User:Jaume Oliver Lafont/Dalzell-type integrals

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Dalzell's integral

$\pi ={\frac {22}{7}}-\int _{0}^{1}{\frac {x^{4}(1-x)^{4}}{1+x^{2}}}dx$

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$\pi =3+2\int _{0}^{1}{\frac {x(1-x)^{2}}{1+x^{2}}}dx$

$\pi =3+4\int _{0}^{1}{\frac {x^{4}(1-x)(2+x)}{1+x+x^{2}+x^{3}}}dx$

$\pi =4-4\int _{0}^{1}{\frac {x^{2}}{1+x^{2}}}dx$

$\pi ={\frac {10}{3}}-\int _{0}^{1}{\frac {(1-x)^{4}}{1+x^{2}}}dx$

$\pi ={\frac {31}{10}}+2\int _{0}^{1}{\frac {x^{2}(1-x)^{2}(1-x+x^{2})}{1+x^{2}}}dx$

$\pi ={\frac {31}{10}}+4\int _{0}^{1}{\frac {x^{8}(1-x)(2+x)}{1+x+x^{2}+x^{3}}}dx$

$\pi ={\frac {16}{5}}-\int _{0}^{1}{\frac {x^{2}(1-x)^{2}(1+2x+x^{2})}{1+x^{2}}}dx$

$\pi ={\frac {19}{6}}-2\int _{0}^{1}{\frac {x^{3}(1-x)^{2}}{1+x^{2}}}dx$

$\pi ={\frac {25}{8}}+{\frac {1}{4}}\int _{0}^{1}{\frac {x(1-x)^{4}(1+4x+x^{2})}{1+x^{2}}}dx$

$\pi ={\frac {22}{7}}-\int _{0}^{1}{\frac {x^{4}(1-x)^{4}}{1+x^{2}}}dx$

$\pi ={\frac {22}{7}}-{\frac {1}{28}}\int _{0}^{1}{\frac {x(1-x)^{8}(2+7x+2x^{2})}{1+x^{2}}}dx$

$\pi ={\frac {47}{15}}+\int _{0}^{1}{\frac {x^{2}(1-x)^{4}}{1+x^{2}}}dx$

$\pi ={\frac {47}{15}}+2\int _{0}^{1}{\frac {x^{5}(1-x)^{2}}{1+x^{2}}}dx$

$\pi ={\frac {333}{106}}+{\frac {1}{53}}\int _{0}^{1}{\frac {x^{4}(1-x)^{4}(7-30x+60x^{2}-30x^{3})}{1+x^{2}}}dx$

$\pi ={\frac {377}{120}}-{\frac {1}{2}}\int _{0}^{1}{\frac {x^{5}(1-x)^{6}}{1+x^{2}}}dx$

$\pi ={\frac {864}{275}}-{\frac {1}{20}}\int _{0}^{1}{\frac {x^{4}(1-x)^{6}(7+10x+7x^{2})}{1+x^{2}}}dx$

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$\pi ={\frac {11{\sqrt {3}}}{6}}-{\sqrt {3}}\int _{0}^{1}{\frac {x^{2}(1-x)^{2}}{1+x+x^{2}}}dx$

$\pi ={\frac {457{\sqrt {3}}}{252}}+{\frac {\sqrt {3}}{3}}\int _{0}^{1}{\frac {x^{4}(1-x)^{4}}{1+x+x^{2}}}dx$

$\pi ={\frac {5387{\sqrt {3}}}{2970}}-{\frac {\sqrt {3}}{9}}\int _{0}^{1}{\frac {x^{6}(1-x)^{6}}{1+x+x^{2}}}dx$

$\pi ={\frac {17647759{\sqrt {3}}}{9729720}}+{\frac {\sqrt {3}}{27}}\int _{0}^{1}{\frac {x^{8}(1-x)^{8}}{1+x+x^{2}}}dx$

$\pi ={\frac {9{\sqrt {3}}}{5}}+{\sqrt {3}}\int _{0}^{1}{\frac {x^{3}(1-x)^{2}(1+x)}{1+x+x^{2}}}dx$

$\pi ={\frac {9{\sqrt {3}}}{5}}+{\frac {6{\sqrt {3}}}{5}}\int _{0}^{1}{\frac {x^{3}(1-x)^{2}}{1+x^{2}+x^{4}}}dx$

$\pi =2{\sqrt {3}}-2{\sqrt {3}}\int _{0}^{1}{\frac {x^{4}}{1+x^{2}+x^{4}}}dx$

$\pi ={\frac {789{\sqrt {3}}}{435}}+{\frac {30{\sqrt {3}}}{145}}\int _{0}^{1}{\frac {x^{7}(1-x)^{6}}{1+x^{2}+x^{4}}}dx$

$\pi ={\frac {1656{\sqrt {3}}}{913}}-{\frac {6{\sqrt {3}}}{83}}\int _{0}^{1}{\frac {x^{6}(1-x)^{8}}{1+x^{2}+x^{4}}}dx$

$\pi =8\left(1+{\sqrt {2}}\right)\int _{0}^{1}{\frac {x^{2}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi =8\left(3+2{\sqrt {2}}\right)\int _{0}^{1}{\frac {x(1-x)^{2}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi =2\left(4+3{\sqrt {2}}\right)\int _{0}^{1}{\frac {(1-x)^{4}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi =8\left({\sqrt {2}}-1\right)-8\left({\sqrt {2}}-1\right)\int _{0}^{1}{\frac {x^{6}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi ={\frac {12}{7}}\left(2{\sqrt {2}}-1\right)+{\frac {8}{7}}\left(2{\sqrt {2}}-1\right)\int _{0}^{1}{\frac {x^{5}(1-x)^{2}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi ={\frac {24{\sqrt {2}}}{11}}+{\frac {8}{11}}\int _{0}^{1}{\frac {x(1-x)^{2}(1+2{\sqrt {2}}x^{4})}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi ={\frac {20{\sqrt {2}}}{9}}-{\frac {2{\sqrt {2}}}{3}}\int _{0}^{1}{\frac {x^{4}(1-x)^{4}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi ={\frac {35}{191}}\left(3+10{\sqrt {2}}\right)+{\frac {4}{191}}\left(3+10{\sqrt {2}}\right)\int _{0}^{1}{\frac {x^{3}(1-x)^{6}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi ={\frac {126\left(8+17{\sqrt {2}}\right)}{1285}}-{\frac {8+17{\sqrt {2}}}{257}}\int _{0}^{1}{\frac {x^{2}(1-x)^{8}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi ={\frac {154\left(33+58{\sqrt {2}}\right)}{5639}}+{\frac {2\left(33+58{\sqrt {2}}\right)}{5639}}\int _{0}^{1}{\frac {x(1-x)^{10}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi ={\frac {858\left(62+99{\sqrt {2}}\right)}{55153}}-{\frac {62+99{\sqrt {2}}}{15758}}\int _{0}^{1}{\frac {(1-x)^{12}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi ={\frac {173\left(16+17{\sqrt {2}}\right)}{2205}}+{\frac {16+17{\sqrt {2}}}{161}}\int _{0}^{1}{\frac {x^{6}(1-x)^{8}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi ={\frac {691\left(11+10{\sqrt {2}}\right)}{5530}}-{\frac {4\left(11+10{\sqrt {2}}\right)}{79}}\int _{0}^{1}{\frac {x^{7}(1-x)^{6}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi ={\frac {1838\left(62+99{\sqrt {2}}\right)}{118185}}-{\frac {62+99{\sqrt {2}}}{15758}}\int _{0}^{1}{\frac {x^{8}(1-x)^{12}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi ={\frac {10810\left(8+17{\sqrt {2}}\right)}{110253}}-{\frac {8+17{\sqrt {2}}}{257}}\int _{0}^{1}{\frac {x^{10}(1-x)^{8}}{1+x^{2}+x^{4}+x^{6}}}dx$

$\pi ={\frac {210798485\left(424+577{\sqrt {2}}\right)}{83203142022}}+{\frac {424+577{\sqrt {2}}}{972164}}\int _{0}^{1}{\frac {x^{10}(1-x)^{16}}{1+x^{2}+x^{4}+x^{6}}}dx$

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$\pi ={\frac {9-e}{2}}+\int _{0}^{1}{\frac {(1-x)^{2}}{4}}\left(e^{x}-1-x-{\frac {x^{2}}{2}}-{\frac {2x^{5}(1-x)^{4}}{1+x^{2}}}\right)dx$