Optimal. Leaf size=33 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}} \]
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Rubi [A]
time = 0.03, 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 = {2157, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}-x^2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 2157
Rubi steps
\begin {align*} \int \frac {\sqrt {-x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx &=\text {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 = 0.15, size = 33, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 0.10, size = 22, normalized size = 0.67
method | result | size |
meijerg | \(-\frac {\sqrt {2}\, \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}, \frac {5}{4}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], -\frac {1}{x^{4}}\right )}{4 x^{2}}\) | \(22\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.80, size = 29, normalized size = 0.88 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \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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Sympy [A]
time = 0.40, size = 15, normalized size = 0.45 \begin {gather*} \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 }} \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*} \text {could not integrate} \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 = -0.03 \begin {gather*} \int \frac {\sqrt {\sqrt {x^4+1}-x^2}}{\sqrt {x^4+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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