Optimal. Leaf size=273 \[ \frac {1}{2} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]
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Rubi [C] time = 1.69, antiderivative size = 309, normalized size of antiderivative = 1.13, number of steps used = 26, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6725, 2133, 725, 206} \begin {gather*} \frac {1}{8} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {1}{8} \sqrt {1-i} \tanh ^{-1}\left (\frac {1-i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )+\frac {1}{8} \sqrt {1-i} \tanh ^{-1}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{8} \sqrt {1+i} \tanh ^{-1}\left (\frac {x+1}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {1}{8} \sqrt {1-i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {1}{8} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )-\frac {1}{8} \sqrt {1+i} \tanh ^{-1}\left (\frac {1+i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )-\frac {1}{8} \sqrt {1-i} \tanh ^{-1}\left (\frac {x+1}{\sqrt {1-i} \sqrt {1+i x^2}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 206
Rule 725
Rule 2133
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^4\right ) \sqrt {1+x^4}} \, dx &=\int \left (-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1-x^2\right ) \sqrt {1+x^4}}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1+x^2\right ) \sqrt {1+x^4}}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx\right )-\frac {1}{2} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 (i-x) \sqrt {1+x^4}}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 (i+x) \sqrt {1+x^4}}\right ) \, dx\right )-\frac {1}{2} \int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 (1-x) \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 (1+x) \sqrt {1+x^4}}\right ) \, dx\\ &=-\left (\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(i-x) \sqrt {1+x^4}} \, dx\right )-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(i+x) \sqrt {1+x^4}} \, dx-\frac {1}{4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1-x) \sqrt {1+x^4}} \, dx-\frac {1}{4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx\\ &=-\left (\left (-\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i-x) \sqrt {1+i x^2}} \, dx\right )-\left (-\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i+x) \sqrt {1+i x^2}} \, dx-\left (\frac {1}{8}-\frac {i}{8}\right ) \int \frac {1}{(1-x) \sqrt {1-i x^2}} \, dx-\left (\frac {1}{8}-\frac {i}{8}\right ) \int \frac {1}{(1+x) \sqrt {1-i x^2}} \, dx-\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i-x) \sqrt {1-i x^2}} \, dx-\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i+x) \sqrt {1-i x^2}} \, dx-\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(1-x) \sqrt {1+i x^2}} \, dx-\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(1+x) \sqrt {1+i x^2}} \, dx\\ &=-\left (\left (-\frac {1}{8}-\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {-1-x}{\sqrt {1-i x^2}}\right )\right )-\left (-\frac {1}{8}-\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {1-x}{\sqrt {1-i x^2}}\right )-\left (-\frac {1}{8}-\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {-1-i x}{\sqrt {1+i x^2}}\right )-\left (-\frac {1}{8}-\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {1-i x}{\sqrt {1+i x^2}}\right )-\left (-\frac {1}{8}+\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {-1+i x}{\sqrt {1-i x^2}}\right )-\left (-\frac {1}{8}+\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {1+i x}{\sqrt {1-i x^2}}\right )-\left (\frac {1}{8}-\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {-1+x}{\sqrt {1+i x^2}}\right )-\left (\frac {1}{8}-\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {1+x}{\sqrt {1+i x^2}}\right )\\ &=\frac {1}{8} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {1}{8} \sqrt {1-i} \tanh ^{-1}\left (\frac {1-i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )+\frac {1}{8} \sqrt {1-i} \tanh ^{-1}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{8} \sqrt {1+i} \tanh ^{-1}\left (\frac {1+x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {1}{8} \sqrt {1-i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {1}{8} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )-\frac {1}{8} \sqrt {1+i} \tanh ^{-1}\left (\frac {1+i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )-\frac {1}{8} \sqrt {1-i} \tanh ^{-1}\left (\frac {1+x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )\\ \end {align*}
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Mathematica [F] time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^4\right ) \sqrt {1+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.60, size = 389, normalized size = 1.42 \begin {gather*} \frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \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 {1}{2} \left (-1+\sqrt {2}\right )} \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 {1}{2} \left (1+\sqrt {2}\right )} \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 {1}{2} \left (1+\sqrt {2}\right )} \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 [A] time = 12.46, size = 373, normalized size = 1.37 \begin {gather*} -\frac {1}{4} \, \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}{16} \, \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}{16} \, \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 ) \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^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{\left (x^{4}-1\right ) \sqrt {x^{4}+1}}\, dx\]
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 {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}\,{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 {\sqrt {x^4+1}+x^2}}{\left (x^4-1\right )\,\sqrt {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}}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{4} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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