Integrand size = 22, antiderivative size = 166 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=-\frac {1}{5} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1+2 x+x^2+\sqrt {1+x^4}}\right )+\frac {4}{5} \text {RootSum}\left [4-4 \text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log (x)+2 \log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}-\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]
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\[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=\int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {1+x^4}}-\frac {2}{\sqrt {1+x^4} \left (1+x^5\right )}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx\right )+\int \frac {1}{\sqrt {1+x^4}} \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-2 \int \left (\frac {1}{5 (1+x) \sqrt {1+x^4}}+\frac {4-3 x+2 x^2-x^3}{5 \sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )}\right ) \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {2}{5} \int \frac {1}{(1+x) \sqrt {1+x^4}} \, dx-\frac {2}{5} \int \frac {4-3 x+2 x^2-x^3}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {2}{5} \int \frac {1}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx+\frac {2}{5} \int \frac {x}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx-\frac {2}{5} \int \left (\frac {4}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )}-\frac {3 x}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )}+\frac {2 x^2}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )}-\frac {x^3}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )}\right ) \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {1}{5} \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{5} \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{5} \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {2}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx+\frac {6}{5} \int \frac {x}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {8}{5} \int \frac {1}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx \\ & = \frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \sqrt {1+x^4}}-\frac {1}{5} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )-\frac {1}{5} \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\frac {-1-x^2}{\sqrt {1+x^4}}\right )+\frac {2}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx+\frac {6}{5} \int \frac {x}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {8}{5} \int \frac {1}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {1+x^2}{\sqrt {2} \sqrt {1+x^4}}\right )}{5 \sqrt {2}}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \sqrt {1+x^4}}+\frac {2}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx+\frac {6}{5} \int \frac {x}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {8}{5} \int \frac {1}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx \\ \end{align*}
Time = 1.57 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=-\frac {1}{5} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1+2 x+x^2+\sqrt {1+x^4}}\right )+\frac {4}{5} \text {RootSum}\left [4-4 \text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log (x)+2 \log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}-\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]
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Time = 4.84 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{10}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {5}\, x^{2}+x^{2}+\sqrt {5}-4 x +1}{\sqrt {-2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right ) \sqrt {2+2 \sqrt {5}}}{5}-\frac {\arctan \left (\frac {\sqrt {5}\, x^{2}-x^{2}+\sqrt {5}+4 x -1}{\sqrt {2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right ) \sqrt {-2+2 \sqrt {5}}}{5}\) | \(121\) |
pseudoelliptic | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{10}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {5}\, x^{2}+x^{2}+\sqrt {5}-4 x +1}{\sqrt {-2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right ) \sqrt {2+2 \sqrt {5}}}{5}-\frac {\arctan \left (\frac {\sqrt {5}\, x^{2}-x^{2}+\sqrt {5}+4 x -1}{\sqrt {2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right ) \sqrt {-2+2 \sqrt {5}}}{5}\) | \(121\) |
elliptic | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4 \sqrt {\left (x^{2}-1\right )^{2}+2 x^{2}}}\right )}{10}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (\left (-x^{2}+\sqrt {x^{4}+1}\right )^{2}+\left (\textit {\_R}^{2}-\textit {\_R} -1\right ) \left (-x^{2}+\sqrt {x^{4}+1}\right )+\textit {\_R}^{2}-\textit {\_R} \right )\right )}{5}+\frac {\left (-\frac {\ln \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}-\frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}+\frac {4 \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}+\frac {\ln \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}\right ) \sqrt {2}}{2}\) | \(206\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+\sqrt {x^{4}+1}}{\left (1+x \right )^{2}}\right )}{10}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) \ln \left (\frac {625 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{4} x +50 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x^{2}+50 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x -150 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} \sqrt {x^{4}+1}+50 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right )+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right ) x^{2}-16 \sqrt {x^{4}+1}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2}-4\right )}{4 x^{2}-4 x +25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x +4}\right )}{5}-\operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right ) \ln \left (\frac {625 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{5} x -50 x^{2} \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{3}-250 x \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{3}-50 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{3}+30 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} \sqrt {x^{4}+1}+12 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right ) x^{2}+24 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right ) x +12 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )-8 \sqrt {x^{4}+1}}{25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}-16\right )^{2} x -4 x^{2}-4}\right )\) | \(559\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.36 (sec) , antiderivative size = 556, normalized size of antiderivative = 3.35 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=\frac {1}{10} \, \sqrt {2} \sqrt {\sqrt {5} + 1} \log \left (-\frac {2 \, {\left (\sqrt {2} {\left (4 \, x^{4} + 7 \, x^{3} - 14 \, x^{2} - \sqrt {5} {\left (2 \, x^{4} + 3 \, x^{3} - 6 \, x^{2} + 3 \, x + 2\right )} + 7 \, x + 4\right )} \sqrt {\sqrt {5} + 1} + 2 \, \sqrt {x^{4} + 1} {\left (7 \, x^{2} - \sqrt {5} {\left (3 \, x^{2} - 5 \, x + 3\right )} - 11 \, x + 7\right )}\right )}}{x^{4} - x^{3} + x^{2} - x + 1}\right ) - \frac {1}{10} \, \sqrt {2} \sqrt {\sqrt {5} + 1} \log \left (\frac {2 \, {\left (\sqrt {2} {\left (4 \, x^{4} + 7 \, x^{3} - 14 \, x^{2} - \sqrt {5} {\left (2 \, x^{4} + 3 \, x^{3} - 6 \, x^{2} + 3 \, x + 2\right )} + 7 \, x + 4\right )} \sqrt {\sqrt {5} + 1} - 2 \, \sqrt {x^{4} + 1} {\left (7 \, x^{2} - \sqrt {5} {\left (3 \, x^{2} - 5 \, x + 3\right )} - 11 \, x + 7\right )}\right )}}{x^{4} - x^{3} + x^{2} - x + 1}\right ) + \frac {1}{20} \, \sqrt {2} \log \left (-\frac {3 \, x^{4} + 4 \, x^{3} + 2 \, \sqrt {2} \sqrt {x^{4} + 1} {\left (x^{2} + x + 1\right )} + 6 \, x^{2} + 4 \, x + 3}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) + \frac {1}{20} \, \sqrt {-8 \, \sqrt {5} + 8} \log \left (-\frac {4 \, \sqrt {x^{4} + 1} {\left (7 \, x^{2} + \sqrt {5} {\left (3 \, x^{2} - 5 \, x + 3\right )} - 11 \, x + 7\right )} + {\left (4 \, x^{4} + 7 \, x^{3} - 14 \, x^{2} + \sqrt {5} {\left (2 \, x^{4} + 3 \, x^{3} - 6 \, x^{2} + 3 \, x + 2\right )} + 7 \, x + 4\right )} \sqrt {-8 \, \sqrt {5} + 8}}{x^{4} - x^{3} + x^{2} - x + 1}\right ) - \frac {1}{20} \, \sqrt {-8 \, \sqrt {5} + 8} \log \left (-\frac {4 \, \sqrt {x^{4} + 1} {\left (7 \, x^{2} + \sqrt {5} {\left (3 \, x^{2} - 5 \, x + 3\right )} - 11 \, x + 7\right )} - {\left (4 \, x^{4} + 7 \, x^{3} - 14 \, x^{2} + \sqrt {5} {\left (2 \, x^{4} + 3 \, x^{3} - 6 \, x^{2} + 3 \, x + 2\right )} + 7 \, x + 4\right )} \sqrt {-8 \, \sqrt {5} + 8}}{x^{4} - x^{3} + x^{2} - x + 1}\right ) \]
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Not integrable
Time = 24.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.25 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )}{\left (x + 1\right ) \sqrt {x^{4} + 1} \left (x^{4} - x^{3} + x^{2} - x + 1\right )}\, dx \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=\int { \frac {x^{5} - 1}{{\left (x^{5} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=\int { \frac {x^{5} - 1}{{\left (x^{5} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
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Not integrable
Time = 6.52 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.13 \[ \int \frac {-1+x^5}{\sqrt {1+x^4} \left (1+x^5\right )} \, dx=\int \frac {x^5-1}{\sqrt {x^4+1}\,\left (x^5+1\right )} \,d x \]
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