Optimal. Leaf size=311 \[ \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]
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Rubi [F] time = 1.79, 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 {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{-1+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{-1+x^4} \, dx &=\int \left (-\frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1-x^2\right )}-\frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1+x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{1-x^2} \, dx\right )-\frac {1}{2} \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {i \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{2 (i-x)}+\frac {i \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{2 (i+x)}\right ) \, dx\right )-\frac {1}{2} \int \left (\frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{2 (1-x)}+\frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{2 (1+x)}\right ) \, dx\\ &=-\left (\frac {1}{4} i \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{i-x} \, dx\right )-\frac {1}{4} i \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{i+x} \, dx-\frac {1}{4} \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{1-x} \, dx-\frac {1}{4} \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{1+x} \, dx\\ \end {align*}
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Mathematica [F] time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{-1+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.01, size = 440, normalized size = 1.41 \begin {gather*} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{\sqrt {2}}-\frac {1}{2}} \sqrt {x^4+1}+\sqrt {\frac {1}{\sqrt {2}}-\frac {1}{2}} x^2-\sqrt {\frac {1}{\sqrt {2}}-\frac {1}{2}}}{x \sqrt {\sqrt {x^4+1}+x^2}}\right )-\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {x^4+1}+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2-\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}}{x \sqrt {\sqrt {x^4+1}+x^2}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt {x^4+1}}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{x \sqrt {\sqrt {x^4+1}+x^2}}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{\sqrt {2}}-\frac {1}{2}} \sqrt {x^4+1}+\sqrt {\frac {1}{\sqrt {2}}-\frac {1}{2}} x^2-\sqrt {\frac {1}{\sqrt {2}}-\frac {1}{2}}}{x \sqrt {\sqrt {x^4+1}+x^2}}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {x^4+1}+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2-\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}}{x \sqrt {\sqrt {x^4+1}+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 5.15, size = 433, normalized size = 1.39 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \arctan \left (\frac {x^{8} - 2 \, x^{4} - 2 \, {\left (2 \, x^{7} - 2 \, x^{3} + \sqrt {2} {\left (3 \, x^{7} + x^{3}\right )} - {\left (4 \, \sqrt {2} x^{5} + 5 \, x^{5} - x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1} - 2 \, \sqrt {2} {\left (x^{8} + 3 \, x^{4}\right )} - 2 \, {\left (3 \, x^{6} + x^{2} + \sqrt {2} {\left (x^{6} - x^{2}\right )}\right )} \sqrt {x^{4} + 1} + 1}{7 \, x^{8} + 10 \, x^{4} - 1}\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \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 (2 \, \sqrt {2} x^{4} + 3 \, x^{4} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} + x^{2}\right )} + 1\right )} \sqrt {\sqrt {2} + 1}}{x^{4} - 1}\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \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 (2 \, \sqrt {2} x^{4} + 3 \, x^{4} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} + x^{2}\right )} + 1\right )} \sqrt {\sqrt {2} + 1}}{x^{4} - 1}\right ) + \frac {1}{4} \, \sqrt {2} \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 ) \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 {\sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{x^{4} - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.49, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}}{x^{4}-1}\, dx \end {gather*}
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 {\sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{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 {\sqrt {x^4+1}\,\sqrt {\sqrt {x^4+1}+x^2}}{x^4-1} \,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 {x^{2} + \sqrt {x^{4} + 1}} \sqrt {x^{4} + 1}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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