Optimal. Leaf size=274 \[ \sqrt {\sqrt {x^4+1}+x^2}-\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {\sqrt {2}-1}}\right )+\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {1+\sqrt {2}}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]
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Rubi [F] time = 0.11, 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 {x^2+\sqrt {1+x^4}}}{1+x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x} \, dx &=\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x} \, dx\\ \end {align*}
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Mathematica [F] time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.51, size = 274, normalized size = 1.00 \begin {gather*} \sqrt {\sqrt {x^4+1}+x^2}-\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^4+1}+x^2}\right )+\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {2}-2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^4+1}+x^2}\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 {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+2 \sqrt {2}} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.26, size = 436, normalized size = 1.59 \begin {gather*} \sqrt {\sqrt {2} - 1} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1}\right )} \sqrt {\sqrt {2} + 1} \sqrt {\sqrt {2} - 1} - 2 \, {\left (x^{3} - x^{2} - \sqrt {2} {\left (x^{2} - x\right )} - \sqrt {x^{4} + 1} {\left (x - \sqrt {2} - 1\right )} + x + 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}}\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 ) + \frac {1}{4} \, \sqrt {\sqrt {2} + 1} \log \left (-\frac {2 \, {\left ({\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 (2 \, x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )} + 2\right )} \sqrt {\sqrt {2} + 1}\right )}}{x^{2} + 2 \, x + 1}\right ) - \frac {1}{4} \, \sqrt {\sqrt {2} + 1} \log \left (-\frac {2 \, {\left ({\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 (2 \, x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )} + 2\right )} \sqrt {\sqrt {2} + 1}\right )}}{x^{2} + 2 \, x + 1}\right ) + \sqrt {x^{2} + \sqrt {x^{4} + 1}} \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}}}{x + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x}\, 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^{2} + \sqrt {x^{4} + 1}}}{x + 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 {\sqrt {x^4+1}+x^2}}{x+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}}}{x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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