Optimal. Leaf size=57 \[ \frac {1}{2} \sqrt {\sqrt {x^4+1}+x^2} x+\frac {\tan ^{-1}\left (\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}\right )}{2 \sqrt {2}} \]
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Rubi [F] time = 0.02, 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 \sqrt {x^2+\sqrt {1+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \sqrt {x^2+\sqrt {1+x^4}} \, dx &=\int \sqrt {x^2+\sqrt {1+x^4}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.18, size = 92, normalized size = 1.61 \begin {gather*} \frac {2 \left (\sqrt {x^4+1}+x^2\right ) x^2+\sqrt {2} \sqrt {x^2 \left (\sqrt {x^4+1}+x^2\right )} \tan ^{-1}\left (\sqrt {\left (\sqrt {x^4+1}+x^2\right )^2-1}\right )}{4 x \sqrt {\sqrt {x^4+1}+x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 57, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \sqrt {x^2+\sqrt {1+x^4}}+\frac {\tan ^{-1}\left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 61, normalized size = 1.07 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + \sqrt {x^{4} + 1}} x - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\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 \sqrt {x^{2} + \sqrt {x^{4} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 22, normalized size = 0.39
method | result | size |
meijerg | \(\frac {\sqrt {2}\, x^{2} \hypergeom \left (\left [-\frac {1}{2}, -\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {1}{2}, \frac {1}{2}\right ], -\frac {1}{x^{4}}\right )}{2}\) | \(22\) |
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 \sqrt {x^{2} + \sqrt {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 = -0.02 \begin {gather*} \int \sqrt {\sqrt {x^4+1}+x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.70, size = 17, normalized size = 0.30 \begin {gather*} - \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} \frac {3}{2}, 1 & 1 \\\frac {1}{4}, \frac {3}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{16 \sqrt {\pi }} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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