Optimal. Leaf size=225 \[ \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {\sqrt {2}-1}}\right )-\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {1+\sqrt {2}}}\right )+\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]
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Rubi [C] time = 0.18, antiderivative size = 81, normalized size of antiderivative = 0.36, number of steps used = 5, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2133, 725, 206} \begin {gather*} -\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-i x}{\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
Rubi steps
\begin {align*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx &=\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{(1+x) \sqrt {1-i x^2}} \, dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(1+x) \sqrt {1+i x^2}} \, dx\\ &=\left (-\frac {1}{2}-\frac {i}{2}\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}{2}+\frac {i}{2}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {1+i x}{\sqrt {1-i x^2}}\right )\\ &=-\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )\\ \end {align*}
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Mathematica [F] time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.40, size = 225, normalized size = 1.00 \begin {gather*} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {-2+2 \sqrt {2}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )+\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+2 \sqrt {2}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 6.48, size = 369, normalized size = 1.64 \begin {gather*} \frac {1}{2} \, \sqrt {2 \, \sqrt {2} - 2} \arctan \left (\frac {{\left (2 \, x^{2} - \sqrt {2} {\left (x^{3} - x^{2} + x + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2\right )} - 2 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {2 \, \sqrt {2} - 2} + {\left (x^{2} + \sqrt {2} \sqrt {x^{4} + 1} + 1\right )} \sqrt {2 \, \sqrt {2} + 2} \sqrt {2 \, \sqrt {2} - 2}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) - \frac {1}{8} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (-\frac {{\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 2\right )} + 1\right )} \sqrt {2 \, \sqrt {2} + 2}}{x^{2} + 2 \, x + 1}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (-\frac {{\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 2\right )} + 1\right )} \sqrt {2 \, \sqrt {2} + 2}}{x^{2} + 2 \, x + 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 + 1\right )}}\,{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 \[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{\left (1+x \right ) \sqrt {x^{4}+1}}\, dx\]
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 + 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}}{\sqrt {x^4+1}\,\left (x+1\right )} \,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 ) \sqrt {x^{4} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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