Integrand size = 31, antiderivative size = 64 \[ \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^7 \left (-1+4 x^6\right )} \, dx=\frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{6 x^6}-\frac {1}{6} \arctan \left (\sqrt {-1+x^6}\right )-\frac {\arctan \left (\frac {2 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {587, 186, 43, 65, 209, 52} \[ \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^7 \left (-1+4 x^6\right )} \, dx=-\frac {1}{6} \arctan \left (\sqrt {x^6-1}\right )-\frac {\arctan \left (\frac {2 \sqrt {x^6-1}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\sqrt {x^6-1}}{6 x^6}+\frac {\sqrt {x^6-1}}{3} \]
[In]
[Out]
Rule 43
Rule 52
Rule 65
Rule 186
Rule 209
Rule 587
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {-1+x} (-1+2 x)^2}{x^2 (-1+4 x)} \, dx,x,x^6\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \left (-\frac {\sqrt {-1+x}}{x^2}+\frac {4 \sqrt {-1+x}}{-1+4 x}\right ) \, dx,x,x^6\right ) \\ & = -\left (\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\right )\right )+\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{-1+4 x} \, dx,x,x^6\right ) \\ & = \frac {1}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{12} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} (-1+4 x)} \, dx,x,x^6\right ) \\ & = \frac {1}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )-\text {Subst}\left (\int \frac {1}{3+4 x^2} \, dx,x,\sqrt {-1+x^6}\right ) \\ & = \frac {1}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{6} \arctan \left (\sqrt {-1+x^6}\right )-\frac {\arctan \left (\frac {2 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^7 \left (-1+4 x^6\right )} \, dx=\frac {1}{6} \left (\frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{x^6}-\arctan \left (\sqrt {-1+x^6}\right )-\sqrt {3} \arctan \left (\frac {2 \sqrt {-1+x^6}}{\sqrt {3}}\right )\right ) \]
[In]
[Out]
Time = 3.00 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.44
method | result | size |
pseudoelliptic | \(\frac {\sqrt {3}\, \arctan \left (\frac {\left (x^{3}+2\right ) \sqrt {3}}{3 \sqrt {x^{6}-1}}\right ) x^{6}-\sqrt {3}\, \arctan \left (\frac {\left (x^{3}-2\right ) \sqrt {3}}{3 \sqrt {x^{6}-1}}\right ) x^{6}+2 \arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{6}+4 \sqrt {x^{6}-1}\, x^{6}+2 \sqrt {x^{6}-1}}{12 x^{6}}\) | \(92\) |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (2 x^{6}+1\right )}{6 x^{6}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{6}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )+12 \sqrt {x^{6}-1}}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{12}\) | \(107\) |
risch | \(\frac {\sqrt {x^{6}-1}}{6 x^{6}}+\frac {\sqrt {x^{6}-1}}{3}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{6}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{6}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )-12 \sqrt {x^{6}-1}}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{12}\) | \(108\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^7 \left (-1+4 x^6\right )} \, dx=-\frac {\sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + x^{6} \arctan \left (\sqrt {x^{6} - 1}\right ) - {\left (2 \, x^{6} + 1\right )} \sqrt {x^{6} - 1}}{6 \, x^{6}} \]
[In]
[Out]
Time = 16.35 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^7 \left (-1+4 x^6\right )} \, dx=\frac {\sqrt {x^{6} - 1}}{3} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x^{6} - 1}}{3} \right )}}{6} - \frac {\operatorname {atan}{\left (\sqrt {x^{6} - 1} \right )}}{6} + \frac {\sqrt {x^{6} - 1}}{6 x^{6}} \]
[In]
[Out]
\[ \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^7 \left (-1+4 x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} - 1\right )}^{2} \sqrt {x^{6} - 1}}{{\left (4 \, x^{6} - 1\right )} x^{7}} \,d x } \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^7 \left (-1+4 x^6\right )} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + \frac {1}{3} \, \sqrt {x^{6} - 1} + \frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} - \frac {1}{6} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]
[In]
[Out]
Time = 5.72 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^7 \left (-1+4 x^6\right )} \, dx=\frac {\sqrt {x^6-1}}{3}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,\sqrt {x^6-1}}{3}\right )}{6}-\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{6}+\frac {\sqrt {x^6-1}}{6\,x^6} \]
[In]
[Out]