Integrand size = 22, antiderivative size = 294 \[ \int \frac {1}{x^3 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {\sqrt [3]{1-x^3}}{2 x^2}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}+\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}-\frac {\log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {\log \left (2 \sqrt [3]{2}+\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}} \] Output:
-1/2*(-x^3+1)^(1/3)/x^2-1/6*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)+1/1 2*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 = 11.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.41 \[ \int \frac {1}{x^3 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {\sqrt [3]{1-x^3} \left (-1+\frac {4 x^3 \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},1,\frac {4}{3},x^3,-x^3\right )}{\left (1+x^3\right ) \left (-4 \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},1,\frac {4}{3},x^3,-x^3\right )+x^3 \left (3 \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{3},2,\frac {7}{3},x^3,-x^3\right )+\operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},x^3,-x^3\right )\right )\right )}\right )}{2 x^2} \] Input:
Integrate[1/(x^3*(1 - x^3)^(2/3)*(1 + x^3)),x]
Output:
((1 - x^3)^(1/3)*(-1 + (4*x^3*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*x^ 2)
Time = 0.73 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {980, 25, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (1-x^3\right )^{2/3} \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 980 |
| \(\displaystyle \frac {1}{2} \int -\frac {\sqrt [3]{1-x^3}}{x^3+1}dx-\frac {\sqrt [3]{1-x^3}}{2 x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} \int \frac {\sqrt [3]{1-x^3}}{x^3+1}dx-\frac {\sqrt [3]{1-x^3}}{2 x^2}\) |
| \(\Big \downarrow \) 927 |
| \(\displaystyle \frac {9}{2} \int \frac {1-x}{\sqrt [3]{1-x^3} \left (4-\frac {(1-x)^3}{1-x^3}\right ) \left (\frac {2 (1-x)^3}{1-x^3}+1\right )}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {\sqrt [3]{1-x^3}}{2 x^2}\) |
| \(\Big \downarrow \) 982 |
| \(\displaystyle \frac {9}{2} \left (\frac {1}{9} \int \frac {1-x}{\sqrt [3]{1-x^3} \left (4-\frac {(1-x)^3}{1-x^3}\right )}d\frac {1-x}{\sqrt [3]{1-x^3}}+\frac {2}{9} \int \frac {1-x}{\sqrt [3]{1-x^3} \left (\frac {2 (1-x)^3}{1-x^3}+1\right )}d\frac {1-x}{\sqrt [3]{1-x^3}}\right )-\frac {\sqrt [3]{1-x^3}}{2 x^2}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\int \frac {1}{\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}\right )+\frac {1}{9} \left (\frac {\int \frac {1}{2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\int \frac {2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}\right )\right )-\frac {\sqrt [3]{1-x^3}}{2 x^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\int \frac {2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {\sqrt [3]{1-x^3}}{2 x^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}+\frac {\int -\frac {\sqrt [3]{2} \left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}-\frac {1}{2} \int \frac {2^{2/3} \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {\sqrt [3]{1-x^3}}{2 x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {\int \frac {\sqrt [3]{2} \left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{2 \sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}-\frac {1}{2} \int \frac {2^{2/3} \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {\sqrt [3]{1-x^3}}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {3 \int \frac {1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}-\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {\sqrt [3]{1-x^3}}{2 x^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {3 \int \frac {1}{-\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}-3}d\left (1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{\sqrt [3]{2}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {-3 \int \frac {1}{-\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}-3}d\left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {\sqrt [3]{1-x^3}}{2 x^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}d\frac {1-x}{\sqrt [3]{1-x^3}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )-\frac {\int \frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}}d\frac {1-x}{\sqrt [3]{1-x^3}}}{\sqrt [3]{2}}}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {\sqrt [3]{1-x^3}}{2 x^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {9}{2} \left (\frac {2}{9} \left (\frac {\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}\right )+\frac {1}{9} \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{3\ 2^{2/3}}-\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\right )\right )-\frac {\sqrt [3]{1-x^3}}{2 x^2}\) |
Input:
Int[1/(x^3*(1 - x^3)^(2/3)*(1 + x^3)),x]
Output:
-1/2*(1 - x^3)^(1/3)/x^2 + (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*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Simp[1/(a*c*e^n*(m + 1)) Int[(e*x)^(m + n)*( a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && 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 = 73.19 (sec) , antiderivative size = 695, normalized size of antiderivative = 2.36
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(695\) |
| trager | \(\text {Expression too large to display}\) | \(1150\) |
Input:
int(1/x^3/(-x^3+1)^(2/3)/(x^3+1),x,method=_RETURNVERBOSE)
Output:
1/2*(x^3-1)/x^2/(-x^3+1)^(2/3)+(1/12*RootOf(_Z^3+2)*ln((6*RootOf(RootOf(_Z ^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*RootOf(_Z^3+2)^3*x^3+36*RootOf(RootOf( _Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)^2*RootOf(_Z^3+2)^2*x^3+RootOf(_Z^3+2 )*x^6+6*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x^6+9*(x^6-2*x ^3+1)^(2/3)*RootOf(_Z^3+2)^2*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9 *_Z^2)*x-6*(x^6-2*x^3+1)^(1/3)*RootOf(_Z^3+2)^2*x^2-6*RootOf(_Z^3+2)*x^3-3 6*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x^3-6*(x^6-2*x^3+1)^ (2/3)*x+RootOf(_Z^3+2)+6*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^ 2))/(1+x)^2/(x^2-x+1)^2)+1/4*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9 *_Z^2)*ln(-(12*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*RootOf( _Z^3+2)^3*x^3+18*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)^2*Roo tOf(_Z^3+2)^2*x^3-2*RootOf(_Z^3+2)*x^6-3*RootOf(RootOf(_Z^3+2)^2+3*_Z*Root Of(_Z^3+2)+9*_Z^2)*x^6+9*(x^6-2*x^3+1)^(2/3)*RootOf(_Z^3+2)^2*RootOf(RootO f(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x-6*(x^6-2*x^3+1)^(1/3)*RootOf(_Z^ 3+2)^2*x^2-18*(x^6-2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3 +2)+9*_Z^2)*RootOf(_Z^3+2)*x^2+4*RootOf(_Z^3+2)*x^3+6*RootOf(RootOf(_Z^3+2 )^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x^3-2*RootOf(_Z^3+2)-3*RootOf(RootOf(_Z^3+ 2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2))/(1+x)^2/(x^2-x+1)^2))/(-x^3+1)^(2/3)*((x ^3-1)^2)^(1/3)
Time = 1.39 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^3 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {12 \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} x^{2} \arctan \left (-\frac {4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (6 \cdot 4^{\frac {2}{3}} {\left (x^{16} - 33 \, x^{13} + 110 \, x^{10} - 110 \, x^{7} + 33 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 48 \, {\left (x^{14} - 2 \, x^{11} - 6 \, x^{8} - 2 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} {\left (x^{18} + 42 \, x^{15} - 417 \, x^{12} + 812 \, x^{9} - 417 \, x^{6} + 42 \, x^{3} + 1\right )}\right )}}{2 \, {\left (x^{18} - 102 \, x^{15} + 447 \, x^{12} - 628 \, x^{9} + 447 \, x^{6} - 102 \, x^{3} + 1\right )}}\right ) + 2 \cdot 4^{\frac {2}{3}} x^{2} \log \left (-\frac {12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{2} - 3 \cdot 4^{\frac {2}{3}} {\left (x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} {\left (x^{6} + 2 \, x^{3} + 1\right )}}{x^{6} + 2 \, x^{3} + 1}\right ) - 4^{\frac {2}{3}} x^{2} \log \left (\frac {24 \cdot 4^{\frac {1}{3}} {\left (x^{8} - 4 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {2}{3}} {\left (x^{12} - 32 \, x^{9} + 78 \, x^{6} - 32 \, x^{3} + 1\right )} + 12 \, {\left (x^{10} - 11 \, x^{7} + 11 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) + 72 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{144 \, x^{2}} \] Input:
integrate(1/x^3/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="fricas")
Output:
-1/144*(12*4^(1/6)*sqrt(1/3)*x^2*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 - 417*x^12 + 812*x^9 - 417*x^6 + 42*x^3 + 1))/(x^18 - 102*x^15 + 447 *x^12 - 628*x^9 + 447*x^6 - 102*x^3 + 1)) + 2*4^(2/3)*x^2*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)) - 4^(2/3)*x^2*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)) + 72*(-x^3 + 1)^(1/3))/x^2
\[ \int \frac {1}{x^3 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {1}{x^{3} \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \] Input:
integrate(1/x**3/(-x**3+1)**(2/3)/(x**3+1),x)
Output:
Integral(1/(x**3*(-(x - 1)*(x**2 + x + 1))**(2/3)*(x + 1)*(x**2 - x + 1)), x)
\[ \int \frac {1}{x^3 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{3}} \,d x } \] Input:
integrate(1/x^3/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="maxima")
Output:
integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^3), x)
\[ \int \frac {1}{x^3 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{3}} \,d x } \] Input:
integrate(1/x^3/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="giac")
Output:
integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^3), x)
Timed out. \[ \int \frac {1}{x^3 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {1}{x^3\,{\left (1-x^3\right )}^{2/3}\,\left (x^3+1\right )} \,d x \] Input:
int(1/(x^3*(1 - x^3)^(2/3)*(x^3 + 1)),x)
Output:
int(1/(x^3*(1 - x^3)^(2/3)*(x^3 + 1)), x)
\[ \int \frac {1}{x^3 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {1}{\left (-x^{3}+1\right )^{\frac {2}{3}} x^{6}+\left (-x^{3}+1\right )^{\frac {2}{3}} x^{3}}d x \] Input:
int(1/x^3/(-x^3+1)^(2/3)/(x^3+1),x)
Output:
int(1/(( - x**3 + 1)**(2/3)*x**6 + ( - x**3 + 1)**(2/3)*x**3),x)