Integrand size = 38, antiderivative size = 100 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx=\frac {3 \left (1+x^4\right )^{2/3} \left (1+5 x^3+x^4\right )}{5 x^5}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^4}}\right )+2 \log \left (-x+\sqrt [3]{1+x^4}\right )-\log \left (x^2+x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx=\int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \left (1+x^4\right )^{2/3}}{x^6}-\frac {6 \left (1+x^4\right )^{2/3}}{x^3}+\frac {\left (1+x^4\right )^{2/3}}{x^2}+\frac {2 (-3+4 x) \left (1+x^4\right )^{2/3}}{1-x^3+x^4}\right ) \, dx \\ & = 2 \int \frac {(-3+4 x) \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx-3 \int \frac {\left (1+x^4\right )^{2/3}}{x^6} \, dx-6 \int \frac {\left (1+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (1+x^4\right )^{2/3}}{x^2} \, dx \\ & = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{5 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}+2 \int \left (-\frac {3 \left (1+x^4\right )^{2/3}}{1-x^3+x^4}+\frac {4 x \left (1+x^4\right )^{2/3}}{1-x^3+x^4}\right ) \, dx-3 \text {Subst}\left (\int \frac {\left (1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right ) \\ & = \frac {3 \left (1+x^4\right )^{2/3}}{x^2}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{5 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}-4 \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^2}} \, dx,x,x^2\right )-6 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+8 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx \\ & = \frac {3 \left (1+x^4\right )^{2/3}}{x^2}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{5 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}-6 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+8 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx-\frac {\left (6 \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{x^2} \\ & = \frac {3 \left (1+x^4\right )^{2/3}}{x^2}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{5 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}-6 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+8 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+\frac {\left (6 \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{x^2}-\frac {\left (6 \left (1+\sqrt {3}\right ) \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{x^2} \\ & = \frac {3 \left (1+x^4\right )^{2/3}}{x^2}+\frac {12 x^2}{1-\sqrt {3}-\sqrt [3]{1+x^4}}-\frac {6 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1+x^4}\right ) \sqrt {\frac {1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1+x^4}}{1-\sqrt {3}-\sqrt [3]{1+x^4}}\right )|-7+4 \sqrt {3}\right )}{x^2 \sqrt {-\frac {1-\sqrt [3]{1+x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}}}+\frac {4 \sqrt {2} 3^{3/4} \left (1-\sqrt [3]{1+x^4}\right ) \sqrt {\frac {1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1+x^4}}{1-\sqrt {3}-\sqrt [3]{1+x^4}}\right ),-7+4 \sqrt {3}\right )}{x^2 \sqrt {-\frac {1-\sqrt [3]{1+x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}}}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{5 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}-6 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+8 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx \\ \end{align*}
Time = 2.50 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx=\frac {3 \left (1+x^4\right )^{2/3} \left (1+5 x^3+x^4\right )}{5 x^5}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^4}}\right )+2 \log \left (-x+\sqrt [3]{1+x^4}\right )-\log \left (x^2+x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
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Time = 4.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.05
method | result | size |
pseudoelliptic | \(\frac {-5 \ln \left (\frac {x^{2}+x \left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+10 \sqrt {3}\, \arctan \left (\frac {\left (x +2 \left (x^{4}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{5}+10 \ln \left (\frac {-x +\left (x^{4}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+3 \left (x^{4}+1\right )^{\frac {2}{3}} \left (x^{4}+5 x^{3}+1\right )}{5 x^{5}}\) | \(105\) |
risch | \(\frac {\frac {3}{5} x^{8}+3 x^{7}+\frac {6}{5} x^{4}+3 x^{3}+\frac {3}{5}}{x^{5} \left (x^{4}+1\right )^{\frac {1}{3}}}+2 \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-\left (x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -\left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}-2 \left (x^{4}+1\right )^{\frac {2}{3}} x +x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{4}-x^{3}+1}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-\left (x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +2 \left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}+\left (x^{4}+1\right )^{\frac {2}{3}} x +x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{4}-x^{3}+1}\right )\) | \(290\) |
trager | \(\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}} \left (x^{4}+5 x^{3}+1\right )}{5 x^{5}}-6 \ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+69 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+51 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +51 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-33 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+24 x^{4}+37 \left (x^{4}+1\right )^{\frac {2}{3}} x +37 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+8 x^{3}+45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+69 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+24}{x^{4}-x^{3}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-20610 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-10989 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}-9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-2067 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2746 x^{4}+8010 \left (x^{4}+1\right )^{\frac {2}{3}} x +8010 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+4119 x^{3}+10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-10989 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+2746}{x^{4}-x^{3}+1}\right )-2 \ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+69 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+51 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +51 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-33 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+24 x^{4}+37 \left (x^{4}+1\right )^{\frac {2}{3}} x +37 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+8 x^{3}+45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+69 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+24}{x^{4}-x^{3}+1}\right )\) | \(616\) |
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Time = 1.85 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.44 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {13034 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} - 686 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (37 \, x^{4} + 6137 \, x^{3} + 37\right )}}{3 \, {\left (x^{4} + 6859 \, x^{3} + 1\right )}}\right ) - 5 \, x^{5} \log \left (\frac {x^{4} - x^{3} + 3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + 1}{x^{4} - x^{3} + 1}\right ) - 3 \, {\left (x^{4} + 5 \, x^{3} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \]
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Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + x^{3} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} - x^{3} + 1\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + x^{3} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} - x^{3} + 1\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx=\int \frac {{\left (x^4+1\right )}^{2/3}\,\left (x^4-3\right )\,\left (x^4+x^3+1\right )}{x^6\,\left (x^4-x^3+1\right )} \,d x \]
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