Optimal. Leaf size=56 \[ \frac {1}{2} x \sqrt {-x^2-y^4+1}-\frac {1}{2} i \left (y^4-1\right ) \log \left (\sqrt {-x^2-y^4+1}-i x\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 52, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {195, 217, 203} \begin {gather*} \frac {1}{2} x \sqrt {-x^2-y^4+1}+\frac {1}{2} \left (1-y^4\right ) \tan ^{-1}\left (\frac {x}{\sqrt {-x^2-y^4+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rubi steps
\begin {align*} \int \sqrt {1-x^2-y^4} \, dx &=\frac {1}{2} x \sqrt {1-x^2-y^4}+\frac {1}{2} \left (1-y^4\right ) \int \frac {1}{\sqrt {1-x^2-y^4}} \, dx\\ &=\frac {1}{2} x \sqrt {1-x^2-y^4}+\frac {1}{2} \left (1-y^4\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {1-x^2-y^4}}\right )\\ &=\frac {1}{2} x \sqrt {1-x^2-y^4}+\frac {1}{2} \left (1-y^4\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-x^2-y^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 0.88 \begin {gather*} \frac {1}{2} \left (x \sqrt {-x^2-y^4+1}-\left (y^4-1\right ) \tan ^{-1}\left (\frac {x}{\sqrt {-x^2-y^4+1}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 56, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \sqrt {1-x^2-y^4}-\frac {1}{2} i \left (-1+y^4\right ) \log \left (-i x+\sqrt {1-x^2-y^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 44, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, {\left (y^{4} - 1\right )} \arctan \left (\frac {\sqrt {-y^{4} - x^{2} + 1}}{x}\right ) + \frac {1}{2} \, \sqrt {-y^{4} - x^{2} + 1} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 37, normalized size = 0.66 \begin {gather*} -\frac {1}{2} \, {\left (y^{4} - 1\right )} \arcsin \left (\frac {x}{\sqrt {-y^{4} + 1}}\right ) + \frac {1}{2} \, \sqrt {-y^{4} - x^{2} + 1} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 45, normalized size = 0.80
method | result | size |
default | \(\frac {x \sqrt {-y^{4}-x^{2}+1}}{2}-\frac {\left (4 y^{4}-4\right ) \arctan \left (\frac {x}{\sqrt {-y^{4}-x^{2}+1}}\right )}{8}\) | \(45\) |
risch | \(-\frac {x \left (y^{4}+x^{2}-1\right )}{2 \sqrt {-y^{4}-x^{2}+1}}-\frac {\arctan \left (\frac {x}{\sqrt {-y^{4}-x^{2}+1}}\right ) y^{4}}{2}+\frac {\arctan \left (\frac {x}{\sqrt {-y^{4}-x^{2}+1}}\right )}{2}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 48, normalized size = 0.86 \begin {gather*} \frac {x\,\sqrt {-x^2-y^4+1}}{2}+\ln \left (\sqrt {-x^2-y^4+1}+x\,1{}\mathrm {i}\right )\,\left (\frac {y^4\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.65, size = 745, normalized size = 13.30
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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