Integrand size = 32, antiderivative size = 101 \[ \int \frac {1+x^2+2 x^4}{\sqrt [4]{1+x^2} \left (2+3 x^2+x^4\right )} \, dx=\frac {4 x}{\sqrt [4]{1+x^2}}+\frac {7}{4} \arctan \left (\frac {-x+\sqrt [4]{1+x^2}}{\sqrt [4]{1+x^2}}\right )-\frac {7}{4} \arctan \left (\frac {x+\sqrt [4]{1+x^2}}{\sqrt [4]{1+x^2}}\right )-\frac {7}{4} \text {arctanh}\left (\frac {2 x \sqrt [4]{1+x^2}}{x^2+2 \sqrt {1+x^2}}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.72, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1702, 6857, 202, 291, 412, 406} \[ \int \frac {1+x^2+2 x^4}{\sqrt [4]{1+x^2} \left (2+3 x^2+x^4\right )} \, dx=\frac {7}{2} \arctan \left (\frac {\sqrt {x^2+1}+1}{x \sqrt [4]{x^2+1}}\right )+\frac {7}{2} \text {arctanh}\left (\frac {1-\sqrt {x^2+1}}{x \sqrt [4]{x^2+1}}\right )+\frac {4 x}{\sqrt [4]{x^2+1}} \]
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Rule 202
Rule 291
Rule 406
Rule 412
Rule 1702
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+x^2+2 x^4}{\left (1+x^2\right )^{5/4} \left (2+x^2\right )} \, dx \\ & = \int \left (-\frac {3}{\left (1+x^2\right )^{5/4}}+\frac {2 x^2}{\left (1+x^2\right )^{5/4}}+\frac {7}{\left (1+x^2\right )^{5/4} \left (2+x^2\right )}\right ) \, dx \\ & = 2 \int \frac {x^2}{\left (1+x^2\right )^{5/4}} \, dx-3 \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx+7 \int \frac {1}{\left (1+x^2\right )^{5/4} \left (2+x^2\right )} \, dx \\ & = \frac {4 x}{\sqrt [4]{1+x^2}}-6 E\left (\left .\frac {\arctan (x)}{2}\right |2\right )-4 \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx+7 \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx-7 \int \frac {1}{\sqrt [4]{1+x^2} \left (2+x^2\right )} \, dx \\ & = \frac {4 x}{\sqrt [4]{1+x^2}}+\frac {7}{2} \arctan \left (\frac {1+\sqrt {1+x^2}}{x \sqrt [4]{1+x^2}}\right )+\frac {7}{2} \text {arctanh}\left (\frac {1-\sqrt {1+x^2}}{x \sqrt [4]{1+x^2}}\right ) \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.81 \[ \int \frac {1+x^2+2 x^4}{\sqrt [4]{1+x^2} \left (2+3 x^2+x^4\right )} \, dx=\frac {1}{4} \left (\frac {16 x}{\sqrt [4]{1+x^2}}+7 \arctan \left (1-\frac {x}{\sqrt [4]{1+x^2}}\right )-7 \arctan \left (1+\frac {x}{\sqrt [4]{1+x^2}}\right )-7 \text {arctanh}\left (\frac {2 x \sqrt [4]{1+x^2}}{x^2+2 \sqrt {1+x^2}}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.70 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.88
method | result | size |
trager | \(\frac {4 x}{\left (x^{2}+1\right )^{\frac {1}{4}}}+14 \operatorname {RootOf}\left (32 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \ln \left (-\frac {8 \left (x^{2}+1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (32 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-x \sqrt {x^{2}+1}+8 \operatorname {RootOf}\left (32 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{4}}-8 \operatorname {RootOf}\left (32 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x -2 \left (x^{2}+1\right )^{\frac {1}{4}}+x}{x^{2}+2}\right )+\frac {7 \ln \left (\frac {8 \left (x^{2}+1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (32 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-2 \left (x^{2}+1\right )^{\frac {3}{4}}+x \sqrt {x^{2}+1}+8 \operatorname {RootOf}\left (32 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{4}}-8 \operatorname {RootOf}\left (32 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x +x}{x^{2}+2}\right )}{2}-14 \ln \left (\frac {8 \left (x^{2}+1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (32 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-2 \left (x^{2}+1\right )^{\frac {3}{4}}+x \sqrt {x^{2}+1}+8 \operatorname {RootOf}\left (32 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{4}}-8 \operatorname {RootOf}\left (32 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x +x}{x^{2}+2}\right ) \operatorname {RootOf}\left (32 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )\) | \(291\) |
risch | \(\frac {4 x}{\left (x^{2}+1\right )^{\frac {1}{4}}}-\frac {7 \ln \left (\frac {4 \left (x^{2}+1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+2 \left (x^{2}+1\right )^{\frac {3}{4}}+x \sqrt {x^{2}+1}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{4}}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x +x}{x^{2}+2}\right )}{2}-7 \ln \left (\frac {4 \left (x^{2}+1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+2 \left (x^{2}+1\right )^{\frac {3}{4}}+x \sqrt {x^{2}+1}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{4}}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x +x}{x^{2}+2}\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+7 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \ln \left (-\frac {4 \left (x^{2}+1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )-x \sqrt {x^{2}+1}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{4}}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x +2 \left (x^{2}+1\right )^{\frac {1}{4}}+x}{x^{2}+2}\right )\) | \(291\) |
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Time = 0.25 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.36 \[ \int \frac {1+x^2+2 x^4}{\sqrt [4]{1+x^2} \left (2+3 x^2+x^4\right )} \, dx=\frac {14 \, {\left (x^{2} + 1\right )} \arctan \left (\frac {x + 2 \, {\left (x^{2} + 1\right )}^{\frac {1}{4}}}{x}\right ) + 14 \, {\left (x^{2} + 1\right )} \arctan \left (-\frac {x - 2 \, {\left (x^{2} + 1\right )}^{\frac {1}{4}}}{x}\right ) - 7 \, {\left (x^{2} + 1\right )} \log \left (\frac {x^{2} + 2 \, {\left (x^{2} + 1\right )}^{\frac {1}{4}} x + 2 \, \sqrt {x^{2} + 1}}{x^{2}}\right ) + 7 \, {\left (x^{2} + 1\right )} \log \left (\frac {x^{2} - 2 \, {\left (x^{2} + 1\right )}^{\frac {1}{4}} x + 2 \, \sqrt {x^{2} + 1}}{x^{2}}\right ) + 32 \, {\left (x^{2} + 1\right )}^{\frac {3}{4}} x}{8 \, {\left (x^{2} + 1\right )}} \]
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\[ \int \frac {1+x^2+2 x^4}{\sqrt [4]{1+x^2} \left (2+3 x^2+x^4\right )} \, dx=\int \frac {2 x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{\frac {5}{4}} \left (x^{2} + 2\right )}\, dx \]
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\[ \int \frac {1+x^2+2 x^4}{\sqrt [4]{1+x^2} \left (2+3 x^2+x^4\right )} \, dx=\int { \frac {2 \, x^{4} + x^{2} + 1}{{\left (x^{4} + 3 \, x^{2} + 2\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {1+x^2+2 x^4}{\sqrt [4]{1+x^2} \left (2+3 x^2+x^4\right )} \, dx=\int { \frac {2 \, x^{4} + x^{2} + 1}{{\left (x^{4} + 3 \, x^{2} + 2\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1+x^2+2 x^4}{\sqrt [4]{1+x^2} \left (2+3 x^2+x^4\right )} \, dx=\int \frac {2\,x^4+x^2+1}{{\left (x^2+1\right )}^{1/4}\,\left (x^4+3\,x^2+2\right )} \,d x \]
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