Integrand size = 13, antiderivative size = 49 \[ \int x^2 \sqrt [4]{1+x^4} \, dx=\frac {1}{4} x^3 \sqrt [4]{1+x^4}-\frac {1}{8} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{8} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {285, 338, 304, 209, 212} \[ \int x^2 \sqrt [4]{1+x^4} \, dx=-\frac {1}{8} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{8} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{4} \sqrt [4]{x^4+1} x^3 \]
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Rule 209
Rule 212
Rule 285
Rule 304
Rule 338
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^3 \sqrt [4]{1+x^4}+\frac {1}{4} \int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx \\ & = \frac {1}{4} x^3 \sqrt [4]{1+x^4}+\frac {1}{4} \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ & = \frac {1}{4} x^3 \sqrt [4]{1+x^4}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{8} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ & = \frac {1}{4} x^3 \sqrt [4]{1+x^4}-\frac {1}{8} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{8} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92 \[ \int x^2 \sqrt [4]{1+x^4} \, dx=\frac {1}{8} \left (2 x^3 \sqrt [4]{1+x^4}-\arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.91 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.35
method | result | size |
meijerg | \(\frac {x^{3} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -x^{4}\right )}{3}\) | \(17\) |
risch | \(\frac {x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}{4}+\frac {x^{3} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -x^{4}\right )}{12}\) | \(30\) |
pseudoelliptic | \(\frac {x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}{4}-\frac {\ln \left (\frac {\left (x^{4}+1\right )^{\frac {1}{4}}-x}{x}\right )}{16}+\frac {\arctan \left (\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\right )}{8}+\frac {\ln \left (\frac {\left (x^{4}+1\right )^{\frac {1}{4}}+x}{x}\right )}{16}\) | \(62\) |
trager | \(\frac {x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}}{4}+\frac {\ln \left (2 \left (x^{4}+1\right )^{\frac {3}{4}} x +2 \sqrt {x^{4}+1}\, x^{2}+2 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}+2 x^{4}+1\right )}{16}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}+1\right )^{\frac {3}{4}} x -2 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{16}\) | \(127\) |
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Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.27 \[ \int x^2 \sqrt [4]{1+x^4} \, dx=\frac {1}{4} \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + \frac {1}{8} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{16} \, \log \left (\frac {x + {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{16} \, \log \left (-\frac {x - {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) \]
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Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.63 \[ \int x^2 \sqrt [4]{1+x^4} \, dx=\frac {x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.47 \[ \int x^2 \sqrt [4]{1+x^4} \, dx=\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{4 \, x {\left (\frac {x^{4} + 1}{x^{4}} - 1\right )}} + \frac {1}{8} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{16} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} + 1\right ) - \frac {1}{16} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} - 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.20 \[ \int x^2 \sqrt [4]{1+x^4} \, dx=\frac {1}{4} \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + \frac {1}{8} \, \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{16} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} + 1\right ) - \frac {1}{16} \, \log \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} - 1\right ) \]
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Timed out. \[ \int x^2 \sqrt [4]{1+x^4} \, dx=\int x^2\,{\left (x^4+1\right )}^{1/4} \,d x \]
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