Optimal. Leaf size=74 \[ \frac {15}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2 x^4-1}}\right )-\frac {15}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2 x^4-1}}\right )+\frac {\sqrt [4]{2 x^4-1} \left (69 x^8-56 x^4-8\right )}{20 x^5 \left (x^4-1\right )} \]
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Rubi [C] time = 2.88, antiderivative size = 198, normalized size of antiderivative = 2.68, number of steps used = 176, number of rules used = 33, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.222, Rules used = {6742, 2153, 1240, 412, 529, 237, 335, 275, 232, 407, 409, 1213, 537, 511, 510, 1248, 733, 844, 234, 220, 747, 400, 442, 444, 63, 212, 206, 203, 1336, 264, 277, 331, 298} \begin {gather*} -\frac {4 \sqrt [4]{2 x^4-1} x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{3 \sqrt [4]{1-2 x^4}}-\frac {\sqrt [4]{2 x^4-1} x^3 \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{3 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}+\frac {4 \sqrt [4]{2 x^4-1}}{x}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4-1}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4-1}}\right )-\frac {2 \left (2 x^4-1\right )^{5/4}}{5 x^5} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 63
Rule 203
Rule 206
Rule 212
Rule 220
Rule 232
Rule 234
Rule 237
Rule 264
Rule 275
Rule 277
Rule 298
Rule 331
Rule 335
Rule 400
Rule 407
Rule 409
Rule 412
Rule 442
Rule 444
Rule 510
Rule 511
Rule 529
Rule 537
Rule 733
Rule 747
Rule 844
Rule 1213
Rule 1240
Rule 1248
Rule 1336
Rule 2153
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-1+2 x^4} \left (-2+x^8\right )}{x^6 \left (-1+x^4\right )^2} \, dx &=\int \left (-\frac {\sqrt [4]{-1+2 x^4}}{16 (-1+x)^2}-\frac {2 \sqrt [4]{-1+2 x^4}}{x^6}-\frac {4 \sqrt [4]{-1+2 x^4}}{x^2}-\frac {\sqrt [4]{-1+2 x^4}}{16 (1+x)^2}+\frac {17 \sqrt [4]{-1+2 x^4}}{8 \left (-1+x^2\right )}+\frac {\sqrt [4]{-1+2 x^4}}{4 \left (1+x^2\right )^2}+\frac {2 \sqrt [4]{-1+2 x^4}}{1+x^2}\right ) \, dx\\ &=-\left (\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{(-1+x)^2} \, dx\right )-\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{(1+x)^2} \, dx+\frac {1}{4} \int \frac {\sqrt [4]{-1+2 x^4}}{\left (1+x^2\right )^2} \, dx-2 \int \frac {\sqrt [4]{-1+2 x^4}}{x^6} \, dx+2 \int \frac {\sqrt [4]{-1+2 x^4}}{1+x^2} \, dx+\frac {17}{8} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^2} \, dx-4 \int \frac {\sqrt [4]{-1+2 x^4}}{x^2} \, dx\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {1}{16} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}-\frac {2 x \sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}+\frac {x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}\right ) \, dx-\frac {1}{16} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}+\frac {2 x \sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}+\frac {x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}\right ) \, dx+\frac {1}{4} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}-\frac {2 x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}+\frac {x^4 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}\right ) \, dx+2 \int \left (\frac {\sqrt [4]{-1+2 x^4}}{1-x^4}+\frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4}\right ) \, dx+\frac {17}{8} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{-1+x^4}+\frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4}\right ) \, dx-8 \int \frac {x^2}{\left (-1+2 x^4\right )^{3/4}} \, dx\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-2 \left (\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2} \, dx\right )-2 \left (\frac {1}{16} \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2} \, dx\right )+\frac {1}{4} \int \frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx+\frac {1}{4} \int \frac {x^4 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx-\frac {1}{2} \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx+2 \int \frac {\sqrt [4]{-1+2 x^4}}{1-x^4} \, dx+2 \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx+\frac {17}{8} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx+\frac {17}{8} \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx-8 \operatorname {Subst}\left (\int \frac {x^2}{1-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}+\frac {1}{16} \int \frac {3-4 x^4}{\left (-1+x^4\right ) \left (-1+2 x^4\right )^{3/4}} \, dx-2 \left (\frac {1}{16} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2}+\frac {\sqrt [4]{-1+2 x^4}}{-1+x^2}\right ) \, dx\right )-2 \left (\frac {1}{16} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}+\frac {2 x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}+\frac {x^4 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}\right ) \, dx\right )-\left (2 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\left (2 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^4 \sqrt [4]{1-2 x^4}}{\left (-1+x^4\right )^2} \, dx}{4 \sqrt [4]{1-2 x^4}}-\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^2 \sqrt [4]{1-2 x^4}}{\left (-1+x^4\right )^2} \, dx}{2 \sqrt [4]{1-2 x^4}}+\frac {\left (2 \sqrt [4]{-1+2 x^4}\right ) \int \frac {x^2 \sqrt [4]{1-2 x^4}}{-1+x^4} \, dx}{\sqrt [4]{1-2 x^4}}+\frac {\left (17 \sqrt [4]{-1+2 x^4}\right ) \int \frac {x^2 \sqrt [4]{1-2 x^4}}{-1+x^4} \, dx}{8 \sqrt [4]{1-2 x^4}}+\left (2 \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (1-x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {1}{8} \left (17 \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (-1+x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^2\right )^2} \, dx+\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^2} \, dx\right )-\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx-\frac {1}{8} \int \frac {1}{\left (-1+2 x^4\right )^{3/4}} \, dx-2 \left (\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx+\frac {1}{16} \int \frac {x^4 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx+\frac {1}{8} \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx\right )+\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {1}{16} \left (17 \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {1}{16} \left (17 \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {1}{16} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}+\frac {2 x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}+\frac {x^4 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2}\right ) \, dx+\frac {1}{16} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{-1+x^4}+\frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4}\right ) \, dx\right )-\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3} \, dx}{8 \left (-1+2 x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {1}{64} \int \frac {3-4 x^4}{\left (-1+x^4\right ) \left (-1+2 x^4\right )^{3/4}} \, dx+\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^4 \sqrt [4]{1-2 x^4}}{\left (-1+x^4\right )^2} \, dx}{16 \sqrt [4]{1-2 x^4}}+\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^2 \sqrt [4]{1-2 x^4}}{\left (-1+x^4\right )^2} \, dx}{8 \sqrt [4]{1-2 x^4}}\right )-\frac {1}{16} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (-1+x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {\left (17 \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1-x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{8 \sqrt {2}}-\frac {\left (17 \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1+x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{8 \sqrt {2}}+\left (\sqrt {2} \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1-x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\left (\sqrt {2} \sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1+x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{16 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{16 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-\frac {1}{64} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx-\frac {1}{32} \int \frac {1}{\left (-1+2 x^4\right )^{3/4}} \, dx\right )-2 \left (\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx+\frac {1}{16} \int \frac {x^4 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx+\frac {1}{16} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx+\frac {1}{16} \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx+\frac {1}{8} \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{\left (-1+x^4\right )^2} \, dx\right )+\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {x^4}{2}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{8 \left (-1+2 x^4\right )^{3/4}}+\frac {1}{32} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {1}{32} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{16 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{16 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}+\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x^2}{2}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{16 \left (-1+2 x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3} \, dx}{32 \left (-1+2 x^4\right )^{3/4}}-\frac {1}{64} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (-1+x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\right )-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {1}{64} \int \frac {3-4 x^4}{\left (-1+x^4\right ) \left (-1+2 x^4\right )^{3/4}} \, dx+\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^4 \sqrt [4]{1-2 x^4}}{\left (-1+x^4\right )^2} \, dx}{16 \sqrt [4]{1-2 x^4}}+\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^2 \sqrt [4]{1-2 x^4}}{-1+x^4} \, dx}{16 \sqrt [4]{1-2 x^4}}+\frac {\sqrt [4]{-1+2 x^4} \int \frac {x^2 \sqrt [4]{1-2 x^4}}{\left (-1+x^4\right )^2} \, dx}{8 \sqrt [4]{1-2 x^4}}+\frac {1}{16} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (-1+x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\right )+\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1-x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{16 \sqrt {2}}+\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1+x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{16 \sqrt {2}}\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{8 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}+\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {x^4}{2}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{32 \left (-1+2 x^4\right )^{3/4}}+\frac {1}{128} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {1}{128} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\right )-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}-\frac {x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{48 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-\frac {1}{64} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx-\frac {1}{32} \int \frac {1}{\left (-1+2 x^4\right )^{3/4}} \, dx-\frac {1}{32} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {1}{32} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{8 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}+\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x^2}{2}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{64 \left (-1+2 x^4\right )^{3/4}}+\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1-x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{64 \sqrt {2}}+\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1+x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{64 \sqrt {2}}\right )-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}-\frac {x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{48 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3} \, dx}{32 \left (-1+2 x^4\right )^{3/4}}-\frac {1}{64} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (-1+x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1-x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{16 \sqrt {2}}-\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1+x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{16 \sqrt {2}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{8 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{32 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}+\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}\right )-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}-\frac {x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{48 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}+\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {x^4}{2}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{32 \left (-1+2 x^4\right )^{3/4}}+\frac {1}{128} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {1}{128} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^4}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{8 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{32 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}+\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}\right )-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}-\frac {x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{48 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}+\frac {\left (\left (1-\frac {1}{2 x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x^2}{2}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{64 \left (-1+2 x^4\right )^{3/4}}+\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1-x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{64 \sqrt {2}}+\frac {\left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sqrt {2}-2 x^2} \left (1+x^2\right ) \sqrt {\sqrt {2}+2 x^2}} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )}{64 \sqrt {2}}\right )\\ &=\frac {4 \sqrt [4]{-1+2 x^4}}{x}+\frac {x \sqrt [4]{-1+2 x^4}}{16 \left (1-x^4\right )}-\frac {2 \left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {11 x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{8 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{20 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{8 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{32 \sqrt [4]{2}}-\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{6 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}-\frac {x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{48 \sqrt [4]{1-2 x^4}}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{32 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}-\frac {3 \sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}-\frac {3 \sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}\right )-2 \left (\frac {x \sqrt [4]{-1+2 x^4}}{64 \left (1-x^4\right )}+\frac {x^5 \sqrt [4]{-1+2 x^4} F_1\left (\frac {5}{4};-\frac {1}{4},2;\frac {9}{4};2 x^4,x^4\right )}{80 \sqrt [4]{1-2 x^4}}+\frac {\left (2-\frac {1}{x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\sqrt {2} x^2\right )\right |2\right )}{32 \sqrt [4]{2} \left (-1+2 x^4\right )^{3/4}}+\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (-\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {\sqrt {\frac {1}{1-2 x^4}} \sqrt {-1+2 x^4} \Pi \left (\frac {1}{\sqrt {2}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )\right |-1\right )}{128 \sqrt [4]{2}}+\frac {x^3 \sqrt [4]{-1+2 x^4} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )}{24 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.23, size = 92, normalized size = 1.24 \begin {gather*} \frac {25 \left (1-2 x^4\right )^{3/4} \sqrt [4]{1-x^4} x^8 \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {x^4}{1-x^4}\right )+138 x^{12}-181 x^8+40 x^4+8}{20 x^5 \left (x^4-1\right ) \left (2 x^4-1\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 74, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-1+2 x^4} \left (-8-56 x^4+69 x^8\right )}{20 x^5 \left (-1+x^4\right )}+\frac {15}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {15}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+2 x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 3.95, size = 151, normalized size = 2.04 \begin {gather*} \frac {75 \, {\left (x^{9} - x^{5}\right )} \arctan \left (\frac {2 \, {\left ({\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + {\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - 1}\right ) + 75 \, {\left (x^{9} - x^{5}\right )} \log \left (-\frac {3 \, x^{4} - 2 \, {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {2 \, x^{4} - 1} x^{2} - 2 \, {\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} - 1}\right ) + 4 \, {\left (69 \, x^{8} - 56 \, x^{4} - 8\right )} {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{80 \, {\left (x^{9} - x^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 108, normalized size = 1.46 \begin {gather*} -\frac {2 \, {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} - 2\right )}}{5 \, x} - \frac {4 \, {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x} + \frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{4 \, x {\left (\frac {1}{x^{4}} - 1\right )}} + \frac {15}{8} \, \arctan \left (\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {15}{16} \, \log \left (\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) - \frac {15}{16} \, \log \left ({\left | \frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.37, size = 196, normalized size = 2.65
method | result | size |
trager | \(\frac {\left (2 x^{4}-1\right )^{\frac {1}{4}} \left (69 x^{8}-56 x^{4}-8\right )}{20 x^{5} \left (x^{4}-1\right )}+\frac {15 \ln \left (\frac {2 \left (2 x^{4}-1\right )^{\frac {3}{4}} x -2 \sqrt {2 x^{4}-1}\, x^{2}+2 \left (2 x^{4}-1\right )^{\frac {1}{4}} x^{3}-3 x^{4}+1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}+\frac {15 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 \sqrt {2 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (2 x^{4}-1\right )^{\frac {3}{4}} x -2 \left (2 x^{4}-1\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}\) | \(196\) |
risch | \(\frac {138 x^{12}-181 x^{8}+40 x^{4}+8}{20 x^{5} \left (x^{4}-1\right ) \left (2 x^{4}-1\right )^{\frac {3}{4}}}+\frac {\left (-\frac {15 \ln \left (\frac {12 x^{12}+8 \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x^{9}+4 \sqrt {8 x^{12}-12 x^{8}+6 x^{4}-1}\, x^{6}-16 x^{8}+2 \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {3}{4}} x^{3}-8 \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x^{5}-2 \sqrt {8 x^{12}-12 x^{8}+6 x^{4}-1}\, x^{2}+7 x^{4}+2 \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x -1}{\left (2 x^{4}-1\right )^{2} \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}+\frac {15 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {12 x^{12}-8 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x^{9}-4 \sqrt {8 x^{12}-12 x^{8}+6 x^{4}-1}\, x^{6}-16 x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {3}{4}} x^{3}+8 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x^{5}+2 \sqrt {8 x^{12}-12 x^{8}+6 x^{4}-1}\, x^{2}+7 x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{12}-12 x^{8}+6 x^{4}-1\right )^{\frac {1}{4}} x -1}{\left (2 x^{4}-1\right )^{2} \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{16}\right ) \left (\left (2 x^{4}-1\right )^{3}\right )^{\frac {1}{4}}}{\left (2 x^{4}-1\right )^{\frac {3}{4}}}\) | \(471\) |
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 \begin {gather*} \int \frac {{\left (x^{8} - 2\right )} {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (x^{4} - 1\right )}^{2} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^8-2\right )\,{\left (2\,x^4-1\right )}^{1/4}}{x^6\,{\left (x^4-1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{2 x^{4} - 1} \left (x^{8} - 2\right )}{x^{6} \left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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