Optimal. Leaf size=272 \[ -\frac {\sqrt {x-1} (1-11 x)}{32 \left (x^2+1\right )}+\frac {\sqrt {x-1} x}{4 \left (x^2+1\right )^2}-\frac {1}{128} \sqrt {\frac {1}{2} \left (527+373 \sqrt {2}\right )} \log \left (-x-\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {x-1}-\sqrt {2}+1\right )+\frac {1}{128} \sqrt {\frac {1}{2} \left (527+373 \sqrt {2}\right )} \log \left (-x+\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {x-1}-\sqrt {2}+1\right )-\frac {1}{64} \sqrt {\frac {1}{2} \left (373 \sqrt {2}-527\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )}-2 \sqrt {x-1}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{64} \sqrt {\frac {1}{2} \left (373 \sqrt {2}-527\right )} \tan ^{-1}\left (\frac {2 \sqrt {x-1}+\sqrt {2 \left (\sqrt {2}-1\right )}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right ) \]
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Rubi [A] time = 0.36, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {737, 823, 827, 1169, 634, 618, 204, 628} \[ -\frac {\sqrt {x-1} (1-11 x)}{32 \left (x^2+1\right )}+\frac {\sqrt {x-1} x}{4 \left (x^2+1\right )^2}-\frac {1}{128} \sqrt {\frac {1}{2} \left (527+373 \sqrt {2}\right )} \log \left (-x-\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {x-1}-\sqrt {2}+1\right )+\frac {1}{128} \sqrt {\frac {1}{2} \left (527+373 \sqrt {2}\right )} \log \left (-x+\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {x-1}-\sqrt {2}+1\right )-\frac {1}{64} \sqrt {\frac {1}{2} \left (373 \sqrt {2}-527\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )}-2 \sqrt {x-1}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{64} \sqrt {\frac {1}{2} \left (373 \sqrt {2}-527\right )} \tan ^{-1}\left (\frac {2 \sqrt {x-1}+\sqrt {2 \left (\sqrt {2}-1\right )}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 737
Rule 823
Rule 827
Rule 1169
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x}}{\left (1+x^2\right )^3} \, dx &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {1}{4} \int \frac {3-\frac {5 x}{2}}{\sqrt {-1+x} \left (1+x^2\right )^2} \, dx\\ &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}+\frac {1}{16} \int \frac {-\frac {25}{4}+\frac {11 x}{4}}{\sqrt {-1+x} \left (1+x^2\right )} \, dx\\ &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {-\frac {7}{2}+\frac {11 x^2}{4}}{2+2 x^2+x^4} \, dx,x,\sqrt {-1+x}\right )\\ &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-7 \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )}-\left (-\frac {7}{2}-\frac {11}{2 \sqrt {2}}\right ) x}{\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {-1+x}\right )}{32 \sqrt {-1+\sqrt {2}}}+\frac {\operatorname {Subst}\left (\int \frac {-7 \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )}+\left (-\frac {7}{2}-\frac {11}{2 \sqrt {2}}\right ) x}{\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {-1+x}\right )}{32 \sqrt {-1+\sqrt {2}}}\\ &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}+\frac {1}{128} \sqrt {219-154 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {-1+x}\right )+\frac {1}{128} \sqrt {219-154 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {-1+x}\right )+\frac {\left (14+11 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {2 \left (-1+\sqrt {2}\right )}+2 x}{\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {-1+x}\right )}{256 \sqrt {-1+\sqrt {2}}}-\frac {\left (14+11 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2 \left (-1+\sqrt {2}\right )}+2 x}{\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {-1+x}\right )}{256 \sqrt {-1+\sqrt {2}}}\\ &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}-\frac {1}{256} \sqrt {1054+746 \sqrt {2}} \log \left (1-\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {-1+x}-x\right )+\frac {1}{256} \sqrt {1054+746 \sqrt {2}} \log \left (1-\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {-1+x}-x\right )-\frac {1}{64} \sqrt {219-154 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{-2 \left (1+\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {2 \left (-1+\sqrt {2}\right )}+2 \sqrt {-1+x}\right )-\frac {1}{64} \sqrt {219-154 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{-2 \left (1+\sqrt {2}\right )-x^2} \, dx,x,\sqrt {2 \left (-1+\sqrt {2}\right )}+2 \sqrt {-1+x}\right )\\ &=\frac {\sqrt {-1+x} x}{4 \left (1+x^2\right )^2}-\frac {(1-11 x) \sqrt {-1+x}}{32 \left (1+x^2\right )}-\frac {1}{64} \sqrt {\frac {1}{2} \left (-527+373 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}-2 \sqrt {-1+x}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{64} \sqrt {\frac {1}{2} \left (-527+373 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}+2 \sqrt {-1+x}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )-\frac {1}{256} \sqrt {1054+746 \sqrt {2}} \log \left (1-\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {-1+x}-x\right )+\frac {1}{256} \sqrt {1054+746 \sqrt {2}} \log \left (1-\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {-1+x}-x\right )\\ \end {align*}
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Mathematica [C] time = 0.09, size = 90, normalized size = 0.33 \[ \frac {1}{64} \left (\frac {2 \sqrt {x-1} \left (11 x^3-x^2+19 x-1\right )}{\left (x^2+1\right )^2}-(7-18 i) \sqrt {1-i} \tan ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {1-i}}\right )-(7+18 i) \sqrt {1+i} \tan ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {1+i}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 436, normalized size = 1.60 \[ -\frac {92 \cdot 278258^{\frac {1}{4}} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \sqrt {-393142 \, \sqrt {2} + 556516} \arctan \left (\frac {1}{109810067572} \cdot 278258^{\frac {3}{4}} \sqrt {46} \sqrt {373 \cdot 278258^{\frac {1}{4}} \sqrt {x - 1} {\left (11 \, \sqrt {2} + 14\right )} \sqrt {-393142 \, \sqrt {2} + 556516} + 6399934 \, x + 6399934 \, \sqrt {2} - 6399934} {\left (7 \, \sqrt {2} + 11\right )} \sqrt {-393142 \, \sqrt {2} + 556516} - \frac {1}{6399934} \cdot 278258^{\frac {3}{4}} \sqrt {x - 1} {\left (7 \, \sqrt {2} + 11\right )} \sqrt {-393142 \, \sqrt {2} + 556516} - \sqrt {2} + 1\right ) + 92 \cdot 278258^{\frac {1}{4}} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \sqrt {-393142 \, \sqrt {2} + 556516} \arctan \left (\frac {1}{109810067572} \cdot 278258^{\frac {3}{4}} \sqrt {46} \sqrt {-373 \cdot 278258^{\frac {1}{4}} \sqrt {x - 1} {\left (11 \, \sqrt {2} + 14\right )} \sqrt {-393142 \, \sqrt {2} + 556516} + 6399934 \, x + 6399934 \, \sqrt {2} - 6399934} {\left (7 \, \sqrt {2} + 11\right )} \sqrt {-393142 \, \sqrt {2} + 556516} - \frac {1}{6399934} \cdot 278258^{\frac {3}{4}} \sqrt {x - 1} {\left (7 \, \sqrt {2} + 11\right )} \sqrt {-393142 \, \sqrt {2} + 556516} + \sqrt {2} - 1\right ) + 278258^{\frac {1}{4}} {\left (746 \, x^{4} + 1492 \, x^{2} + 527 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} + 746\right )} \sqrt {-393142 \, \sqrt {2} + 556516} \log \left (\frac {373}{46} \cdot 278258^{\frac {1}{4}} \sqrt {x - 1} {\left (11 \, \sqrt {2} + 14\right )} \sqrt {-393142 \, \sqrt {2} + 556516} + 139129 \, x + 139129 \, \sqrt {2} - 139129\right ) - 278258^{\frac {1}{4}} {\left (746 \, x^{4} + 1492 \, x^{2} + 527 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} + 746\right )} \sqrt {-393142 \, \sqrt {2} + 556516} \log \left (-\frac {373}{46} \cdot 278258^{\frac {1}{4}} \sqrt {x - 1} {\left (11 \, \sqrt {2} + 14\right )} \sqrt {-393142 \, \sqrt {2} + 556516} + 139129 \, x + 139129 \, \sqrt {2} - 139129\right ) - 137264 \, {\left (11 \, x^{3} - x^{2} + 19 \, x - 1\right )} \sqrt {x - 1}}{4392448 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 206, normalized size = 0.76 \[ \frac {1}{128} \, \sqrt {746 \, \sqrt {2} - 1054} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} + 2 \, \sqrt {x - 1}\right )}}{2 \, \sqrt {\sqrt {2} + 2}}\right ) + \frac {1}{128} \, \sqrt {746 \, \sqrt {2} - 1054} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} - 2 \, \sqrt {x - 1}\right )}}{2 \, \sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{256} \, \sqrt {746 \, \sqrt {2} + 1054} \log \left (2^{\frac {1}{4}} \sqrt {x - 1} \sqrt {-\sqrt {2} + 2} + x + \sqrt {2} - 1\right ) + \frac {1}{256} \, \sqrt {746 \, \sqrt {2} + 1054} \log \left (-2^{\frac {1}{4}} \sqrt {x - 1} \sqrt {-\sqrt {2} + 2} + x + \sqrt {2} - 1\right ) + \frac {11 \, {\left (x - 1\right )}^{\frac {7}{2}} + 32 \, {\left (x - 1\right )}^{\frac {5}{2}} + 50 \, {\left (x - 1\right )}^{\frac {3}{2}} + 28 \, \sqrt {x - 1}}{32 \, {\left ({\left (x - 1\right )}^{2} + 2 \, x\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.07, size = 639, normalized size = 2.35 \[ \frac {\sqrt {2}\, \arctan \left (\frac {2 \sqrt {x -1}-\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right )}{64 \left (3+2 \sqrt {2}\right ) \sqrt {2+2 \sqrt {2}}}+\frac {5 \arctan \left (\frac {2 \sqrt {x -1}-\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right )}{64 \left (3+2 \sqrt {2}\right ) \sqrt {2+2 \sqrt {2}}}+\frac {\sqrt {2}\, \arctan \left (\frac {2 \sqrt {x -1}+\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right )}{64 \left (3+2 \sqrt {2}\right ) \sqrt {2+2 \sqrt {2}}}+\frac {5 \arctan \left (\frac {2 \sqrt {x -1}+\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right )}{64 \left (3+2 \sqrt {2}\right ) \sqrt {2+2 \sqrt {2}}}+\frac {13 \sqrt {2}\, \sqrt {-2+2 \sqrt {2}}\, \ln \left (x -1-\sqrt {x -1}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )}{32 \left (3+2 \sqrt {2}\right )}+\frac {147 \sqrt {-2+2 \sqrt {2}}\, \ln \left (x -1-\sqrt {x -1}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )}{256 \left (3+2 \sqrt {2}\right )}-\frac {13 \sqrt {2}\, \sqrt {-2+2 \sqrt {2}}\, \ln \left (x -1+\sqrt {x -1}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )}{32 \left (3+2 \sqrt {2}\right )}-\frac {147 \sqrt {-2+2 \sqrt {2}}\, \ln \left (x -1+\sqrt {x -1}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )}{256 \left (3+2 \sqrt {2}\right )}-\frac {-\frac {4 \left (-759-506 \sqrt {2}\right ) \left (x -1\right )^{\frac {3}{2}}}{23 \left (-6-4 \sqrt {2}\right )}-\frac {\left (-5336-3588 \sqrt {2}\right ) \sqrt {-2+2 \sqrt {2}}\, \left (x -1\right )}{23 \left (-6-4 \sqrt {2}\right )}-\frac {2 \left (-2392 \sqrt {2}-3036\right ) \sqrt {x -1}}{23 \left (-6-4 \sqrt {2}\right )}-\frac {\left (-3312 \sqrt {2}-4416\right ) \sqrt {-2+2 \sqrt {2}}}{46 \left (-6-4 \sqrt {2}\right )}}{128 \left (x -1+\sqrt {x -1}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )^{2}}+\frac {\frac {4 \left (-759-506 \sqrt {2}\right ) \left (x -1\right )^{\frac {3}{2}}}{23 \left (-6-4 \sqrt {2}\right )}-\frac {\left (-5336-3588 \sqrt {2}\right ) \sqrt {-2+2 \sqrt {2}}\, \left (x -1\right )}{23 \left (-6-4 \sqrt {2}\right )}+\frac {2 \left (-2392 \sqrt {2}-3036\right ) \sqrt {x -1}}{23 \left (-6-4 \sqrt {2}\right )}-\frac {\left (-3312 \sqrt {2}-4416\right ) \sqrt {-2+2 \sqrt {2}}}{46 \left (-6-4 \sqrt {2}\right )}}{128 \left (x -1-\sqrt {x -1}\, \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x - 1}}{{\left (x^{2} + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 440, normalized size = 1.62 \[ \mathrm {atanh}\left (\frac {275\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {x-1}}{64\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}-\frac {207}{4096}\right )}+\frac {275\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}\,\sqrt {x-1}}{64\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}-\frac {207}{4096}\right )}+\frac {373\,\sqrt {2}\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {x-1}}{128\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}-\frac {207}{4096}\right )}-\frac {373\,\sqrt {2}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}\,\sqrt {x-1}}{128\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}-\frac {207}{4096}\right )}\right )\,\left (2\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}+2\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}\right )-\mathrm {atanh}\left (\frac {275\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {x-1}}{64\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}+\frac {207}{4096}\right )}-\frac {275\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}\,\sqrt {x-1}}{64\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}+\frac {207}{4096}\right )}+\frac {373\,\sqrt {2}\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {x-1}}{128\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}+\frac {207}{4096}\right )}+\frac {373\,\sqrt {2}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}\,\sqrt {x-1}}{128\,\left (28\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}+\frac {207}{4096}\right )}\right )\,\left (2\,\sqrt {\frac {527}{32768}-\frac {373\,\sqrt {2}}{32768}}-2\,\sqrt {\frac {373\,\sqrt {2}}{32768}+\frac {527}{32768}}\right )+\frac {\frac {7\,\sqrt {x-1}}{8}+\frac {25\,{\left (x-1\right )}^{3/2}}{16}+{\left (x-1\right )}^{5/2}+\frac {11\,{\left (x-1\right )}^{7/2}}{32}}{8\,x+8\,{\left (x-1\right )}^2+4\,{\left (x-1\right )}^3+{\left (x-1\right )}^4-4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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