Integrand size = 15, antiderivative size = 68 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=-\frac {\sqrt [4]{3+4 x^4}}{x}-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{\sqrt {2}} \]
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Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {283, 338, 304, 209, 212} \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{\sqrt {2}}-\frac {\sqrt [4]{4 x^4+3}}{x} \]
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Rule 209
Rule 212
Rule 283
Rule 304
Rule 338
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt [4]{3+4 x^4}}{x}+4 \int \frac {x^2}{\left (3+4 x^4\right )^{3/4}} \, dx \\ & = -\frac {\sqrt [4]{3+4 x^4}}{x}+4 \text {Subst}\left (\int \frac {x^2}{1-4 x^4} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right ) \\ & = -\frac {\sqrt [4]{3+4 x^4}}{x}+\text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )-\text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right ) \\ & = -\frac {\sqrt [4]{3+4 x^4}}{x}-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=-\frac {\sqrt [4]{3+4 x^4}}{x}-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{\sqrt {2}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.29
method | result | size |
meijerg | \(-\frac {3^{\frac {1}{4}} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{4},-\frac {1}{4};\frac {3}{4};-\frac {4 x^{4}}{3}\right )}{x}\) | \(20\) |
risch | \(-\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}}}{x}+\frac {4 \,3^{\frac {1}{4}} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-\frac {4 x^{4}}{3}\right )}{9}\) | \(35\) |
pseudoelliptic | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}} \sqrt {2}}{2 x}\right ) x +\sqrt {2}\, \arctan \left (\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}} \sqrt {2}}{2 x}\right ) x -2 \left (4 x^{4}+3\right )^{\frac {1}{4}}}{2 x}\) | \(64\) |
trager | \(-\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}}}{x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {4 x^{4}+3}\, x^{2}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{4}-4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x +8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \sqrt {4 x^{4}+3}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{4}-4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x -8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{4}\) | \(166\) |
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Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (55) = 110\).
Time = 2.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=-\frac {2 \, \sqrt {2} x \arctan \left (\frac {4}{3} \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} x^{3} + \frac {2}{3} \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {3}{4}} x\right ) - \sqrt {2} x \log \left (-256 \, x^{8} - 192 \, x^{4} - 4 \, \sqrt {2} {\left (16 \, x^{5} + 3 \, x\right )} {\left (4 \, x^{4} + 3\right )}^{\frac {3}{4}} - 8 \, \sqrt {2} {\left (16 \, x^{7} + 9 \, x^{3}\right )} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} - 16 \, {\left (8 \, x^{6} + 3 \, x^{2}\right )} \sqrt {4 \, x^{4} + 3} - 9\right ) + 8 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{8 \, x} \]
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Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=\frac {\sqrt [4]{3} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {4 x^{4} e^{i \pi }}{3}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x} \]
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Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x} \]
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Time = 0.48 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.26 \[ \int \frac {\sqrt [4]{3+4 x^4}}{x^2} \, dx=-\frac {3^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ -\frac {4\,x^4}{3}\right )}{x} \]
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