Integrand size = 14, antiderivative size = 385 \[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {2 a^3 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {55}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{3} a^3 \left (1-\frac {1}{a x}\right )^{9/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {55 a^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}-\frac {55 a^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {55 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}}-\frac {55 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}} \]
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Time = 0.25 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6306, 91, 81, 52, 65, 246, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {55 a^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{8 \sqrt {2}}-\frac {55 a^3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{8 \sqrt {2}}+\frac {1}{3} a^3 \left (1-\frac {1}{a x}\right )^{9/4} \left (\frac {1}{a x}+1\right )^{3/4}+\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (\frac {1}{a x}+1\right )^{3/4}+\frac {55}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}+\frac {2 a^3 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{\frac {1}{a x}+1}}+\frac {55 a^3 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{16 \sqrt {2}}-\frac {55 a^3 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{16 \sqrt {2}} \]
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Rule 52
Rule 65
Rule 81
Rule 91
Rule 210
Rule 217
Rule 246
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6306
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^2 \left (1-\frac {x}{a}\right )^{5/4}}{\left (1+\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {2 a^3 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}-\left (2 a^3\right ) \text {Subst}\left (\int \frac {\left (-\frac {5}{2 a}+\frac {x}{2 a^2}\right ) \left (1-\frac {x}{a}\right )^{5/4}}{\sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {2 a^3 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {1}{3} a^3 \left (1-\frac {1}{a x}\right )^{9/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{2} \left (11 a^2\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/4}}{\sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {2 a^3 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{3} a^3 \left (1-\frac {1}{a x}\right )^{9/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{8} \left (55 a^2\right ) \text {Subst}\left (\int \frac {\sqrt [4]{1-\frac {x}{a}}}{\sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {2 a^3 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {55}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{3} a^3 \left (1-\frac {1}{a x}\right )^{9/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{16} \left (55 a^2\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {2 a^3 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {55}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{3} a^3 \left (1-\frac {1}{a x}\right )^{9/4} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{4} \left (55 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-\frac {1}{a x}}\right ) \\ & = \frac {2 a^3 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {55}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{3} a^3 \left (1-\frac {1}{a x}\right )^{9/4} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{4} \left (55 a^3\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right ) \\ & = \frac {2 a^3 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {55}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{3} a^3 \left (1-\frac {1}{a x}\right )^{9/4} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{8} \left (55 a^3\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )-\frac {1}{8} \left (55 a^3\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right ) \\ & = \frac {2 a^3 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {55}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{3} a^3 \left (1-\frac {1}{a x}\right )^{9/4} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{16} \left (55 a^3\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )-\frac {1}{16} \left (55 a^3\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )+\frac {\left (55 a^3\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}}+\frac {\left (55 a^3\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}} \\ & = \frac {2 a^3 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {55}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{3} a^3 \left (1-\frac {1}{a x}\right )^{9/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {55 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}}-\frac {55 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}}-\frac {\left (55 a^3\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {\left (55 a^3\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}} \\ & = \frac {2 a^3 \left (1-\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {55}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {11}{4} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {1}{3} a^3 \left (1-\frac {1}{a x}\right )^{9/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {55 a^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}-\frac {55 a^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {55 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}}-\frac {55 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.27 \[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=a^3 \left (\frac {e^{-\frac {1}{2} \coth ^{-1}(a x)} \left (96+425 e^{2 \coth ^{-1}(a x)}+462 e^{4 \coth ^{-1}(a x)}+165 e^{6 \coth ^{-1}(a x)}\right )}{12 \left (1+e^{2 \coth ^{-1}(a x)}\right )^3}-\frac {55}{32} \text {RootSum}\left [1+\text {$\#$1}^4\&,\frac {\coth ^{-1}(a x)+2 \log \left (e^{-\frac {1}{2} \coth ^{-1}(a x)}-\text {$\#$1}\right )}{\text {$\#$1}^3}\&\right ]\right ) \]
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\[\int \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}}{x^{4}}d x\]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.56 \[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {165 \, \left (-a^{12}\right )^{\frac {1}{4}} x^{3} \log \left (55 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 55 \, \left (-a^{12}\right )^{\frac {1}{4}}\right ) + 165 i \, \left (-a^{12}\right )^{\frac {1}{4}} x^{3} \log \left (55 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 55 i \, \left (-a^{12}\right )^{\frac {1}{4}}\right ) - 165 i \, \left (-a^{12}\right )^{\frac {1}{4}} x^{3} \log \left (55 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 55 i \, \left (-a^{12}\right )^{\frac {1}{4}}\right ) - 165 \, \left (-a^{12}\right )^{\frac {1}{4}} x^{3} \log \left (55 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 55 \, \left (-a^{12}\right )^{\frac {1}{4}}\right ) - 2 \, {\left (287 \, a^{3} x^{3} + 61 \, a^{2} x^{2} - 26 \, a x + 8\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{48 \, x^{3}} \]
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\[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}{x^{4}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {1}{96} \, {\left (330 \, \sqrt {2} a^{2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 330 \, \sqrt {2} a^{2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 165 \, \sqrt {2} a^{2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 165 \, \sqrt {2} a^{2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 768 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \frac {8 \, {\left (137 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} + 174 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + 69 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}\right )} a \]
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Time = 0.31 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {1}{96} \, {\left (330 \, \sqrt {2} a^{2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 330 \, \sqrt {2} a^{2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 165 \, \sqrt {2} a^{2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 165 \, \sqrt {2} a^{2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 768 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \frac {8 \, {\left (\frac {174 \, {\left (a x - 1\right )} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} + \frac {137 \, {\left (a x - 1\right )}^{2} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{2}} + 69 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{3}}\right )} a \]
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Time = 0.08 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.49 \[ \int \frac {e^{-\frac {5}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {\frac {23\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{4}+\frac {29\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{2}+\frac {137\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}}{12}}{\frac {3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {3\,\left (a\,x-1\right )}{a\,x+1}+1}+8\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}+\frac {{\left (-1\right )}^{1/4}\,a^3\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )\,55{}\mathrm {i}}{8}+\frac {55\,{\left (-1\right )}^{1/4}\,a^3\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\,1{}\mathrm {i}\right )}{8} \]
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