Integrand size = 30, antiderivative size = 61 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\frac {1}{6} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3+x^6}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+\text {$\#$1}^3}\&\right ] \]
Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\frac {1}{6} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3+x^6}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+\text {$\#$1}^3}\&\right ] \]
RootSum[2 - 2*#1^3 + #1^6 & , (-(Log[x]*#1^2) + Log[(-1 + x^3 + x^6)^(1/3) - x*#1]*#1^2)/(-1 + #1^3) & ]/6
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.
\(\displaystyle \int \frac {\left (x^6+1\right ) \left (x^6+x^3-1\right )^{2/3}}{x^{12}-x^6+1} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {\left (1-i \sqrt {3}\right ) \left (x^6+x^3-1\right )^{2/3}}{2 x^6-i \sqrt {3}-1}+\frac {\left (1+i \sqrt {3}\right ) \left (x^6+x^3-1\right )^{2/3}}{2 x^6+i \sqrt {3}-1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (1+i \sqrt {3}\right ) \int \frac {\left (x^6+x^3-1\right )^{2/3}}{\sqrt {1-i \sqrt {3}}-\sqrt {2} x^3}dx}{2 \sqrt {1-i \sqrt {3}}}-\frac {\left (1-i \sqrt {3}\right ) \int \frac {\left (x^6+x^3-1\right )^{2/3}}{\sqrt {1+i \sqrt {3}}-\sqrt {2} x^3}dx}{2 \sqrt {1+i \sqrt {3}}}-\frac {\left (1+i \sqrt {3}\right ) \int \frac {\left (x^6+x^3-1\right )^{2/3}}{\sqrt {2} x^3+\sqrt {1-i \sqrt {3}}}dx}{2 \sqrt {1-i \sqrt {3}}}-\frac {\left (1-i \sqrt {3}\right ) \int \frac {\left (x^6+x^3-1\right )^{2/3}}{\sqrt {2} x^3+\sqrt {1+i \sqrt {3}}}dx}{2 \sqrt {1+i \sqrt {3}}}\) |
3.9.8.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 21.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{3}+1\right )}{\sum }\textit {\_R} \ln \left (\frac {4 \left (x^{6}+x^{3}-1\right )^{\frac {1}{3}} \textit {\_R}^{2} x -2 \textit {\_R} \,x^{2}+\left (x^{6}+x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{3}\) | \(54\) |
trager | \(\text {Expression too large to display}\) | \(2936\) |
1/3*sum(_R*ln((4*(x^6+x^3-1)^(1/3)*_R^2*x-2*_R*x^2+(x^6+x^3-1)^(2/3))/x^2) ,_R=RootOf(4*_Z^3+1))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 19.06 (sec) , antiderivative size = 583, normalized size of antiderivative = 9.56 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\frac {1}{18} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{6}} {\left (6 \cdot 4^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} {\left (x^{31} - 14 \, x^{28} - 75 \, x^{25} + 82 \, x^{22} + 293 \, x^{19} - 132 \, x^{16} - 293 \, x^{13} + 82 \, x^{10} + 75 \, x^{7} - 14 \, x^{4} - x\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} - 12 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} {\left (3 \, x^{32} + 49 \, x^{29} - 51 \, x^{26} - 344 \, x^{23} + 99 \, x^{20} + 609 \, x^{17} - 99 \, x^{14} - 344 \, x^{11} + 51 \, x^{8} + 49 \, x^{5} - 3 \, x^{2}\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (x^{36} + 54 \, x^{33} - 129 \, x^{30} - 846 \, x^{27} + 258 \, x^{24} + 2502 \, x^{21} - 169 \, x^{18} - 2502 \, x^{15} + 258 \, x^{12} + 846 \, x^{9} - 129 \, x^{6} - 54 \, x^{3} + 1\right )}\right )}}{6 \, {\left (x^{36} - 54 \, x^{33} - 489 \, x^{30} + 270 \, x^{27} + 2922 \, x^{24} - 54 \, x^{21} - 4921 \, x^{18} + 54 \, x^{15} + 2922 \, x^{12} - 270 \, x^{9} - 489 \, x^{6} + 54 \, x^{3} + 1\right )}}\right ) + \frac {1}{36} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{7} + 4 \, x^{4} - x\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{12} + 18 \, x^{9} + 17 \, x^{6} - 18 \, x^{3} + 1\right )} - 6 \, {\left (3 \, x^{8} + 5 \, x^{5} - 3 \, x^{2}\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {1}{3}}}{x^{12} - x^{6} + 1}\right ) - \frac {1}{72} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (3 \, x^{20} - 5 \, x^{17} - 12 \, x^{14} + 11 \, x^{11} + 12 \, x^{8} - 5 \, x^{5} - 3 \, x^{2}\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {1}{3}} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{24} - 18 \, x^{21} - 2 \, x^{18} + 72 \, x^{15} + 3 \, x^{12} - 72 \, x^{9} - 2 \, x^{6} + 18 \, x^{3} + 1\right )} - 12 \, {\left (x^{19} - 5 \, x^{16} - 2 \, x^{13} + 11 \, x^{10} + 2 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}}}{x^{24} - 2 \, x^{18} + 3 \, x^{12} - 2 \, x^{6} + 1}\right ) \]
1/18*4^(1/6)*sqrt(3)*(-1)^(1/3)*arctan(1/6*4^(1/6)*(6*4^(2/3)*sqrt(3)*(-1) ^(2/3)*(x^31 - 14*x^28 - 75*x^25 + 82*x^22 + 293*x^19 - 132*x^16 - 293*x^1 3 + 82*x^10 + 75*x^7 - 14*x^4 - x)*(x^6 + x^3 - 1)^(2/3) - 12*sqrt(3)*(-1) ^(1/3)*(3*x^32 + 49*x^29 - 51*x^26 - 344*x^23 + 99*x^20 + 609*x^17 - 99*x^ 14 - 344*x^11 + 51*x^8 + 49*x^5 - 3*x^2)*(x^6 + x^3 - 1)^(1/3) + 4^(1/3)*s qrt(3)*(x^36 + 54*x^33 - 129*x^30 - 846*x^27 + 258*x^24 + 2502*x^21 - 169* x^18 - 2502*x^15 + 258*x^12 + 846*x^9 - 129*x^6 - 54*x^3 + 1))/(x^36 - 54* x^33 - 489*x^30 + 270*x^27 + 2922*x^24 - 54*x^21 - 4921*x^18 + 54*x^15 + 2 922*x^12 - 270*x^9 - 489*x^6 + 54*x^3 + 1)) + 1/36*4^(2/3)*(-1)^(1/3)*log( (3*4^(2/3)*(-1)^(1/3)*(x^7 + 4*x^4 - x)*(x^6 + x^3 - 1)^(2/3) - 4^(1/3)*(- 1)^(2/3)*(x^12 + 18*x^9 + 17*x^6 - 18*x^3 + 1) - 6*(3*x^8 + 5*x^5 - 3*x^2) *(x^6 + x^3 - 1)^(1/3))/(x^12 - x^6 + 1)) - 1/72*4^(2/3)*(-1)^(1/3)*log((6 *4^(1/3)*(-1)^(2/3)*(3*x^20 - 5*x^17 - 12*x^14 + 11*x^11 + 12*x^8 - 5*x^5 - 3*x^2)*(x^6 + x^3 - 1)^(1/3) - 4^(2/3)*(-1)^(1/3)*(x^24 - 18*x^21 - 2*x^ 18 + 72*x^15 + 3*x^12 - 72*x^9 - 2*x^6 + 18*x^3 + 1) - 12*(x^19 - 5*x^16 - 2*x^13 + 11*x^10 + 2*x^7 - 5*x^4 - x)*(x^6 + x^3 - 1)^(2/3))/(x^24 - 2*x^ 18 + 3*x^12 - 2*x^6 + 1))
Timed out. \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\text {Timed out} \]
Not integrable
Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\int { \frac {{\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{6} + 1\right )}}{x^{12} - x^{6} + 1} \,d x } \]
Not integrable
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\int { \frac {{\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{6} + 1\right )}}{x^{12} - x^{6} + 1} \,d x } \]
Not integrable
Time = 6.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\int \frac {\left (x^6+1\right )\,{\left (x^6+x^3-1\right )}^{2/3}}{x^{12}-x^6+1} \,d x \]