Optimal. Leaf size=362 \[ \frac {x \left (x^4-3\right )}{2 \left (x^4-1\right ) \sqrt {\sqrt {x^4+1}+x^2}}+\frac {1}{2} \sqrt {\frac {1}{2} \left (7+5 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (7+5 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{\sqrt {2}}-\frac {1}{2} \sqrt {\frac {1}{2} \left (5 \sqrt {2}-7\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (5 \sqrt {2}-7\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]
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Rubi [F] time = 1.70, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx &=\int \left (\frac {1}{\sqrt {x^2+\sqrt {1+x^4}}}+\frac {1}{4 (-1+x)^2 \sqrt {x^2+\sqrt {1+x^4}}}+\frac {1}{4 (1+x)^2 \sqrt {x^2+\sqrt {1+x^4}}}+\frac {1}{\left (-1-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\frac {1}{2 \left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\frac {1}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx\\ &=\frac {1}{4} \int \frac {1}{(-1+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} \int \frac {1}{(1+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{2} \int \frac {1}{\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\left (-1-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=\frac {1}{4} \int \frac {1}{(-1+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} \int \frac {1}{(1+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{2} \int \left (-\frac {1}{2 (1-x) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {1}{2 (1+x) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \left (-\frac {i}{2 (i-x) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {i}{2 (i+x) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+\int \left (-\frac {1}{4 (i-x)^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {1}{4 (i+x)^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {1}{2 \left (-1-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx\\ &=-\left (\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx\right )-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} \int \frac {1}{(i-x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} \int \frac {1}{(1-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} \int \frac {1}{(-1+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} \int \frac {1}{(i+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} \int \frac {1}{(1+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} \int \frac {1}{(1+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{2} \int \frac {1}{\left (-1-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-\left (\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx\right )-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} \int \frac {1}{(i-x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} \int \frac {1}{(1-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} \int \frac {1}{(-1+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} \int \frac {1}{(i+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} \int \frac {1}{(1+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} \int \frac {1}{(1+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{2} \int \left (-\frac {i}{2 (i-x) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {i}{2 (i+x) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=\frac {1}{4} i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} i \int \frac {1}{(i+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} \int \frac {1}{(i-x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} \int \frac {1}{(1-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} \int \frac {1}{(-1+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} \int \frac {1}{(i+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} \int \frac {1}{(1+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} \int \frac {1}{(1+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 6.85, size = 483, normalized size = 1.33 \begin {gather*} \frac {x \left (-3+x^4\right )}{2 \left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\frac {1}{2} \sqrt {\frac {7}{2}+\frac {5}{\sqrt {2}}} \tan ^{-1}\left (\frac {-\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\frac {1}{2} \sqrt {\frac {7}{2}+\frac {5}{\sqrt {2}}} \tan ^{-1}\left (\frac {-\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {\tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}-\frac {1}{2} \sqrt {-\frac {7}{2}+\frac {5}{\sqrt {2}}} \tanh ^{-1}\left (\frac {-\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\frac {1}{2} \sqrt {-\frac {7}{2}+\frac {5}{\sqrt {2}}} \tanh ^{-1}\left (\frac {-\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 6.35, size = 553, normalized size = 1.53 \begin {gather*} -\frac {4 \, \sqrt {2} {\left (x^{4} - 1\right )} \sqrt {5 \, \sqrt {2} + 7} \arctan \left (\frac {2 \, {\left (6 \, x^{7} + 10 \, x^{3} - \sqrt {2} {\left (5 \, x^{7} + 7 \, x^{3}\right )} - {\left (x^{5} - 2 \, \sqrt {2} {\left (x^{5} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {5 \, \sqrt {2} + 7} - {\left (5 \, x^{8} + 10 \, x^{4} - \sqrt {2} {\left (3 \, x^{8} + 4 \, x^{4} + 1\right )} - 2 \, {\left (x^{6} + 3 \, x^{2} - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}\right )} \sqrt {x^{4} + 1} + 1\right )} \sqrt {5 \, \sqrt {2} + 7} \sqrt {\sqrt {2} - 1}}{7 \, x^{8} + 10 \, x^{4} - 1}\right ) + \sqrt {2} {\left (x^{4} - 1\right )} \sqrt {5 \, \sqrt {2} - 7} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (17 \, x^{4} + 2 \, \sqrt {2} {\left (6 \, x^{4} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (5 \, \sqrt {2} x^{2} + 7 \, x^{2}\right )} + 3\right )} \sqrt {5 \, \sqrt {2} - 7}}{x^{4} - 1}\right ) - \sqrt {2} {\left (x^{4} - 1\right )} \sqrt {5 \, \sqrt {2} - 7} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (17 \, x^{4} + 2 \, \sqrt {2} {\left (6 \, x^{4} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (5 \, \sqrt {2} x^{2} + 7 \, x^{2}\right )} + 3\right )} \sqrt {5 \, \sqrt {2} - 7}}{x^{4} - 1}\right ) - 2 \, \sqrt {2} {\left (x^{4} - 1\right )} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) + 8 \, {\left (x^{7} - 3 \, x^{3} - {\left (x^{5} - 3 \, x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{16 \, {\left (x^{4} - 1\right )}} \end {gather*}
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 {{\left (x^{4} + 1\right )}^{2}}{{\left (x^{4} - 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}+1\right )^{2}}{\left (x^{4}-1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}\, dx\]
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^{4} + 1\right )}^{2}}{{\left (x^{4} - 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}\,{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 {{\left (x^4+1\right )}^2}{{\left (x^4-1\right )}^2\,\sqrt {\sqrt {x^4+1}+x^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 {\left (x^{4} + 1\right )^{2}}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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