Integrand size = 25, antiderivative size = 94 \[ \int \frac {\left (-1+x^4\right )^{3/4} \left (4+x^4\right )}{x^8 \left (-4+x^4\right )} \, dx=\frac {\left (-1+x^4\right )^{3/4} \left (6+x^4\right )}{42 x^7}-\frac {3^{3/4} \arctan \left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}-\frac {3^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {594, 597, 12, 385, 218, 212, 209} \[ \int \frac {\left (-1+x^4\right )^{3/4} \left (4+x^4\right )}{x^8 \left (-4+x^4\right )} \, dx=-\frac {3^{3/4} \arctan \left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{8 \sqrt {2}}-\frac {3^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{8 \sqrt {2}}+\frac {\left (x^4-1\right )^{3/4}}{7 x^7}+\frac {\left (x^4-1\right )^{3/4}}{42 x^3} \]
[In]
[Out]
Rule 12
Rule 209
Rule 212
Rule 218
Rule 385
Rule 594
Rule 597
Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^4\right )^{3/4}}{7 x^7}-\frac {1}{28} \int \frac {8-44 x^4}{x^4 \left (-4+x^4\right ) \sqrt [4]{-1+x^4}} \, dx \\ & = \frac {\left (-1+x^4\right )^{3/4}}{7 x^7}+\frac {\left (-1+x^4\right )^{3/4}}{42 x^3}+\frac {1}{336} \int \frac {504}{\left (-4+x^4\right ) \sqrt [4]{-1+x^4}} \, dx \\ & = \frac {\left (-1+x^4\right )^{3/4}}{7 x^7}+\frac {\left (-1+x^4\right )^{3/4}}{42 x^3}+\frac {3}{2} \int \frac {1}{\left (-4+x^4\right ) \sqrt [4]{-1+x^4}} \, dx \\ & = \frac {\left (-1+x^4\right )^{3/4}}{7 x^7}+\frac {\left (-1+x^4\right )^{3/4}}{42 x^3}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{-4+3 x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {\left (-1+x^4\right )^{3/4}}{7 x^7}+\frac {\left (-1+x^4\right )^{3/4}}{42 x^3}-\frac {3}{8} \text {Subst}\left (\int \frac {1}{2-\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {3}{8} \text {Subst}\left (\int \frac {1}{2+\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {\left (-1+x^4\right )^{3/4}}{7 x^7}+\frac {\left (-1+x^4\right )^{3/4}}{42 x^3}-\frac {3^{3/4} \arctan \left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}-\frac {3^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^4\right )^{3/4} \left (4+x^4\right )}{x^8 \left (-4+x^4\right )} \, dx=\frac {\left (-1+x^4\right )^{3/4} \left (6+x^4\right )}{42 x^7}-\frac {3^{3/4} \arctan \left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}-\frac {3^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}} \]
[In]
[Out]
Time = 14.16 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.17
method | result | size |
pseudoelliptic | \(\frac {42 \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (x^{4}-1\right )^{\frac {1}{4}}}{3 x}\right ) \sqrt {2}\, 3^{\frac {3}{4}} x^{7}-21 \ln \left (\frac {-\sqrt {2}\, 3^{\frac {1}{4}} x -2 \left (x^{4}-1\right )^{\frac {1}{4}}}{\sqrt {2}\, 3^{\frac {1}{4}} x -2 \left (x^{4}-1\right )^{\frac {1}{4}}}\right ) \sqrt {2}\, 3^{\frac {3}{4}} x^{7}+16 \left (x^{4}-1\right )^{\frac {3}{4}} x^{4}+96 \left (x^{4}-1\right )^{\frac {3}{4}}}{672 x^{7}}\) | \(110\) |
trager | \(\frac {\left (x^{4}-1\right )^{\frac {3}{4}} \left (x^{4}+6\right )}{42 x^{7}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2}\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2} \sqrt {x^{4}-1}\, x^{2}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}-21 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2}\right ) x^{4}-72 \left (x^{4}-1\right )^{\frac {3}{4}} x +12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2}\right )}{\left (x^{2}-2\right ) \left (x^{2}+2\right )}\right )}{32}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{3} \sqrt {x^{4}-1}\, x^{2}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+21 \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right ) x^{4}+72 \left (x^{4}-1\right )^{\frac {3}{4}} x -12 \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )}{\left (x^{2}-2\right ) \left (x^{2}+2\right )}\right )}{32}\) | \(242\) |
risch | \(\frac {x^{8}+5 x^{4}-6}{42 x^{7} \left (x^{4}-1\right )^{\frac {1}{4}}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2}\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2} \sqrt {x^{4}-1}\, x^{2}-6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}-21 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2}\right ) x^{4}+72 \left (x^{4}-1\right )^{\frac {3}{4}} x +12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2}\right )}{\left (x^{2}-2\right ) \left (x^{2}+2\right )}\right )}{32}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{3} \sqrt {x^{4}-1}\, x^{2}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )^{2} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+21 \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right ) x^{4}+72 \left (x^{4}-1\right )^{\frac {3}{4}} x -12 \operatorname {RootOf}\left (\textit {\_Z}^{4}-108\right )}{\left (x^{2}-2\right ) \left (x^{2}+2\right )}\right )}{32}\) | \(247\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 3.75 (sec) , antiderivative size = 340, normalized size of antiderivative = 3.62 \[ \int \frac {\left (-1+x^4\right )^{3/4} \left (4+x^4\right )}{x^8 \left (-4+x^4\right )} \, dx=-\frac {21 \cdot 27^{\frac {1}{4}} \sqrt {2} x^{7} \log \left (\frac {2 \, {\left (4 \cdot 27^{\frac {3}{4}} \sqrt {2} \sqrt {x^{4} - 1} x^{2} + 36 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 3 \cdot 27^{\frac {1}{4}} \sqrt {2} {\left (7 \, x^{4} - 4\right )} + 72 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - 4}\right ) - 21 \cdot 27^{\frac {1}{4}} \sqrt {2} x^{7} \log \left (-\frac {2 \, {\left (4 \cdot 27^{\frac {3}{4}} \sqrt {2} \sqrt {x^{4} - 1} x^{2} - 36 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 3 \cdot 27^{\frac {1}{4}} \sqrt {2} {\left (7 \, x^{4} - 4\right )} - 72 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - 4}\right ) + 21 i \cdot 27^{\frac {1}{4}} \sqrt {2} x^{7} \log \left (-\frac {2 \, {\left (4 i \cdot 27^{\frac {3}{4}} \sqrt {2} \sqrt {x^{4} - 1} x^{2} + 36 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 3 \cdot 27^{\frac {1}{4}} \sqrt {2} {\left (-7 i \, x^{4} + 4 i\right )} - 72 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - 4}\right ) - 21 i \cdot 27^{\frac {1}{4}} \sqrt {2} x^{7} \log \left (-\frac {2 \, {\left (-4 i \cdot 27^{\frac {3}{4}} \sqrt {2} \sqrt {x^{4} - 1} x^{2} + 36 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 3 \cdot 27^{\frac {1}{4}} \sqrt {2} {\left (7 i \, x^{4} - 4 i\right )} - 72 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - 4}\right ) - 32 \, {\left (x^{4} + 6\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}}}{1344 \, x^{7}} \]
[In]
[Out]
\[ \int \frac {\left (-1+x^4\right )^{3/4} \left (4+x^4\right )}{x^8 \left (-4+x^4\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{4}} \left (x^{2} - 2 x + 2\right ) \left (x^{2} + 2 x + 2\right )}{x^{8} \left (x^{2} - 2\right ) \left (x^{2} + 2\right )}\, dx \]
[In]
[Out]
\[ \int \frac {\left (-1+x^4\right )^{3/4} \left (4+x^4\right )}{x^8 \left (-4+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 4\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}}}{{\left (x^{4} - 4\right )} x^{8}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (-1+x^4\right )^{3/4} \left (4+x^4\right )}{x^8 \left (-4+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 4\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}}}{{\left (x^{4} - 4\right )} x^{8}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (-1+x^4\right )^{3/4} \left (4+x^4\right )}{x^8 \left (-4+x^4\right )} \, dx=\int \frac {{\left (x^4-1\right )}^{3/4}\,\left (x^4+4\right )}{x^8\,\left (x^4-4\right )} \,d x \]
[In]
[Out]