Optimal. Leaf size=152 \[ \frac {2 x}{\sqrt {\sqrt {x^2+1}+1}}-2 \tan ^{-1}\left (\frac {x}{\sqrt {\sqrt {x^2+1}+1}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right )-\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+1}}\right )-\sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+1}}\right ) \]
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Rubi [F] time = 0.88, 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 {1+x^4}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx &=\int \left (\frac {1}{\sqrt {1+\sqrt {1+x^2}}}+\frac {2}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}\right ) \, dx\\ &=2 \int \frac {1}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx+\int \frac {1}{\sqrt {1+\sqrt {1+x^2}}} \, dx\\ &=2 \int \left (-\frac {1}{2 \left (1-x^2\right ) \sqrt {1+\sqrt {1+x^2}}}-\frac {1}{2 \left (1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}\right ) \, dx+\int \frac {1}{\sqrt {1+\sqrt {1+x^2}}} \, dx\\ &=\int \frac {1}{\sqrt {1+\sqrt {1+x^2}}} \, dx-\int \frac {1}{\left (1-x^2\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx-\int \frac {1}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx\\ &=\int \frac {1}{\sqrt {1+\sqrt {1+x^2}}} \, dx-\int \left (\frac {i}{2 (i-x) \sqrt {1+\sqrt {1+x^2}}}+\frac {i}{2 (i+x) \sqrt {1+\sqrt {1+x^2}}}\right ) \, dx-\int \left (\frac {1}{2 (1-x) \sqrt {1+\sqrt {1+x^2}}}+\frac {1}{2 (1+x) \sqrt {1+\sqrt {1+x^2}}}\right ) \, dx\\ &=-\left (\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {1+\sqrt {1+x^2}}} \, dx\right )-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {1+\sqrt {1+x^2}}} \, dx-\frac {1}{2} \int \frac {1}{(1-x) \sqrt {1+\sqrt {1+x^2}}} \, dx-\frac {1}{2} \int \frac {1}{(1+x) \sqrt {1+\sqrt {1+x^2}}} \, dx+\int \frac {1}{\sqrt {1+\sqrt {1+x^2}}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.49, size = 152, normalized size = 1.00 \begin {gather*} \frac {2 x}{\sqrt {1+\sqrt {1+x^2}}}-2 \tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}\right )-\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {-1+\sqrt {2}} x}{\sqrt {1+\sqrt {1+x^2}}}\right )-\sqrt {-1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {2}} x}{\sqrt {1+\sqrt {1+x^2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 21.35, size = 520, normalized size = 3.42 \begin {gather*} \frac {4 \, x \sqrt {\sqrt {2} + 1} \arctan \left (-\frac {{\left (71 \, x^{5} - 874 \, x^{3} - \sqrt {2} {\left (61 \, x^{5} - 548 \, x^{3} - 81 \, x\right )} - 2 \, {\left (51 \, x^{3} - 2 \, \sqrt {2} {\left (5 \, x^{3} - 127 \, x\right )} - 335 \, x\right )} \sqrt {x^{2} + 1} - 173 \, x\right )} \sqrt {3821 \, \sqrt {2} + 4841} \sqrt {\sqrt {2} + 1} - 4802 \, {\left (x^{4} - 6 \, x^{2} + \sqrt {2} {\left (3 \, x^{2} + 1\right )} + {\left (x^{2} + \sqrt {2} {\left (x^{2} - 1\right )} + 3\right )} \sqrt {x^{2} + 1} - 3\right )} \sqrt {\sqrt {2} + 1} \sqrt {\sqrt {x^{2} + 1} + 1}}{2401 \, {\left (x^{5} - 10 \, x^{3} - 7 \, x\right )}}\right ) - 4 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^{2} + 1} + 1}}{x}\right ) + x \sqrt {\sqrt {2} - 1} \log \left (-\frac {{\left (51 \, x^{3} - 2 \, \sqrt {2} {\left (5 \, x^{3} + 66 \, x\right )} + 2 \, \sqrt {x^{2} + 1} {\left (61 \, \sqrt {2} x - 71 \, x\right )} + 193 \, x\right )} \sqrt {\sqrt {2} - 1} + 2 \, {\left (71 \, x^{2} - \sqrt {2} {\left (61 \, x^{2} + 132\right )} + \sqrt {x^{2} + 1} {\left (132 \, \sqrt {2} - 193\right )} + 193\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{3} - x}\right ) - x \sqrt {\sqrt {2} - 1} \log \left (\frac {{\left (51 \, x^{3} - 2 \, \sqrt {2} {\left (5 \, x^{3} + 66 \, x\right )} + 2 \, \sqrt {x^{2} + 1} {\left (61 \, \sqrt {2} x - 71 \, x\right )} + 193 \, x\right )} \sqrt {\sqrt {2} - 1} - 2 \, {\left (71 \, x^{2} - \sqrt {2} {\left (61 \, x^{2} + 132\right )} + \sqrt {x^{2} + 1} {\left (132 \, \sqrt {2} - 193\right )} + 193\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{3} - x}\right ) + 2 \, x \arctan \left (\frac {4 \, {\left (x^{4} - 12 \, x^{2} + {\left (5 \, x^{2} - 3\right )} \sqrt {x^{2} + 1} + 3\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{5} - 46 \, x^{3} + 17 \, x}\right ) + 8 \, \sqrt {\sqrt {x^{2} + 1} + 1} {\left (\sqrt {x^{2} + 1} - 1\right )}}{4 \, x} \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 {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{4}+1}{\left (x^{4}-1\right ) \sqrt {1+\sqrt {x^{2}+1}}}\, dx\]
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 {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 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 = -0.01 \begin {gather*} \int \frac {x^4+1}{\left (x^4-1\right )\,\sqrt {\sqrt {x^2+1}+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 {x^{4} + 1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {\sqrt {x^{2} + 1} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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