Integrand size = 17, antiderivative size = 74 \[ \int \frac {\left (1+x^4\right )^{3/4}}{\left (2+x^4\right )^2} \, dx=\frac {x \left (1+x^4\right )^{3/4}}{8 \left (2+x^4\right )}+\frac {3 \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )}{16\ 2^{3/4}}+\frac {3 \text {arctanh}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )}{16\ 2^{3/4}} \]
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Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {386, 385, 218, 212, 209} \[ \int \frac {\left (1+x^4\right )^{3/4}}{\left (2+x^4\right )^2} \, dx=\frac {3 \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}}+\frac {3 \text {arctanh}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}}+\frac {\left (x^4+1\right )^{3/4} x}{8 \left (x^4+2\right )} \]
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Rule 209
Rule 212
Rule 218
Rule 385
Rule 386
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (1+x^4\right )^{3/4}}{8 \left (2+x^4\right )}+\frac {3}{8} \int \frac {1}{\sqrt [4]{1+x^4} \left (2+x^4\right )} \, dx \\ & = \frac {x \left (1+x^4\right )^{3/4}}{8 \left (2+x^4\right )}+\frac {3}{8} \text {Subst}\left (\int \frac {1}{2-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ & = \frac {x \left (1+x^4\right )^{3/4}}{8 \left (2+x^4\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{16 \sqrt {2}}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{16 \sqrt {2}} \\ & = \frac {x \left (1+x^4\right )^{3/4}}{8 \left (2+x^4\right )}+\frac {3 \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )}{16\ 2^{3/4}}+\frac {3 \text {arctanh}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )}{16\ 2^{3/4}} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^4\right )^{3/4}}{\left (2+x^4\right )^2} \, dx=\frac {x \left (1+x^4\right )^{3/4}}{8 \left (2+x^4\right )}+\frac {3 \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )}{16\ 2^{3/4}}+\frac {3 \text {arctanh}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )}{16\ 2^{3/4}} \]
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Time = 2.65 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.18
method | result | size |
pseudoelliptic | \(\frac {-3 \left (x^{4}+2\right ) \left (2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}{x}\right )-\ln \left (\frac {-x 2^{\frac {3}{4}}-2 \left (x^{4}+1\right )^{\frac {1}{4}}}{x 2^{\frac {3}{4}}-2 \left (x^{4}+1\right )^{\frac {1}{4}}}\right )\right ) 2^{\frac {1}{4}}+8 \left (x^{4}+1\right )^{\frac {3}{4}} x}{64 x^{4}+128}\) | \(87\) |
trager | \(\frac {\left (x^{4}+1\right )^{\frac {3}{4}} x}{8 x^{4}+16}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {-2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-2 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{4}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )}{x^{4}+2}\right )}{64}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{4}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )}{x^{4}+2}\right )}{64}\) | \(229\) |
risch | \(\frac {\left (x^{4}+1\right )^{\frac {3}{4}} x}{8 x^{4}+16}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-2 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{4}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )}{x^{4}+2}\right )}{64}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+2 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{4}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )}{x^{4}+2}\right )}{64}\) | \(229\) |
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Result contains complex when optimal does not.
Time = 3.41 (sec) , antiderivative size = 313, normalized size of antiderivative = 4.23 \[ \int \frac {\left (1+x^4\right )^{3/4}}{\left (2+x^4\right )^2} \, dx=\frac {3 \cdot 8^{\frac {3}{4}} {\left (x^{4} + 2\right )} \log \left (\frac {8 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 8^{\frac {3}{4}} {\left (3 \, x^{4} + 2\right )} + 16 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) - 3 \cdot 8^{\frac {3}{4}} {\left (-i \, x^{4} - 2 i\right )} \log \left (-\frac {8 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 8 i \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 8^{\frac {3}{4}} {\left (3 i \, x^{4} + 2 i\right )} - 16 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) - 3 \cdot 8^{\frac {3}{4}} {\left (i \, x^{4} + 2 i\right )} \log \left (-\frac {8 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 8 i \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 8^{\frac {3}{4}} {\left (-3 i \, x^{4} - 2 i\right )} - 16 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) - 3 \cdot 8^{\frac {3}{4}} {\left (x^{4} + 2\right )} \log \left (\frac {8 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 8^{\frac {3}{4}} {\left (3 \, x^{4} + 2\right )} + 16 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) + 64 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{512 \, {\left (x^{4} + 2\right )}} \]
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\[ \int \frac {\left (1+x^4\right )^{3/4}}{\left (2+x^4\right )^2} \, dx=\int \frac {\left (x^{4} + 1\right )^{\frac {3}{4}}}{\left (x^{4} + 2\right )^{2}}\, dx \]
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\[ \int \frac {\left (1+x^4\right )^{3/4}}{\left (2+x^4\right )^2} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {3}{4}}}{{\left (x^{4} + 2\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (1+x^4\right )^{3/4}}{\left (2+x^4\right )^2} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {3}{4}}}{{\left (x^{4} + 2\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (1+x^4\right )^{3/4}}{\left (2+x^4\right )^2} \, dx=\int \frac {{\left (x^4+1\right )}^{3/4}}{{\left (x^4+2\right )}^2} \,d x \]
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