Integrand size = 29, antiderivative size = 126 \[ \int \frac {\left (1+2 x^6\right ) \sqrt [3]{x+x^3-x^7}}{\left (-1+x^6\right )^2} \, dx=-\frac {x \sqrt [3]{x+x^3-x^7}}{2 \left (-1+x^6\right )}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3-x^7}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x+\sqrt [3]{x+x^3-x^7}\right )+\frac {1}{12} \log \left (x^2+x \sqrt [3]{x+x^3-x^7}+\left (x+x^3-x^7\right )^{2/3}\right ) \] Output:
-x*(-x^7+x^3+x)^(1/3)/(2*x^6-2)-1/6*arctan(3^(1/2)*x/(x+2*(-x^7+x^3+x)^(1/ 3)))*3^(1/2)-1/6*ln(-x+(-x^7+x^3+x)^(1/3))+1/12*ln(x^2+x*(-x^7+x^3+x)^(1/3 )+(-x^7+x^3+x)^(2/3))
Time = 3.03 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.83 \[ \int \frac {\left (1+2 x^6\right ) \sqrt [3]{x+x^3-x^7}}{\left (-1+x^6\right )^2} \, dx=\frac {\sqrt [3]{x+x^3-x^7} \left (-6 x^{4/3} \sqrt [3]{-1-x^2+x^6}+2 \sqrt {3} \left (-1+x^6\right ) \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{-1-x^2+x^6}}\right )+2 \left (-1+x^6\right ) \log \left (x^{2/3}+\sqrt [3]{-1-x^2+x^6}\right )+\log \left (x^{4/3}-x^{2/3} \sqrt [3]{-1-x^2+x^6}+\left (-1-x^2+x^6\right )^{2/3}\right )-x^6 \log \left (x^{4/3}-x^{2/3} \sqrt [3]{-1-x^2+x^6}+\left (-1-x^2+x^6\right )^{2/3}\right )\right )}{12 \sqrt [3]{x} \left (-1+x^6\right ) \sqrt [3]{-1-x^2+x^6}} \] Input:
Integrate[((1 + 2*x^6)*(x + x^3 - x^7)^(1/3))/(-1 + x^6)^2,x]
Output:
((x + x^3 - x^7)^(1/3)*(-6*x^(4/3)*(-1 - x^2 + x^6)^(1/3) + 2*Sqrt[3]*(-1 + x^6)*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) - 2*(-1 - x^2 + x^6)^(1/3))] + 2* (-1 + x^6)*Log[x^(2/3) + (-1 - x^2 + x^6)^(1/3)] + Log[x^(4/3) - x^(2/3)*( -1 - x^2 + x^6)^(1/3) + (-1 - x^2 + x^6)^(2/3)] - x^6*Log[x^(4/3) - x^(2/3 )*(-1 - x^2 + x^6)^(1/3) + (-1 - x^2 + x^6)^(2/3)]))/(12*x^(1/3)*(-1 + x^6 )*(-1 - x^2 + x^6)^(1/3))
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.
| \(\displaystyle \int \frac {\left (2 x^6+1\right ) \sqrt [3]{-x^7+x^3+x}}{\left (x^6-1\right )^2} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{-x^7+x^3+x} \int \frac {\sqrt [3]{x} \sqrt [3]{-x^6+x^2+1} \left (2 x^6+1\right )}{\left (1-x^6\right )^2}dx}{\sqrt [3]{x} \sqrt [3]{-x^6+x^2+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{-x^7+x^3+x} \int \frac {x \sqrt [3]{-x^6+x^2+1} \left (2 x^6+1\right )}{\left (1-x^6\right )^2}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{-x^6+x^2+1}}\) |
| \(\Big \downarrow \) 7283 |
| \(\displaystyle \frac {3 \sqrt [3]{-x^7+x^3+x} \int \frac {x^{2/3} \sqrt [3]{-x^3+x+1} \left (2 x^3+1\right )}{\left (1-x^3\right )^2}dx^{2/3}}{2 \sqrt [3]{x} \sqrt [3]{-x^6+x^2+1}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 \sqrt [3]{-x^7+x^3+x} \int \left (\frac {\sqrt [3]{-x^3+x+1} \left (x^{2/3}+1\right )}{27 \left (2 x^{2/3}+1\right )}-\frac {\sqrt [3]{-x^3+x+1}}{27 \left (x^{2/3}-1\right )}-\frac {x^{2/3} \sqrt [3]{-x^3+x+1}}{3 \left (x^2+x+1\right )}+\frac {\sqrt [3]{-x^3+x+1}}{27 \left (x^{2/3}-1\right )^2}-\frac {\sqrt [3]{-x^3+x+1}}{9 \left (2 x^{2/3}+1\right )^2}+\frac {x^{2/3} (x+1) \sqrt [3]{-x^3+x+1}}{\left (x^2+x+1\right )^2}\right )dx^{2/3}}{2 \sqrt [3]{x} \sqrt [3]{-x^6+x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{-x^7+x^3+x} \left (\frac {4}{27} \int \frac {\sqrt [3]{-x^3+x+1}}{\left (-2 x^{2/3}+i \sqrt {3}-1\right )^2}dx^{2/3}-\frac {4 i \int \frac {\sqrt [3]{-x^3+x+1}}{-2 x^{2/3}+i \sqrt {3}-1}dx^{2/3}}{27 \sqrt {3}}+\frac {1}{27} \int \frac {\sqrt [3]{-x^3+x+1}}{\left (x^{2/3}-1\right )^2}dx^{2/3}-\frac {1}{27} \int \frac {\sqrt [3]{-x^3+x+1}}{x^{2/3}-1}dx^{2/3}+\frac {1}{81} \left (3-i \sqrt {3}\right ) \int \frac {\sqrt [3]{-x^3+x+1}}{2 x^{2/3}-i \sqrt {3}+1}dx^{2/3}+\frac {4}{27} \int \frac {\sqrt [3]{-x^3+x+1}}{\left (2 x^{2/3}+i \sqrt {3}+1\right )^2}dx^{2/3}+\frac {1}{81} \left (3+i \sqrt {3}\right ) \int \frac {\sqrt [3]{-x^3+x+1}}{2 x^{2/3}+i \sqrt {3}+1}dx^{2/3}-\frac {4 i \int \frac {\sqrt [3]{-x^3+x+1}}{2 x^{2/3}+i \sqrt {3}+1}dx^{2/3}}{27 \sqrt {3}}+\frac {2}{3} \left (1-i \sqrt {3}\right ) \int \frac {x^{2/3} \sqrt [3]{-x^3+x+1}}{\left (-2 x+i \sqrt {3}-1\right )^2}dx^{2/3}-\frac {4}{3} \int \frac {x^{2/3} \sqrt [3]{-x^3+x+1}}{\left (-2 x+i \sqrt {3}-1\right )^2}dx^{2/3}+\frac {2}{3} \left (1+i \sqrt {3}\right ) \int \frac {x^{2/3} \sqrt [3]{-x^3+x+1}}{\left (2 x+i \sqrt {3}+1\right )^2}dx^{2/3}-\frac {4}{3} \int \frac {x^{2/3} \sqrt [3]{-x^3+x+1}}{\left (2 x+i \sqrt {3}+1\right )^2}dx^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{-x^6+x^2+1}}\) |
Input:
Int[((1 + 2*x^6)*(x + x^3 - x^7)^(1/3))/(-1 + x^6)^2,x]
Output:
$Aborted
Time = 33.65 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.50
| method | result | size |
| pseudoelliptic | \(\frac {x \left (-6 {\left (-x \left (x^{6}-x^{2}-1\right )\right )}^{\frac {1}{3}} x +\left (x^{6}-1\right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\left (2 {\left (-x \left (x^{6}-x^{2}-1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\ln \left (\frac {{\left (-x \left (x^{6}-x^{2}-1\right )\right )}^{\frac {2}{3}}+{\left (-x \left (x^{6}-x^{2}-1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )-2 \ln \left (\frac {{\left (-x \left (x^{6}-x^{2}-1\right )\right )}^{\frac {1}{3}}-x}{x}\right )\right )\right )}{12 \left ({\left (-x \left (x^{6}-x^{2}-1\right )\right )}^{\frac {2}{3}}+{\left (-x \left (x^{6}-x^{2}-1\right )\right )}^{\frac {1}{3}} x +x^{2}\right ) \left (-{\left (-x \left (x^{6}-x^{2}-1\right )\right )}^{\frac {1}{3}}+x \right )}\) | \(189\) |
| trager | \(-\frac {x \left (-x^{7}+x^{3}+x \right )^{\frac {1}{3}}}{2 \left (x^{6}-1\right )}-\frac {\ln \left (\frac {23084281021492945478344725970280579664 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )^{2} x^{6}+779765626827203872050827736565976550 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) x^{6}-2180370336718022168439941441690525 x^{6}-363577426088513891283929434031919129708 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )^{2} x^{2}+1726084667130388450019245844753992776 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) \left (-x^{7}+x^{3}+x \right )^{\frac {2}{3}}-929663874243506986438828114966559238 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) \left (-x^{7}+x^{3}+x \right )^{\frac {1}{3}} x +165424249675324598017279185640923948 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) x^{2}-2459428238739436472060391838535871 \left (-x^{7}+x^{3}+x \right )^{\frac {2}{3}}-2106933314515559427461422565575221 x \left (-x^{7}+x^{3}+x \right )^{\frac {1}{3}}-23084281021492945478344725970280579664 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )^{2}+2021797948593075101644309336840305 x^{2}-779765626827203872050827736565976550 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )+2180370336718022168439941441690525}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{6}+63 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) \ln \left (-\frac {426823916648008786317628300861745184 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )^{2} x^{6}+43285196651552399719158942411926172 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) x^{6}-2338942724842969235235573546540745 x^{6}-6722476687206138384502645738572486648 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )^{2} x^{2}+1726084667130388450019245844753992776 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) \left (-x^{7}+x^{3}+x \right )^{\frac {2}{3}}-796420792886881463580417729787433538 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) \left (-x^{7}+x^{3}+x \right )^{\frac {1}{3}} x -911879544383173287008926935763986522 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right ) x^{2}-2106933314515559427461422565575221 \left (-x^{7}+x^{3}+x \right )^{\frac {2}{3}}-2459428238739436472060391838535871 x \left (-x^{7}+x^{3}+x \right )^{\frac {1}{3}}-426823916648008786317628300861745184 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )^{2}+4519313061560991403675514988231270 x^{2}-43285196651552399719158942411926172 \operatorname {RootOf}\left (142884 \textit {\_Z}^{2}-378 \textit {\_Z} +1\right )+2338942724842969235235573546540745}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )\) | \(476\) |
| risch | \(\text {Expression too large to display}\) | \(1138\) |
Input:
int((2*x^6+1)*(-x^7+x^3+x)^(1/3)/(x^6-1)^2,x,method=_RETURNVERBOSE)
Output:
1/12*x*(-6*(-x*(x^6-x^2-1))^(1/3)*x+(x^6-1)*(2*3^(1/2)*arctan(1/3*(2*(-x*( x^6-x^2-1))^(1/3)+x)*3^(1/2)/x)+ln(((-x*(x^6-x^2-1))^(2/3)+(-x*(x^6-x^2-1) )^(1/3)*x+x^2)/x^2)-2*ln(((-x*(x^6-x^2-1))^(1/3)-x)/x)))/((-x*(x^6-x^2-1)) ^(2/3)+(-x*(x^6-x^2-1))^(1/3)*x+x^2)/(-(-x*(x^6-x^2-1))^(1/3)+x)
Time = 0.50 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.20 \[ \int \frac {\left (1+2 x^6\right ) \sqrt [3]{x+x^3-x^7}}{\left (-1+x^6\right )^2} \, dx=\frac {2 \, \sqrt {3} {\left (x^{6} - 1\right )} \arctan \left (-\frac {4 \, \sqrt {3} {\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (x^{6} - x^{2} - 1\right )} - 2 \, \sqrt {3} {\left (-x^{7} + x^{3} + x\right )}^{\frac {2}{3}}}{x^{6} - 9 \, x^{2} - 1}\right ) - {\left (x^{6} - 1\right )} \log \left (\frac {x^{6} - 3 \, {\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (-x^{7} + x^{3} + x\right )}^{\frac {2}{3}} - 1}{x^{6} - 1}\right ) - 6 \, {\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} x}{12 \, {\left (x^{6} - 1\right )}} \] Input:
integrate((2*x^6+1)*(-x^7+x^3+x)^(1/3)/(x^6-1)^2,x, algorithm="fricas")
Output:
1/12*(2*sqrt(3)*(x^6 - 1)*arctan(-(4*sqrt(3)*(-x^7 + x^3 + x)^(1/3)*x - sq rt(3)*(x^6 - x^2 - 1) - 2*sqrt(3)*(-x^7 + x^3 + x)^(2/3))/(x^6 - 9*x^2 - 1 )) - (x^6 - 1)*log((x^6 - 3*(-x^7 + x^3 + x)^(1/3)*x + 3*(-x^7 + x^3 + x)^ (2/3) - 1)/(x^6 - 1)) - 6*(-x^7 + x^3 + x)^(1/3)*x)/(x^6 - 1)
\[ \int \frac {\left (1+2 x^6\right ) \sqrt [3]{x+x^3-x^7}}{\left (-1+x^6\right )^2} \, dx=\int \frac {\sqrt [3]{- x \left (x^{6} - x^{2} - 1\right )} \left (2 x^{6} + 1\right )}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} - x + 1\right )^{2} \left (x^{2} + x + 1\right )^{2}}\, dx \] Input:
integrate((2*x**6+1)*(-x**7+x**3+x)**(1/3)/(x**6-1)**2,x)
Output:
Integral((-x*(x**6 - x**2 - 1))**(1/3)*(2*x**6 + 1)/((x - 1)**2*(x + 1)**2 *(x**2 - x + 1)**2*(x**2 + x + 1)**2), x)
\[ \int \frac {\left (1+2 x^6\right ) \sqrt [3]{x+x^3-x^7}}{\left (-1+x^6\right )^2} \, dx=\int { \frac {{\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} + 1\right )}}{{\left (x^{6} - 1\right )}^{2}} \,d x } \] Input:
integrate((2*x^6+1)*(-x^7+x^3+x)^(1/3)/(x^6-1)^2,x, algorithm="maxima")
Output:
integrate((-x^7 + x^3 + x)^(1/3)*(2*x^6 + 1)/(x^6 - 1)^2, x)
\[ \int \frac {\left (1+2 x^6\right ) \sqrt [3]{x+x^3-x^7}}{\left (-1+x^6\right )^2} \, dx=\int { \frac {{\left (-x^{7} + x^{3} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} + 1\right )}}{{\left (x^{6} - 1\right )}^{2}} \,d x } \] Input:
integrate((2*x^6+1)*(-x^7+x^3+x)^(1/3)/(x^6-1)^2,x, algorithm="giac")
Output:
integrate((-x^7 + x^3 + x)^(1/3)*(2*x^6 + 1)/(x^6 - 1)^2, x)
Timed out. \[ \int \frac {\left (1+2 x^6\right ) \sqrt [3]{x+x^3-x^7}}{\left (-1+x^6\right )^2} \, dx=\int \frac {\left (2\,x^6+1\right )\,{\left (-x^7+x^3+x\right )}^{1/3}}{{\left (x^6-1\right )}^2} \,d x \] Input:
int(((2*x^6 + 1)*(x + x^3 - x^7)^(1/3))/(x^6 - 1)^2,x)
Output:
int(((2*x^6 + 1)*(x + x^3 - x^7)^(1/3))/(x^6 - 1)^2, x)
\[ \int \frac {\left (1+2 x^6\right ) \sqrt [3]{x+x^3-x^7}}{\left (-1+x^6\right )^2} \, dx=\frac {-3 x^{\frac {4}{3}} \left (-x^{6}+x^{2}+1\right )^{\frac {1}{3}}+4 \left (\int \frac {x^{\frac {19}{3}} \left (-x^{6}+x^{2}+1\right )^{\frac {1}{3}}}{x^{12}-x^{8}-2 x^{6}+x^{2}+1}d x \right ) x^{6}-4 \left (\int \frac {x^{\frac {19}{3}} \left (-x^{6}+x^{2}+1\right )^{\frac {1}{3}}}{x^{12}-x^{8}-2 x^{6}+x^{2}+1}d x \right )+2 \left (\int \frac {x^{\frac {1}{3}} \left (-x^{6}+x^{2}+1\right )^{\frac {1}{3}}}{x^{12}-x^{8}-2 x^{6}+x^{2}+1}d x \right ) x^{6}-2 \left (\int \frac {x^{\frac {1}{3}} \left (-x^{6}+x^{2}+1\right )^{\frac {1}{3}}}{x^{12}-x^{8}-2 x^{6}+x^{2}+1}d x \right )}{6 x^{6}-6} \] Input:
int((2*x^6+1)*(-x^7+x^3+x)^(1/3)/(x^6-1)^2,x)
Output:
( - 3*x**(1/3)*( - x**6 + x**2 + 1)**(1/3)*x + 4*int((x**(1/3)*( - x**6 + x**2 + 1)**(1/3)*x**6)/(x**12 - x**8 - 2*x**6 + x**2 + 1),x)*x**6 - 4*int( (x**(1/3)*( - x**6 + x**2 + 1)**(1/3)*x**6)/(x**12 - x**8 - 2*x**6 + x**2 + 1),x) + 2*int((x**(1/3)*( - x**6 + x**2 + 1)**(1/3))/(x**12 - x**8 - 2*x **6 + x**2 + 1),x)*x**6 - 2*int((x**(1/3)*( - x**6 + x**2 + 1)**(1/3))/(x* *12 - x**8 - 2*x**6 + x**2 + 1),x))/(6*(x**6 - 1))