Optimal. Leaf size=293 \[ \frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^4-1}}{\sqrt {x^4-1}-x^2}\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}}-\frac {2}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt [4]{x^4-1}}{\sqrt {x^4-1}-x^2}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^4-1}}{\sqrt {x^4-1}+x^2}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^4-1}}{\sqrt {x^4-1}+x^2}\right ) \]
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Rubi [C] time = 0.28, antiderivative size = 298, normalized size of antiderivative = 1.02, number of steps used = 28, number of rules used = 15, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.441, Rules used = {28, 1428, 416, 530, 240, 212, 206, 203, 377, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\left (\frac {1}{8}+\frac {i}{8}\right ) \sqrt [8]{-1} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \tan ^{-1}\left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt {2}}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \tan ^{-1}\left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}+1\right )}{\sqrt {2}}+\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\left (\frac {1}{8}+\frac {i}{8}\right ) \sqrt [8]{-1} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8} \log \left (\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4-1}}+\frac {x^2}{\sqrt {x^4-1}}+\sqrt [4]{-1}\right )}{\sqrt {2}}+\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8} \log \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}-\frac {(-1)^{3/4} x^2}{\sqrt {x^4-1}}+1\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 203
Rule 204
Rule 206
Rule 211
Rule 212
Rule 240
Rule 377
Rule 416
Rule 530
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1428
Rubi steps
\begin {align*} \int \frac {1-2 x^4+x^8}{\sqrt [4]{-1+x^4} \left (1-2 x^4+2 x^8\right )} \, dx &=\int \frac {\left (-1+x^4\right )^{7/4}}{1-2 x^4+2 x^8} \, dx\\ &=-\left (2 i \int \frac {\left (-1+x^4\right )^{7/4}}{(-2-2 i)+4 x^4} \, dx\right )+2 i \int \frac {\left (-1+x^4\right )^{7/4}}{(-2+2 i)+4 x^4} \, dx\\ &=-\left (\frac {1}{8} i \int \frac {(14-2 i)-(20-8 i) x^4}{\sqrt [4]{-1+x^4} \left ((-2-2 i)+4 x^4\right )} \, dx\right )+\frac {1}{8} i \int \frac {(14+2 i)-(20+8 i) x^4}{\sqrt [4]{-1+x^4} \left ((-2+2 i)+4 x^4\right )} \, dx\\ &=-\left (\left (-\frac {1}{4}-\frac {5 i}{8}\right ) \int \frac {1}{\sqrt [4]{-1+x^4}} \, dx\right )+\left (\frac {1}{4}-\frac {5 i}{8}\right ) \int \frac {1}{\sqrt [4]{-1+x^4}} \, dx-\int \frac {1}{\sqrt [4]{-1+x^4} \left ((-2-2 i)+4 x^4\right )} \, dx-\int \frac {1}{\sqrt [4]{-1+x^4} \left ((-2+2 i)+4 x^4\right )} \, dx\\ &=-\left (\left (-\frac {1}{4}-\frac {5 i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\right )+\left (\frac {1}{4}-\frac {5 i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\operatorname {Subst}\left (\int \frac {1}{(-2+2 i)-(2+2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\operatorname {Subst}\left (\int \frac {1}{(-2-2 i)-(2-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=-\left (\left (-\frac {1}{8}-\frac {5 i}{16}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\right )-\left (-\frac {1}{8}-\frac {5 i}{16}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\left (\frac {1}{8}-\frac {5 i}{16}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\left (\frac {1}{8}-\frac {5 i}{16}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} (-1)^{3/4} \operatorname {Subst}\left (\int \frac {\sqrt [4]{-1}-x^2}{(-2-2 i)-(2-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} (-1)^{3/4} \operatorname {Subst}\left (\int \frac {\sqrt [4]{-1}+x^2}{(-2-2 i)-(2-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {i \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {i \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}\\ &=\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {(-1)^{3/8} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {(-1)^{3/8} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-\sqrt [8]{-1} \sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+\sqrt [8]{-1} \sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}+\frac {\left (\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [8]{-1} \sqrt {2}+2 x}{-\sqrt [4]{-1}-\sqrt [8]{-1} \sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}+\frac {\left (\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [8]{-1} \sqrt {2}-2 x}{-\sqrt [4]{-1}+\sqrt [8]{-1} \sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}\\ &=\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {(-1)^{3/8} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {(-1)^{3/8} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8} \log \left (\sqrt [4]{-1}+\frac {x^2}{\sqrt {-1+x^4}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}+\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8} \log \left (1-\frac {(-1)^{3/4} x^2}{\sqrt {-1+x^4}}+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}+\frac {1}{8} (-1)^{7/8} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{8} (-1)^{7/8} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {(-1)^{3/8} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {1}{8} (-1)^{7/8} \tan ^{-1}\left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{8} (-1)^{7/8} \tan ^{-1}\left (1+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {(-1)^{3/8} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8} \log \left (\sqrt [4]{-1}+\frac {x^2}{\sqrt {-1+x^4}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}+\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8} \log \left (1-\frac {(-1)^{3/4} x^2}{\sqrt {-1+x^4}}+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [F] time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-2 x^4+x^8}{\sqrt [4]{-1+x^4} \left (1-2 x^4+2 x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.11, size = 273, normalized size = 0.93 \begin {gather*} \frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^4}}{-x^2+\sqrt {-1+x^4}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^4}}{-x^2+\sqrt {-1+x^4}}\right )+\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^4}}{x^2+\sqrt {-1+x^4}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^4}}{x^2+\sqrt {-1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 1982, normalized size = 6.76
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} - 2 \, x^{4} + 1}{{\left (2 \, x^{8} - 2 \, x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 13.14, size = 1006, normalized size = 3.43
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x -2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{8}+\frac {\ln \left (2 \left (x^{4}-1\right )^{\frac {3}{4}} x +2 x^{2} \sqrt {x^{4}-1}+2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}+2 x^{4}-1\right )}{8}-\frac {\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {\sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} x^{2}-\left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{2} x^{3}-\left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+x^{4}-1}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-\sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} x^{2}+\left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{2} x^{3}-\left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{4}-2 \left (x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+x^{4}-1}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} \RootOf \left (\textit {\_Z}^{2}+1\right )^{2} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+4 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-1}\, x^{2}+4 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{2}+8 \left (x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3}-\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3}}{\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-x^{4}+1}\right )}{32}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} \RootOf \left (\textit {\_Z}^{2}+1\right )^{2} x^{4}-4 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-1}\, x^{2}-4 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{2}+8 \left (x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3}-\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3}}{\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-x^{4}+1}\right )}{32}\) | \(1006\) |
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 {x^{8} - 2 \, x^{4} + 1}{{\left (2 \, x^{8} - 2 \, x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}\,{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 = -0.00 \begin {gather*} \int \frac {x^8-2\,x^4+1}{{\left (x^4-1\right )}^{1/4}\,\left (2\,x^8-2\,x^4+1\right )} \,d x \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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