Integrand size = 36, antiderivative size = 121 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{4} \text {RootSum}\left [1-5 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+3 \log (x) \text {$\#$1}^4-3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-5 \text {$\#$1}+4 \text {$\#$1}^5}\&\right ] \]
[Out]
Time = 0.37 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.91, number of steps used = 16, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6860, 246, 218, 212, 209, 385} \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{2047-439 \sqrt {17}} \arctan \left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}}-\frac {\sqrt [4]{2047+439 \sqrt {17}} \arctan \left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}}+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{2047-439 \sqrt {17}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}}-\frac {\sqrt [4]{2047+439 \sqrt {17}} \text {arctanh}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {17}} \]
[In]
[Out]
Rule 209
Rule 212
Rule 218
Rule 246
Rule 385
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt [4]{-1+x^4}}+\frac {3-x^4}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )}\right ) \, dx \\ & = \int \frac {1}{\sqrt [4]{-1+x^4}} \, dx+\int \frac {3-x^4}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx \\ & = \int \left (\frac {-1+\frac {11}{\sqrt {17}}}{\sqrt [4]{-1+x^4} \left (-1-\sqrt {17}+4 x^4\right )}+\frac {-1-\frac {11}{\sqrt {17}}}{\sqrt [4]{-1+x^4} \left (-1+\sqrt {17}+4 x^4\right )}\right ) \, dx+\text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{17} \left (-17+11 \sqrt {17}\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (-1-\sqrt {17}+4 x^4\right )} \, dx-\frac {1}{17} \left (17+11 \sqrt {17}\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (-1+\sqrt {17}+4 x^4\right )} \, dx \\ & = \frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{17} \left (-17+11 \sqrt {17}\right ) \text {Subst}\left (\int \frac {1}{-1-\sqrt {17}-\left (3-\sqrt {17}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{17} \left (17+11 \sqrt {17}\right ) \text {Subst}\left (\int \frac {1}{-1+\sqrt {17}-\left (3+\sqrt {17}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{34} \left (37-\sqrt {17}\right )} \text {Subst}\left (\int \frac {1}{\sqrt {5+\sqrt {17}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{34} \left (37-\sqrt {17}\right )} \text {Subst}\left (\int \frac {1}{\sqrt {5+\sqrt {17}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{34} \left (37+\sqrt {17}\right )} \text {Subst}\left (\int \frac {1}{\sqrt {5-\sqrt {17}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{34} \left (37+\sqrt {17}\right )} \text {Subst}\left (\int \frac {1}{\sqrt {5-\sqrt {17}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{2047-439 \sqrt {17}} \arctan \left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}}-\frac {\sqrt [4]{2047+439 \sqrt {17}} \arctan \left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}}+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\sqrt [4]{2047-439 \sqrt {17}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}}-\frac {\sqrt [4]{2047+439 \sqrt {17}} \text {arctanh}\left (\frac {\sqrt [4]{5+\sqrt {17}} x}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {17}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\frac {1}{4} \left (2 \left (\arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )\right )-\text {RootSum}\left [1-5 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+3 \log (x) \text {$\#$1}^4-3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-5 \text {$\#$1}+4 \text {$\#$1}^5}\&\right ]\right ) \]
[In]
[Out]
Time = 7.68 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-5 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (3 \textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )}{4 \textit {\_R}^{5}-5 \textit {\_R}}\right )}{4}+\frac {\ln \left (\frac {x +\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )}{4}-\frac {\ln \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}-x}{x}\right )}{4}-\frac {\arctan \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )}{2}\) | \(104\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.29 (sec) , antiderivative size = 514, normalized size of antiderivative = 4.25 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\frac {1}{136} \, \sqrt {17} \sqrt {-\sqrt {439 \, \sqrt {17} + 2047}} \log \left (\frac {{\left (59 \, \sqrt {17} x - 155 \, x\right )} \sqrt {439 \, \sqrt {17} + 2047} \sqrt {-\sqrt {439 \, \sqrt {17} + 2047}} + 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{136} \, \sqrt {17} \sqrt {-\sqrt {439 \, \sqrt {17} + 2047}} \log \left (-\frac {{\left (59 \, \sqrt {17} x - 155 \, x\right )} \sqrt {439 \, \sqrt {17} + 2047} \sqrt {-\sqrt {439 \, \sqrt {17} + 2047}} - 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{136} \, \sqrt {17} \sqrt {-\sqrt {-439 \, \sqrt {17} + 2047}} \log \left (\frac {{\left (59 \, \sqrt {17} x + 155 \, x\right )} \sqrt {-439 \, \sqrt {17} + 2047} \sqrt {-\sqrt {-439 \, \sqrt {17} + 2047}} + 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{136} \, \sqrt {17} \sqrt {-\sqrt {-439 \, \sqrt {17} + 2047}} \log \left (-\frac {{\left (59 \, \sqrt {17} x + 155 \, x\right )} \sqrt {-439 \, \sqrt {17} + 2047} \sqrt {-\sqrt {-439 \, \sqrt {17} + 2047}} - 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{136} \, \sqrt {17} {\left (439 \, \sqrt {17} + 2047\right )}^{\frac {1}{4}} \log \left (\frac {{\left (59 \, \sqrt {17} x - 155 \, x\right )} {\left (439 \, \sqrt {17} + 2047\right )}^{\frac {3}{4}} + 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{136} \, \sqrt {17} {\left (439 \, \sqrt {17} + 2047\right )}^{\frac {1}{4}} \log \left (-\frac {{\left (59 \, \sqrt {17} x - 155 \, x\right )} {\left (439 \, \sqrt {17} + 2047\right )}^{\frac {3}{4}} - 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{136} \, \sqrt {17} {\left (-439 \, \sqrt {17} + 2047\right )}^{\frac {1}{4}} \log \left (\frac {{\left (59 \, \sqrt {17} x + 155 \, x\right )} {\left (-439 \, \sqrt {17} + 2047\right )}^{\frac {3}{4}} + 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{136} \, \sqrt {17} {\left (-439 \, \sqrt {17} + 2047\right )}^{\frac {1}{4}} \log \left (-\frac {{\left (59 \, \sqrt {17} x + 155 \, x\right )} {\left (-439 \, \sqrt {17} + 2047\right )}^{\frac {3}{4}} - 35152 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \log \left (\frac {x + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \log \left (-\frac {x - {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) \]
[In]
[Out]
Not integrable
Time = 134.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.31 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\int \frac {2 x^{8} - 2 x^{4} + 1}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (2 x^{8} - x^{4} - 2\right )}\, dx \]
[In]
[Out]
Not integrable
Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.30 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\int { \frac {2 \, x^{8} - 2 \, x^{4} + 1}{{\left (2 \, x^{8} - x^{4} - 2\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
[In]
[Out]
Not integrable
Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.30 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\int { \frac {2 \, x^{8} - 2 \, x^{4} + 1}{{\left (2 \, x^{8} - x^{4} - 2\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
[In]
[Out]
Not integrable
Time = 6.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.29 \[ \int \frac {1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-2-x^4+2 x^8\right )} \, dx=\int -\frac {2\,x^8-2\,x^4+1}{{\left (x^4-1\right )}^{1/4}\,\left (-2\,x^8+x^4+2\right )} \,d x \]
[In]
[Out]