Integrand size = 41, antiderivative size = 154 \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\frac {4 \sqrt [4]{2+3 x^4} \left (-1+6 x^4\right )}{5 x^5}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )+\frac {5 \arctan \left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{-x^2+\sqrt {2+3 x^4}}\right )}{2 \sqrt {2}}-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )-\frac {5 \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{x^2+\sqrt {2+3 x^4}}\right )}{2 \sqrt {2}} \]
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Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.62 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {6857, 270, 283, 338, 304, 209, 212, 524} \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\frac {10}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-x^4,-\frac {3 x^4}{2}\right )+3 \sqrt [4]{3} \arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{3 x^4+2}}\right )-3 \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{3 x^4+2}}\right )+\frac {6 \sqrt [4]{3 x^4+2}}{x}-\frac {2 \left (3 x^4+2\right )^{5/4}}{5 x^5} \]
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Rule 209
Rule 212
Rule 270
Rule 283
Rule 304
Rule 338
Rule 524
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4 \sqrt [4]{2+3 x^4}}{x^6}-\frac {6 \sqrt [4]{2+3 x^4}}{x^2}+\frac {x^2 \sqrt [4]{2+3 x^4}}{1+x^4}+\frac {10 x^2 \sqrt [4]{2+3 x^4}}{1+2 x^4}\right ) \, dx \\ & = 4 \int \frac {\sqrt [4]{2+3 x^4}}{x^6} \, dx-6 \int \frac {\sqrt [4]{2+3 x^4}}{x^2} \, dx+10 \int \frac {x^2 \sqrt [4]{2+3 x^4}}{1+2 x^4} \, dx+\int \frac {x^2 \sqrt [4]{2+3 x^4}}{1+x^4} \, dx \\ & = \frac {6 \sqrt [4]{2+3 x^4}}{x}-\frac {2 \left (2+3 x^4\right )^{5/4}}{5 x^5}+\frac {10}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-x^4,-\frac {3 x^4}{2}\right )-18 \int \frac {x^2}{\left (2+3 x^4\right )^{3/4}} \, dx \\ & = \frac {6 \sqrt [4]{2+3 x^4}}{x}-\frac {2 \left (2+3 x^4\right )^{5/4}}{5 x^5}+\frac {10}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-x^4,-\frac {3 x^4}{2}\right )-18 \text {Subst}\left (\int \frac {x^2}{1-3 x^4} \, dx,x,\frac {x}{\sqrt [4]{2+3 x^4}}\right ) \\ & = \frac {6 \sqrt [4]{2+3 x^4}}{x}-\frac {2 \left (2+3 x^4\right )^{5/4}}{5 x^5}+\frac {10}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-x^4,-\frac {3 x^4}{2}\right )-\left (3 \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+3 x^4}}\right )+\left (3 \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+3 x^4}}\right ) \\ & = \frac {6 \sqrt [4]{2+3 x^4}}{x}-\frac {2 \left (2+3 x^4\right )^{5/4}}{5 x^5}+\frac {10}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-2 x^4,-\frac {3 x^4}{2}\right )+\frac {1}{3} \sqrt [4]{2} x^3 \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-x^4,-\frac {3 x^4}{2}\right )+3 \sqrt [4]{3} \arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2+3 x^4}}\right )-3 \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{2+3 x^4}}\right ) \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\frac {1}{20} \left (\frac {16 \sqrt [4]{2+3 x^4} \left (-1+6 x^4\right )}{x^5}+10 \arctan \left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )+25 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{-x^2+\sqrt {2+3 x^4}}\right )-10 \text {arctanh}\left (\frac {x}{\sqrt [4]{2+3 x^4}}\right )-25 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{2+3 x^4}}{x^2+\sqrt {2+3 x^4}}\right )\right ) \]
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Time = 8.36 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.43
method | result | size |
pseudoelliptic | \(\frac {-25 \ln \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {3 x^{4}+2}}{\sqrt {3 x^{4}+2}-\left (3 x^{4}+2\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}}\right ) \sqrt {2}\, x^{5}-50 \arctan \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}\, x^{5}-50 \arctan \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}\, x^{5}+10 \ln \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}}-x}{x}\right ) x^{5}-10 \ln \left (\frac {x +\left (3 x^{4}+2\right )^{\frac {1}{4}}}{x}\right ) x^{5}-20 \arctan \left (\frac {\left (3 x^{4}+2\right )^{\frac {1}{4}}}{x}\right ) x^{5}+192 \left (3 x^{4}+2\right )^{\frac {1}{4}} x^{4}-32 \left (3 x^{4}+2\right )^{\frac {1}{4}}}{40 x^{5}}\) | \(220\) |
trager | \(\frac {4 \left (3 x^{4}+2\right )^{\frac {1}{4}} \left (6 x^{4}-1\right )}{5 x^{5}}+\frac {\ln \left (-\frac {\left (3 x^{4}+2\right )^{\frac {3}{4}} x -x^{2} \sqrt {3 x^{4}+2}+\left (3 x^{4}+2\right )^{\frac {1}{4}} x^{3}-2 x^{4}-1}{x^{4}+1}\right )}{4}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+\left (3 x^{4}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\sqrt {3 x^{4}+2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+\left (3 x^{4}+2\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{2 x^{4}+1}\right )}{4}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\sqrt {3 x^{4}+2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\left (3 x^{4}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+\left (3 x^{4}+2\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{2 x^{4}+1}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (-\frac {-\sqrt {3 x^{4}+2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+\left (3 x^{4}+2\right )^{\frac {3}{4}} x -\left (3 x^{4}+2\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{x^{4}+1}\right )}{4}\) | \(360\) |
risch | \(\text {Expression too large to display}\) | \(945\) |
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Result contains complex when optimal does not.
Time = 4.01 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.73 \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\frac {-\left (25 i + 25\right ) \, \sqrt {2} x^{5} \log \left (\frac {2 i \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + \left (i + 1\right ) \, \sqrt {2} \sqrt {3 \, x^{4} + 2} x^{2} + 2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{4} - i + 1\right )}}{2 \, x^{4} + 1}\right ) + \left (25 i + 25\right ) \, \sqrt {2} x^{5} \log \left (\frac {2 i \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} - \left (i + 1\right ) \, \sqrt {2} \sqrt {3 \, x^{4} + 2} x^{2} + 2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{4} + i - 1\right )}}{2 \, x^{4} + 1}\right ) + \left (25 i - 25\right ) \, \sqrt {2} x^{5} \log \left (\frac {-2 i \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} - \left (i - 1\right ) \, \sqrt {2} \sqrt {3 \, x^{4} + 2} x^{2} + 2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{4} + i + 1\right )}}{2 \, x^{4} + 1}\right ) - \left (25 i - 25\right ) \, \sqrt {2} x^{5} \log \left (\frac {-2 i \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + \left (i - 1\right ) \, \sqrt {2} \sqrt {3 \, x^{4} + 2} x^{2} + 2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{4} - i - 1\right )}}{2 \, x^{4} + 1}\right ) + 20 \, x^{5} \arctan \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) + 20 \, x^{5} \log \left (-\frac {2 \, x^{4} - {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + \sqrt {3 \, x^{4} + 2} x^{2} - {\left (3 \, x^{4} + 2\right )}^{\frac {3}{4}} x + 1}{x^{4} + 1}\right ) + 64 \, {\left (6 \, x^{4} - 1\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{80 \, x^{5}} \]
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\[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\int \frac {\sqrt [4]{3 x^{4} + 2} \left (x^{8} + 6 x^{4} + 4\right )}{x^{6} \left (x^{4} + 1\right ) \left (2 x^{4} + 1\right )}\, dx \]
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\[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\int { \frac {{\left (x^{8} + 6 \, x^{4} + 4\right )} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{{\left (2 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )} x^{6}} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=-\frac {5}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right )}\right ) - \frac {5}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right )}\right ) - \frac {5}{8} \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {3 \, x^{4} + 2}}{x^{2}} + 1\right ) + \frac {5}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {3 \, x^{4} + 2}}{x^{2}} + 1\right ) - \frac {2 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}} {\left (\frac {2}{x^{4}} + 3\right )}}{5 \, x} + \frac {6 \, {\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} - \frac {1}{2} \, \arctan \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \log \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {1}{4} \, \log \left (\frac {{\left (3 \, x^{4} + 2\right )}^{\frac {1}{4}}}{x} - 1\right ) \]
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Timed out. \[ \int \frac {\sqrt [4]{2+3 x^4} \left (4+6 x^4+x^8\right )}{x^6 \left (1+x^4\right ) \left (1+2 x^4\right )} \, dx=\int \frac {{\left (3\,x^4+2\right )}^{1/4}\,\left (x^8+6\,x^4+4\right )}{x^6\,\left (x^4+1\right )\,\left (2\,x^4+1\right )} \,d x \]
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