Integrand size = 18, antiderivative size = 54 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\sqrt {-1+x^6} \left (6-8 x^3-3 x^6+8 x^9\right )}{72 x^{12}}+\frac {1}{12} \arctan \left (\frac {1+x^3}{\sqrt {-1+x^6}}\right ) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1489, 849, 821, 272, 43, 65, 209} \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=-\frac {1}{24} \arctan \left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1}}{24 x^6}-\frac {\left (x^6-1\right )^{3/2}}{12 x^{12}}+\frac {\left (x^6-1\right )^{3/2}}{9 x^9} \]
[In]
[Out]
Rule 43
Rule 65
Rule 209
Rule 272
Rule 821
Rule 849
Rule 1489
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(-1+x) \sqrt {-1+x^2}}{x^5} \, dx,x,x^3\right ) \\ & = -\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {1}{12} \text {Subst}\left (\int \frac {(4-x) \sqrt {-1+x^2}}{x^4} \, dx,x,x^3\right ) \\ & = -\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{12} \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{x^3} \, dx,x,x^3\right ) \\ & = -\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{24} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {-1+x^6}}{24 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{48} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {-1+x^6}}{24 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{24} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right ) \\ & = \frac {\sqrt {-1+x^6}}{24 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{24} \arctan \left (\sqrt {-1+x^6}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\sqrt {-1+x^6} \left (6-8 x^3-3 x^6+8 x^9\right )}{72 x^{12}}+\frac {1}{12} \arctan \left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right ) \]
[In]
[Out]
Time = 1.99 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {3 \arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{12}+8 \left (x^{9}-\frac {3}{8} x^{6}-x^{3}+\frac {3}{4}\right ) \sqrt {x^{6}-1}}{72 x^{12}}\) | \(44\) |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (8 x^{9}-3 x^{6}-8 x^{3}+6\right )}{72 x^{12}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{24}\) | \(60\) |
risch | \(\frac {8 x^{15}-3 x^{12}-16 x^{9}+9 x^{6}+8 x^{3}-6}{72 x^{12} \sqrt {x^{6}-1}}-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{48 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(101\) |
meijerg | \(\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-\frac {\sqrt {\pi }\, \left (x^{12}-8 x^{6}+8\right )}{8 x^{12}}+\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right ) \sqrt {-x^{6}+1}}{8 x^{12}}-\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )}{2}+\frac {\left (\frac {1}{2}-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{x^{12}}-\frac {\sqrt {\pi }}{x^{6}}\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-x^{6}+1\right )^{\frac {3}{2}}}{9 \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, x^{9}}\) | \(153\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=-\frac {6 \, x^{12} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - 8 \, x^{12} - {\left (8 \, x^{9} - 3 \, x^{6} - 8 \, x^{3} + 6\right )} \sqrt {x^{6} - 1}}{72 \, x^{12}} \]
[In]
[Out]
Time = 2.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\begin {cases} \frac {\left (x^{6} - 1\right )^{\frac {3}{2}}}{3 x^{9}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} - \frac {\begin {cases} \frac {\operatorname {acos}{\left (\frac {1}{x^{3}} \right )}}{8} - \frac {\left (-1 + \frac {2}{x^{6}}\right ) \sqrt {1 - \frac {1}{x^{6}}}}{8 x^{3}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=-\frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}} - \sqrt {x^{6} - 1}}{24 \, {\left (2 \, x^{6} + {\left (x^{6} - 1\right )}^{2} - 1\right )}} + \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} - \frac {1}{24} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]
[In]
[Out]
Exception generated. \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\text {Exception raised: NotImplementedError} \]
[In]
[Out]
Time = 6.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\frac {\sqrt {x^6-1}}{24}-\frac {{\left (x^6-1\right )}^{3/2}}{24}}{x^{12}}-\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{24}+\frac {{\left (x^6-1\right )}^{3/2}}{9\,x^9} \]
[In]
[Out]