Optimal. Leaf size=33 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}-x^2}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2132, 203} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}-x^2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 2132
Rubi steps
\begin {align*} \int \frac {\sqrt {-x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {-x^2+\sqrt {1+x^4}}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}-x^2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.62, size = 29, normalized size = 0.88 \[ -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) \]
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 \[ \int \frac {\sqrt {-x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 22, normalized size = 0.67 \[ -\frac {\sqrt {2}\, \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}, \frac {5}{4}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], -\frac {1}{x^{4}}\right )}{4 x^{2}} \]
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 \[ \int \frac {\sqrt {-x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {\sqrt {x^4+1}-x^2}}{\sqrt {x^4+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.86, size = 15, normalized size = 0.45 \[ \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} \frac {1}{2}, 1 & 1 \\\frac {1}{4}, \frac {3}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{4 \sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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