Integrand size = 24, antiderivative size = 67 \[ \int \frac {-1+2 x^4}{\sqrt [4]{-1+x^4} \left (-1+x^8\right )} \, dx=-\frac {x}{2 \sqrt [4]{-1+x^4}}+\frac {3 \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt [4]{2}}+\frac {3 \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt [4]{2}} \]
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Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1469, 541, 12, 385, 218, 212, 209} \[ \int \frac {-1+2 x^4}{\sqrt [4]{-1+x^4} \left (-1+x^8\right )} \, dx=\frac {3 \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt [4]{2}}+\frac {3 \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt [4]{2}}-\frac {x}{2 \sqrt [4]{x^4-1}} \]
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Rule 12
Rule 209
Rule 212
Rule 218
Rule 385
Rule 541
Rule 1469
Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+2 x^4}{\left (-1+x^4\right )^{5/4} \left (1+x^4\right )} \, dx \\ & = -\frac {x}{2 \sqrt [4]{-1+x^4}}+\frac {1}{2} \int \frac {3}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx \\ & = -\frac {x}{2 \sqrt [4]{-1+x^4}}+\frac {3}{2} \int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx \\ & = -\frac {x}{2 \sqrt [4]{-1+x^4}}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = -\frac {x}{2 \sqrt [4]{-1+x^4}}+\frac {3}{4} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {3}{4} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = -\frac {x}{2 \sqrt [4]{-1+x^4}}+\frac {3 \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt [4]{2}}+\frac {3 \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt [4]{2}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x^4}{\sqrt [4]{-1+x^4} \left (-1+x^8\right )} \, dx=-\frac {x}{2 \sqrt [4]{-1+x^4}}+\frac {3 \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt [4]{2}}+\frac {3 \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt [4]{2}} \]
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Time = 3.74 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.27
method | result | size |
pseudoelliptic | \(\frac {-6 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}-1\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {3}{4}} \left (x^{4}-1\right )^{\frac {1}{4}}+3 \ln \left (\frac {2^{\frac {1}{4}} x +\left (x^{4}-1\right )^{\frac {1}{4}}}{-2^{\frac {1}{4}} x +\left (x^{4}-1\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}} \left (x^{4}-1\right )^{\frac {1}{4}}-8 x}{16 \left (x^{4}-1\right )^{\frac {1}{4}}}\) | \(85\) |
trager | \(-\frac {x}{2 \left (x^{4}-1\right )^{\frac {1}{4}}}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}-4 \left (x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{x^{4}+1}\right )}{16}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{2}-2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+3 x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )-4 \left (x^{4}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{x^{4}+1}\right )}{16}\) | \(218\) |
risch | \(-\frac {x}{2 \left (x^{4}-1\right )^{\frac {1}{4}}}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}-4 \left (x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{x^{4}+1}\right )}{16}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{2}-2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+3 x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )-4 \left (x^{4}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{x^{4}+1}\right )}{16}\) | \(218\) |
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Result contains complex when optimal does not.
Time = 3.19 (sec) , antiderivative size = 313, normalized size of antiderivative = 4.67 \[ \int \frac {-1+2 x^4}{\sqrt [4]{-1+x^4} \left (-1+x^8\right )} \, dx=\frac {3 \cdot 2^{\frac {3}{4}} {\left (x^{4} - 1\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - 1} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{4} - 1\right )} + 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) - 3 \cdot 2^{\frac {3}{4}} {\left (-i \, x^{4} + i\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - 1} x^{2} - 2^{\frac {3}{4}} {\left (3 i \, x^{4} - i\right )} - 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) - 3 \cdot 2^{\frac {3}{4}} {\left (i \, x^{4} - i\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - 1} x^{2} - 2^{\frac {3}{4}} {\left (-3 i \, x^{4} + i\right )} - 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) - 3 \cdot 2^{\frac {3}{4}} {\left (x^{4} - 1\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - 1} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{4} - 1\right )} + 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) - 16 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x}{32 \, {\left (x^{4} - 1\right )}} \]
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\[ \int \frac {-1+2 x^4}{\sqrt [4]{-1+x^4} \left (-1+x^8\right )} \, dx=\int \frac {2 x^{4} - 1}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \]
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\[ \int \frac {-1+2 x^4}{\sqrt [4]{-1+x^4} \left (-1+x^8\right )} \, dx=\int { \frac {2 \, x^{4} - 1}{{\left (x^{8} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {-1+2 x^4}{\sqrt [4]{-1+x^4} \left (-1+x^8\right )} \, dx=\int { \frac {2 \, x^{4} - 1}{{\left (x^{8} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {-1+2 x^4}{\sqrt [4]{-1+x^4} \left (-1+x^8\right )} \, dx=\int \frac {2\,x^4-1}{{\left (x^4-1\right )}^{1/4}\,\left (x^8-1\right )} \,d x \]
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