Integrand size = 29, antiderivative size = 81 \[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=\frac {x}{3 \sqrt [4]{1+x^4}}-\frac {5 \arctan \left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}-\frac {5 \text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}} \]
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Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1468, 541, 12, 385, 218, 212, 209} \[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=-\frac {5 \arctan \left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4+1}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}-\frac {5 \text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4+1}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}+\frac {x}{3 \sqrt [4]{x^4+1}} \]
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Rule 12
Rule 209
Rule 212
Rule 218
Rule 385
Rule 541
Rule 1468
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+2 x^4}{\left (-2+x^4\right ) \left (1+x^4\right )^{5/4}} \, dx \\ & = \frac {x}{3 \sqrt [4]{1+x^4}}+\frac {1}{3} \int \frac {5}{\left (-2+x^4\right ) \sqrt [4]{1+x^4}} \, dx \\ & = \frac {x}{3 \sqrt [4]{1+x^4}}+\frac {5}{3} \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{1+x^4}} \, dx \\ & = \frac {x}{3 \sqrt [4]{1+x^4}}+\frac {5}{3} \text {Subst}\left (\int \frac {1}{-2+3 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ & = \frac {x}{3 \sqrt [4]{1+x^4}}-\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt {2}}-\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt {2}} \\ & = \frac {x}{3 \sqrt [4]{1+x^4}}-\frac {5 \arctan \left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}-\frac {5 \text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=\frac {x}{3 \sqrt [4]{1+x^4}}-\frac {5 \arctan \left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}-\frac {5 \text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}} \]
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Time = 6.91 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.28
method | result | size |
pseudoelliptic | \(\frac {10 \arctan \left (\frac {3^{\frac {3}{4}} 2^{\frac {1}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}{3 x}\right ) 2^{\frac {1}{4}} 3^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}-5 \ln \left (\frac {-2^{\frac {3}{4}} 3^{\frac {1}{4}} x -2 \left (x^{4}+1\right )^{\frac {1}{4}}}{2^{\frac {3}{4}} 3^{\frac {1}{4}} x -2 \left (x^{4}+1\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}} 3^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+24 x}{72 \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(104\) |
trager | \(\frac {x}{3 \left (x^{4}+1\right )^{\frac {1}{4}}}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right ) \ln \left (-\frac {-2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{3} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2} x^{3}-15 \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x -6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )}{x^{4}-2}\right )}{72}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2} x^{2}-6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2} x^{3}-15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x -6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right )}{x^{4}-2}\right )}{72}\) | \(222\) |
risch | \(\frac {x}{3 \left (x^{4}+1\right )^{\frac {1}{4}}}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right ) \ln \left (-\frac {-2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{3} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2} x^{3}-15 \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x -6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )}{x^{4}-2}\right )}{72}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2} x^{2}-6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2} x^{3}-15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x -6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-54\right )^{2}\right )}{x^{4}-2}\right )}{72}\) | \(222\) |
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Result contains complex when optimal does not.
Time = 3.65 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.86 \[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=-\frac {5 \cdot 24^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {24 \, \sqrt {6} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 24 \cdot 24^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 24^{\frac {3}{4}} {\left (5 \, x^{4} + 2\right )} + 48 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 2}\right ) + 5 \cdot 24^{\frac {3}{4}} {\left (i \, x^{4} + i\right )} \log \left (-\frac {24 \, \sqrt {6} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 24 i \cdot 24^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 24^{\frac {3}{4}} {\left (5 i \, x^{4} + 2 i\right )} - 48 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 2}\right ) + 5 \cdot 24^{\frac {3}{4}} {\left (-i \, x^{4} - i\right )} \log \left (-\frac {24 \, \sqrt {6} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 24 i \cdot 24^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 24^{\frac {3}{4}} {\left (-5 i \, x^{4} - 2 i\right )} - 48 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 2}\right ) - 5 \cdot 24^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {24 \, \sqrt {6} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 24 \cdot 24^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 24^{\frac {3}{4}} {\left (5 \, x^{4} + 2\right )} + 48 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 2}\right ) - 192 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{576 \, {\left (x^{4} + 1\right )}} \]
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\[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=\int \frac {2 x^{4} + 1}{\left (x^{4} - 2\right ) \left (x^{4} + 1\right )^{\frac {5}{4}}}\, dx \]
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\[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{4} + 1}{{\left (x^{8} - x^{4} - 2\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{4} + 1}{{\left (x^{8} - x^{4} - 2\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx=\int -\frac {2\,x^4+1}{{\left (x^4+1\right )}^{1/4}\,\left (-x^8+x^4+2\right )} \,d x \]
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