Integrand size = 30, antiderivative size = 61 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\frac {1}{6} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3+x^6}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+\text {$\#$1}^3}\&\right ] \] Output:
Unintegrable
Time = 0.00 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\frac {1}{6} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3+x^6}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+\text {$\#$1}^3}\&\right ] \] Input:
Integrate[((1 + x^6)*(-1 + x^3 + x^6)^(2/3))/(1 - x^6 + x^12),x]
Output:
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 |
\(\displaystyle -\frac {\left (1+i \sqrt {3}\right ) \int \frac {\left (x^6+x^3-1\right )^{2/3}}{\sqrt {1-i \sqrt {3}}-\sqrt {2} x^3}dx}{2 \sqrt {1-i \sqrt {3}}}-\frac {\left (1-i \sqrt {3}\right ) \int \frac {\left (x^6+x^3-1\right )^{2/3}}{\sqrt {1+i \sqrt {3}}-\sqrt {2} x^3}dx}{2 \sqrt {1+i \sqrt {3}}}-\frac {\left (1+i \sqrt {3}\right ) \int \frac {\left (x^6+x^3-1\right )^{2/3}}{\sqrt {2} x^3+\sqrt {1-i \sqrt {3}}}dx}{2 \sqrt {1-i \sqrt {3}}}-\frac {\left (1-i \sqrt {3}\right ) \int \frac {\left (x^6+x^3-1\right )^{2/3}}{\sqrt {2} x^3+\sqrt {1+i \sqrt {3}}}dx}{2 \sqrt {1+i \sqrt {3}}}\) |
Input:
Int[((1 + x^6)*(-1 + x^3 + x^6)^(2/3))/(1 - x^6 + x^12),x]
Output:
$Aborted
Time = 17.26 (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 | \(\text {Expression too large to display}\) | \(2936\) |
Input:
int((x^6+1)*(x^6+x^3-1)^(2/3)/(x^12-x^6+1),x,method=_RETURNVERBOSE)
Output:
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 = 18.61 (sec) , antiderivative size = 549, normalized size of antiderivative = 9.00 \[ \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} \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^{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}} + 4^{\frac {1}{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 )} + 12 \, {\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}}\right )}}{2 \, {\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}} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{7} + 4 \, x^{4} - x\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{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}} \log \left (\frac {4^{\frac {2}{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 )} + 6 \cdot 4^{\frac {1}{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}} - 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 ) \] Input:
integrate((x^6+1)*(x^6+x^3-1)^(2/3)/(x^12-x^6+1),x, algorithm="fricas")
Output:
-1/6*4^(1/6)*sqrt(1/3)*arctan(1/2*4^(1/6)*sqrt(1/3)*(6*4^(2/3)*(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)*(x^6 + x^3 - 1)^(2/3) + 4^(1/3)*(x^36 + 54*x^33 - 129*x^3 0 - 846*x^27 + 258*x^24 + 2502*x^21 - 169*x^18 - 2502*x^15 + 258*x^12 + 84 6*x^9 - 129*x^6 - 54*x^3 + 1) + 12*(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))/(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)) - 1/36*4^(2/3)*log(-(3*4^(2/3)*(x^7 + 4*x^4 - x)*(x^6 + x^3 - 1)^(2/3) + 4^ (1/3)*(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)*log((4^(2/3)*(x^24 - 18*x^21 - 2*x^18 + 72*x^15 + 3*x^12 - 72*x^9 - 2*x^6 + 18*x^3 + 1) + 6*4^ (1/3)*(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) - 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. \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\text {Timed out} \] Input:
integrate((x**6+1)*(x**6+x**3-1)**(2/3)/(x**12-x**6+1),x)
Output:
Timed out
Not integrable
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\int { \frac {{\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{6} + 1\right )}}{x^{12} - x^{6} + 1} \,d x } \] Input:
integrate((x^6+1)*(x^6+x^3-1)^(2/3)/(x^12-x^6+1),x, algorithm="maxima")
Output:
integrate((x^6 + x^3 - 1)^(2/3)*(x^6 + 1)/(x^12 - x^6 + 1), x)
Not integrable
Time = 0.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\int { \frac {{\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{6} + 1\right )}}{x^{12} - x^{6} + 1} \,d x } \] Input:
integrate((x^6+1)*(x^6+x^3-1)^(2/3)/(x^12-x^6+1),x, algorithm="giac")
Output:
integrate((x^6 + x^3 - 1)^(2/3)*(x^6 + 1)/(x^12 - x^6 + 1), x)
Not integrable
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\int \frac {\left (x^6+1\right )\,{\left (x^6+x^3-1\right )}^{2/3}}{x^{12}-x^6+1} \,d x \] Input:
int(((x^6 + 1)*(x^3 + x^6 - 1)^(2/3))/(x^12 - x^6 + 1),x)
Output:
int(((x^6 + 1)*(x^3 + x^6 - 1)^(2/3))/(x^12 - x^6 + 1), x)
Not integrable
Time = 0.17 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\int \frac {\left (x^{6}+x^{3}-1\right )^{\frac {2}{3}}}{x^{12}-x^{6}+1}d x +\int \frac {\left (x^{6}+x^{3}-1\right )^{\frac {2}{3}} x^{6}}{x^{12}-x^{6}+1}d x \] Input:
int((x^6+1)*(x^6+x^3-1)^(2/3)/(x^12-x^6+1),x)
Output:
int((x**6 + x**3 - 1)**(2/3)/(x**12 - x**6 + 1),x) + int(((x**6 + x**3 - 1 )**(2/3)*x**6)/(x**12 - x**6 + 1),x)