2} \left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}-\frac {1}{2} \int \frac {2^{2/3} \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {1}{2} \sqrt [3]{1-x^3} x\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {\int \frac {\sqrt [3]{2} \left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}-\frac {1}{2} \int \frac {2^{2/3} \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {1}{2} \sqrt [3]{1-x^3} x\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}-\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {1}{2} \sqrt [3]{1-x^3} x\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {3 \int \frac {1}{-\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}-3}d\left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\sqrt [3]{2}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {-3 \int \frac {1}{-\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}-3}d\left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {1}{2} \sqrt [3]{1-x^3} x\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {9}{2} \left (\frac {2}{9} \left (\frac {-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )-\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {1}{2} \sqrt [3]{1-x^3} x\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {1}{2} \sqrt [3]{1-x^3} x\) |
Input:
Int[x^6/((1 - x^3)^(2/3)*(1 + x^3)),x]
Output:
-1/2*(x*(1 - x^3)^(1/3)) - (9*((2*((-((Sqrt[3]*ArcTan[(1 - (2*2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]])/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)]/(2*2^(1/3)))/(3*2^(1/3)) - Log[1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)]/(3*2^(2/3))))/9 + (-1/3*Log[ 2^(2/3) - (1 - x)/(1 - x^3)^(1/3)]/2^(2/3) - (Sqrt[3]*ArcTan[(1 + (2^(1/3) *(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]] - Log[2*2^(1/3) + (1 - x)^2/(1 - x^3)^ (2/3) + (2^(2/3)*(1 - x))/(1 - x^3)^(1/3)]/2)/(3*2^(2/3)))/9))/2
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^3)^(1/3)/((c_) + (d_.)*(x_)^3), x_Symbol] :> With[{q = Rt[b/a, 3]}, Simp[9*(a/(c*q)) Subst[Int[x/((4 - a*x^3)*(1 + 2*a*x^3)), x], x, (1 + q*x)/(a + b*x^3)^(1/3)], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[ b*c - a*d, 0] && EqQ[b*c + a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q) + 1))), x] - Simp[e^(2*n)/(b*d *(m + n*(p + q) + 1)) Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Sim p[a*c*(m - 2*n + 1) + (a*d*(m + n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x ^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && I GtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x ]
Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[(e*x)^m/(a + b*x^n), x], x] - Simp[d /(b*c - a*d) Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 22.78 (sec) , antiderivative size = 696, normalized size of antiderivative = 2.39
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(696\) |
| trager | \(\text {Expression too large to display}\) | \(1163\) |
Input:
int(x^6/(-x^3+1)^(2/3)/(x^3+1),x,method=_RETURNVERBOSE)
Output:
1/2*x*(x^3-1)/(-x^3+1)^(2/3)+(1/4*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3 -2)+9*_Z^2)*ln(-(18*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)^2* RootOf(_Z^3-2)^2*x^3+12*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2 )*RootOf(_Z^3-2)^3*x^3+3*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^ 2)*x^6+2*RootOf(_Z^3-2)*x^6-9*(x^6-2*x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+ 3*_Z*RootOf(_Z^3-2)+9*_Z^2)*RootOf(_Z^3-2)^2*x-18*(x^6-2*x^3+1)^(1/3)*Root Of(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)*RootOf(_Z^3-2)*x^2-6*(x^6- 2*x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x^2-6*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf( _Z^3-2)+9*_Z^2)*x^3-4*RootOf(_Z^3-2)*x^3+3*RootOf(RootOf(_Z^3-2)^2+3*_Z*Ro otOf(_Z^3-2)+9*_Z^2)+2*RootOf(_Z^3-2))/(1+x)^2/(x^2-x+1)^2)+1/12*RootOf(_Z ^3-2)*ln((36*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)^2*RootOf( _Z^3-2)^2*x^3+6*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)*RootOf (_Z^3-2)^3*x^3-6*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)*x^6-R ootOf(_Z^3-2)*x^6-9*(x^6-2*x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootO f(_Z^3-2)+9*_Z^2)*RootOf(_Z^3-2)^2*x-6*(x^6-2*x^3+1)^(1/3)*RootOf(_Z^3-2)^ 2*x^2+36*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf(_Z^3-2)+9*_Z^2)*x^3+6*RootOf( _Z^3-2)*x^3-6*(x^6-2*x^3+1)^(2/3)*x-6*RootOf(RootOf(_Z^3-2)^2+3*_Z*RootOf( _Z^3-2)+9*_Z^2)-RootOf(_Z^3-2))/(1+x)^2/(x^2-x+1)^2))/(-x^3+1)^(2/3)*((x^3 -1)^2)^(1/3)
Time = 1.40 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.20 \[ \int \frac {x^6}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{12} \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} \arctan \left (-\frac {4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (6 \cdot 4^{\frac {2}{3}} {\left (x^{16} - 33 \, x^{13} + 110 \, x^{10} - 110 \, x^{7} + 33 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 48 \, {\left (x^{14} - 2 \, x^{11} - 6 \, x^{8} - 2 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} {\left (x^{18} + 42 \, x^{15} - 417 \, x^{12} + 812 \, x^{9} - 417 \, x^{6} + 42 \, x^{3} + 1\right )}\right )}}{2 \, {\left (x^{18} - 102 \, x^{15} + 447 \, x^{12} - 628 \, x^{9} + 447 \, x^{6} - 102 \, x^{3} + 1\right )}}\right ) + \frac {1}{72} \cdot 4^{\frac {2}{3}} \log \left (-\frac {12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{2} - 3 \cdot 4^{\frac {2}{3}} {\left (x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} {\left (x^{6} + 2 \, x^{3} + 1\right )}}{x^{6} + 2 \, x^{3} + 1}\right ) - \frac {1}{144} \cdot 4^{\frac {2}{3}} \log \left (\frac {24 \cdot 4^{\frac {1}{3}} {\left (x^{8} - 4 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {2}{3}} {\left (x^{12} - 32 \, x^{9} + 78 \, x^{6} - 32 \, x^{3} + 1\right )} + 12 \, {\left (x^{10} - 11 \, x^{7} + 11 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) - \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x \] Input:
integrate(x^6/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="fricas")
Output:
1/12*4^(1/6)*sqrt(1/3)*arctan(-1/2*4^(1/6)*sqrt(1/3)*(6*4^(2/3)*(x^16 - 33 *x^13 + 110*x^10 - 110*x^7 + 33*x^4 - x)*(-x^3 + 1)^(1/3) - 48*(x^14 - 2*x ^11 - 6*x^8 - 2*x^5 + x^2)*(-x^3 + 1)^(2/3) - 4^(1/3)*(x^18 + 42*x^15 - 41 7*x^12 + 812*x^9 - 417*x^6 + 42*x^3 + 1))/(x^18 - 102*x^15 + 447*x^12 - 62 8*x^9 + 447*x^6 - 102*x^3 + 1)) + 1/72*4^(2/3)*log(-(12*(-x^3 + 1)^(2/3)*x ^2 - 3*4^(2/3)*(x^4 - x)*(-x^3 + 1)^(1/3) + 4^(1/3)*(x^6 + 2*x^3 + 1))/(x^ 6 + 2*x^3 + 1)) - 1/144*4^(2/3)*log((24*4^(1/3)*(x^8 - 4*x^5 + x^2)*(-x^3 + 1)^(2/3) + 4^(2/3)*(x^12 - 32*x^9 + 78*x^6 - 32*x^3 + 1) + 12*(x^10 - 11 *x^7 + 11*x^4 - x)*(-x^3 + 1)^(1/3))/(x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 1)) - 1/2*(-x^3 + 1)^(1/3)*x
\[ \int \frac {x^6}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^{6}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \] Input:
integrate(x**6/(-x**3+1)**(2/3)/(x**3+1),x)
Output:
Integral(x**6/((-(x - 1)*(x**2 + x + 1))**(2/3)*(x + 1)*(x**2 - x + 1)), x )
\[ \int \frac {x^6}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {x^{6}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(x^6/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="maxima")
Output:
integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(2/3)), x)
\[ \int \frac {x^6}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {x^{6}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(x^6/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="giac")
Output:
integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(2/3)), x)
Timed out. \[ \int \frac {x^6}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^6}{{\left (1-x^3\right )}^{2/3}\,\left (x^3+1\right )} \,d x \] Input:
int(x^6/((1 - x^3)^(2/3)*(x^3 + 1)),x)
Output:
int(x^6/((1 - x^3)^(2/3)*(x^3 + 1)), x)
\[ \int \frac {x^6}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^{6}}{\left (-x^{3}+1\right )^{\frac {2}{3}} x^{3}+\left (-x^{3}+1\right )^{\frac {2}{3}}}d x \] Input:
int(x^6/(-x^3+1)^(2/3)/(x^3+1),x)
Output:
int(x**6/(( - x**3 + 1)**(2/3)*x**3 + ( - x**3 + 1)**(2/3)),x)