Integrand size = 45, antiderivative size = 128 \[ \int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx=-\sqrt {\frac {1}{2} \left (3-i \sqrt {3}\right )} \arctan \left (\frac {\sqrt {-\frac {3}{2}-\frac {i \sqrt {3}}{2}} x}{\sqrt {-1+x^2+x^4}}\right )-\sqrt {\frac {1}{2} \left (3+i \sqrt {3}\right )} \arctan \left (\frac {\sqrt {-\frac {3}{2}+\frac {i \sqrt {3}}{2}} x}{\sqrt {-1+x^2+x^4}}\right )-\text {arctanh}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right ) \]
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\[ \int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx=\int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-1+x^2+x^4\right )^{3/2}}{-1-x^2}+\frac {\left (-1+x^2+x^4\right )^{3/2}}{-1+x^2}+\frac {\left (1+2 x^2-2 x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8}\right ) \, dx \\ & = \int \frac {\left (-1+x^2+x^4\right )^{3/2}}{-1-x^2} \, dx+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{-1+x^2} \, dx+\int \frac {\left (1+2 x^2-2 x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx \\ & = -\int x^2 \sqrt {-1+x^2+x^4} \, dx-\int \left (-2-x^2\right ) \sqrt {-1+x^2+x^4} \, dx-\int \frac {\sqrt {-1+x^2+x^4}}{-1-x^2} \, dx+\int \frac {\sqrt {-1+x^2+x^4}}{-1+x^2} \, dx+\int \left (\frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8}+\frac {2 x^2 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8}-\frac {2 x^4 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8}\right ) \, dx \\ & = -\frac {1}{15} x \left (1+3 x^2\right ) \sqrt {-1+x^2+x^4}+\frac {1}{15} x \left (11+3 x^2\right ) \sqrt {-1+x^2+x^4}-\frac {1}{15} \int \frac {19-2 x^2}{\sqrt {-1+x^2+x^4}} \, dx+\frac {1}{15} \int \frac {-1+8 x^2}{\sqrt {-1+x^2+x^4}} \, dx+2 \int \frac {x^2 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-2 \int \frac {x^4 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx+\int \frac {x^2}{\sqrt {-1+x^2+x^4}} \, dx-\int \frac {-2-x^2}{\sqrt {-1+x^2+x^4}} \, dx+\int \frac {1}{\left (-1-x^2\right ) \sqrt {-1+x^2+x^4}} \, dx+\int \frac {1}{\left (-1+x^2\right ) \sqrt {-1+x^2+x^4}} \, dx+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx \\ & = -\frac {1}{15} x \left (1+3 x^2\right ) \sqrt {-1+x^2+x^4}+\frac {1}{15} x \left (11+3 x^2\right ) \sqrt {-1+x^2+x^4}+\frac {1}{15} \int \frac {1-\sqrt {5}+2 x^2}{\sqrt {-1+x^2+x^4}} \, dx+\frac {4}{15} \int \frac {1-\sqrt {5}+2 x^2}{\sqrt {-1+x^2+x^4}} \, dx+2 \left (\frac {1}{2} \int \frac {1-\sqrt {5}+2 x^2}{\sqrt {-1+x^2+x^4}} \, dx\right )+2 \int \frac {x^2 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-2 \int \frac {x^4 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-\frac {1}{2} \left (-3-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx-\frac {1}{15} \left (20-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx-\frac {\int \frac {1-\sqrt {5}+2 x^2}{\left (-1+x^2\right ) \sqrt {-1+x^2+x^4}} \, dx}{-3+\sqrt {5}}+\frac {2 \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx}{-3+\sqrt {5}}+\frac {1}{2} \left (-1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx-\frac {\int \frac {1-\sqrt {5}+2 x^2}{\left (-1-x^2\right ) \sqrt {-1+x^2+x^4}} \, dx}{1+\sqrt {5}}-\frac {2 \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx}{1+\sqrt {5}}+\frac {1}{15} \left (-5+4 \sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 1.14 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.97 \[ \int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx=\frac {1}{2} \left (-\sqrt {6-2 i \sqrt {3}} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (-3-i \sqrt {3}\right )} x}{\sqrt {-1+x^2+x^4}}\right )-\sqrt {6+2 i \sqrt {3}} \arctan \left (\frac {\sqrt {\frac {1}{2} i \left (3 i+\sqrt {3}\right )} x}{\sqrt {-1+x^2+x^4}}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.69 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.75
method | result | size |
default | \(\frac {3 \left (\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (-1+2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )-\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (1-2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )-6 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-7+24 i+\textit {\_Z}^{8}+\left (-8-8 i\right ) \textit {\_Z}^{6}+\left (6+24 i\right ) \textit {\_Z}^{4}+\left (-8+56 i\right ) \textit {\_Z}^{2}\right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{2}+2 i-1\right ) \ln \left (\frac {-\textit {\_R} x -x^{2}+\sqrt {x^{4}+x^{2}-1}-i}{x}\right )}{2-14 i-\textit {\_R}^{6}+\left (6+6 i\right ) \textit {\_R}^{4}+\left (-3-12 i\right ) \textit {\_R}^{2}}\right )\right ) \left (\left (\frac {2}{3}-\frac {2 x^{4}}{3}+\left (-\frac {1}{6}-i\right ) x^{2}\right ) \sqrt {x^{4}+x^{2}-1}+\frac {2 x^{6}}{3}+\left (\frac {1}{2}+i\right ) x^{4}+\left (-1+\frac {i}{2}\right ) x^{2}-\frac {2 i}{3}\right )}{\left (x^{2}-\sqrt {x^{4}+x^{2}-1}+i\right )^{3}}\) | \(224\) |
pseudoelliptic | \(\frac {3 \left (\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (-1+2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )-\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (1-2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )-6 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-7+24 i+\textit {\_Z}^{8}+\left (-8-8 i\right ) \textit {\_Z}^{6}+\left (6+24 i\right ) \textit {\_Z}^{4}+\left (-8+56 i\right ) \textit {\_Z}^{2}\right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{2}+2 i-1\right ) \ln \left (\frac {-\textit {\_R} x -x^{2}+\sqrt {x^{4}+x^{2}-1}-i}{x}\right )}{2-14 i-\textit {\_R}^{6}+\left (6+6 i\right ) \textit {\_R}^{4}+\left (-3-12 i\right ) \textit {\_R}^{2}}\right )\right ) \left (\left (\frac {2}{3}-\frac {2 x^{4}}{3}+\left (-\frac {1}{6}-i\right ) x^{2}\right ) \sqrt {x^{4}+x^{2}-1}+\frac {2 x^{6}}{3}+\left (\frac {1}{2}+i\right ) x^{4}+\left (-1+\frac {i}{2}\right ) x^{2}-\frac {2 i}{3}\right )}{\left (x^{2}-\sqrt {x^{4}+x^{2}-1}+i\right )^{3}}\) | \(224\) |
trager | \(-\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right ) \ln \left (\frac {8 x^{2} \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{5}+2 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{3} x^{4}+8 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{3} x^{2}+6 \sqrt {x^{4}+x^{2}-1}\, \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} x -2 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{3}+3 x \sqrt {x^{4}+x^{2}-1}}{4 x^{2} \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}-x^{4}+2 x^{2}+1}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \ln \left (-\frac {16 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{4} x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} x^{4}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) x^{4}+24 \sqrt {x^{4}+x^{2}-1}\, \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} x +4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right )-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) x^{2}+6 x \sqrt {x^{4}+x^{2}-1}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right )}{4 x^{2} \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+x^{4}+x^{2}-1}\right )}{2}-\frac {\ln \left (-\frac {x^{4}+2 x \sqrt {x^{4}+x^{2}-1}+2 x^{2}-1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}\) | \(567\) |
elliptic | \(\frac {\left (-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}+x^{2}-1}}{x}\right )-\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}+\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {4 \sqrt {3}+6}}{2 x}+\sqrt {3}\right ) \sqrt {4 \sqrt {3}+6}\, \sqrt {3}}{4}+\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}+\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {4 \sqrt {3}+6}}{2 x}+\sqrt {3}\right ) \sqrt {4 \sqrt {3}+6}}{2}+\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}+\sqrt {4 \sqrt {3}+6}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (4 \sqrt {3}+6\right ) \sqrt {3}}{2 \sqrt {4 \sqrt {3}-6}}-\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}+\sqrt {4 \sqrt {3}+6}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (4 \sqrt {3}+6\right )}{\sqrt {4 \sqrt {3}-6}}+\frac {2 \arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}+\sqrt {4 \sqrt {3}+6}}{\sqrt {4 \sqrt {3}-6}}\right ) \sqrt {3}}{\sqrt {4 \sqrt {3}-6}}+\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}-\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {4 \sqrt {3}+6}}{2 x}+\sqrt {3}\right ) \sqrt {4 \sqrt {3}+6}\, \sqrt {3}}{4}-\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}-\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {4 \sqrt {3}+6}}{2 x}+\sqrt {3}\right ) \sqrt {4 \sqrt {3}+6}}{2}+\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}-\sqrt {4 \sqrt {3}+6}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (4 \sqrt {3}+6\right ) \sqrt {3}}{2 \sqrt {4 \sqrt {3}-6}}-\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}-\sqrt {4 \sqrt {3}+6}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (4 \sqrt {3}+6\right )}{\sqrt {4 \sqrt {3}-6}}+\frac {2 \arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}-\sqrt {4 \sqrt {3}+6}}{\sqrt {4 \sqrt {3}-6}}\right ) \sqrt {3}}{\sqrt {4 \sqrt {3}-6}}\right ) \sqrt {2}}{2}\) | \(599\) |
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Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (84) = 168\).
Time = 0.37 (sec) , antiderivative size = 550, normalized size of antiderivative = 4.30 \[ \int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx=\frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {-3} - 3} \log \left (\frac {\sqrt {2} {\left (3 \, x^{8} + 9 \, x^{6} - 9 \, x^{4} - 9 \, x^{2} - \sqrt {-3} {\left (x^{8} - x^{6} - 7 \, x^{4} + x^{2} + 1\right )} + 3\right )} \sqrt {\sqrt {-3} - 3} + 12 \, {\left (x^{5} - 2 \, x^{3} + \sqrt {-3} {\left (x^{5} - x\right )} - x\right )} \sqrt {x^{4} + x^{2} - 1}}{x^{8} - x^{6} - x^{4} + x^{2} + 1}\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {-3} - 3} \log \left (-\frac {\sqrt {2} {\left (3 \, x^{8} + 9 \, x^{6} - 9 \, x^{4} - 9 \, x^{2} - \sqrt {-3} {\left (x^{8} - x^{6} - 7 \, x^{4} + x^{2} + 1\right )} + 3\right )} \sqrt {\sqrt {-3} - 3} - 12 \, {\left (x^{5} - 2 \, x^{3} + \sqrt {-3} {\left (x^{5} - x\right )} - x\right )} \sqrt {x^{4} + x^{2} - 1}}{x^{8} - x^{6} - x^{4} + x^{2} + 1}\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {-3} - 3} \log \left (\frac {\sqrt {2} {\left (3 \, x^{8} + 9 \, x^{6} - 9 \, x^{4} - 9 \, x^{2} + \sqrt {-3} {\left (x^{8} - x^{6} - 7 \, x^{4} + x^{2} + 1\right )} + 3\right )} \sqrt {-\sqrt {-3} - 3} + 12 \, {\left (x^{5} - 2 \, x^{3} - \sqrt {-3} {\left (x^{5} - x\right )} - x\right )} \sqrt {x^{4} + x^{2} - 1}}{x^{8} - x^{6} - x^{4} + x^{2} + 1}\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {-3} - 3} \log \left (-\frac {\sqrt {2} {\left (3 \, x^{8} + 9 \, x^{6} - 9 \, x^{4} - 9 \, x^{2} + \sqrt {-3} {\left (x^{8} - x^{6} - 7 \, x^{4} + x^{2} + 1\right )} + 3\right )} \sqrt {-\sqrt {-3} - 3} - 12 \, {\left (x^{5} - 2 \, x^{3} - \sqrt {-3} {\left (x^{5} - x\right )} - x\right )} \sqrt {x^{4} + x^{2} - 1}}{x^{8} - x^{6} - x^{4} + x^{2} + 1}\right ) + \frac {1}{2} \, \log \left (-\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} + x^{2} - 1} x - 1}{x^{4} - 1}\right ) \]
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Timed out. \[ \int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{2}} {\left (x^{4} + 1\right )}}{{\left (x^{8} - x^{6} - x^{4} + x^{2} + 1\right )} {\left (x^{4} - 1\right )}} \,d x } \]
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\[ \int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{2}} {\left (x^{4} + 1\right )}}{{\left (x^{8} - x^{6} - x^{4} + x^{2} + 1\right )} {\left (x^{4} - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx=\int \frac {\left (x^4+1\right )\,{\left (x^4+x^2-1\right )}^{3/2}}{\left (x^4-1\right )\,\left (x^8-x^6-x^4+x^2+1\right )} \,d x \]
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