Optimal. Leaf size=21 \[ \frac {1}{2} \sqrt {x^2+1} x+\frac {1}{2} \sinh ^{-1}(x) \]
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Rubi [A] time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {26, 195, 215} \[ \frac {1}{2} \sqrt {x^2+1} x+\frac {1}{2} \sinh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 26
Rule 195
Rule 215
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^4}}{\sqrt {1-x^2}} \, dx &=\int \sqrt {1+x^2} \, dx\\ &=\frac {1}{2} x \sqrt {1+x^2}+\frac {1}{2} \int \frac {1}{\sqrt {1+x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {1+x^2}+\frac {1}{2} \sinh ^{-1}(x)\\ \end {align*}
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Mathematica [B] time = 0.06, size = 70, normalized size = 3.33 \[ \frac {1}{2} \left (\log \left (1-x^2\right )+\frac {\sqrt {1-x^4} x}{\sqrt {1-x^2}}-\log \left (x^3+\sqrt {1-x^2} \sqrt {1-x^4}-x\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 120, normalized size = 5.71 \[ -\frac {2 \, \sqrt {-x^{4} + 1} \sqrt {-x^{2} + 1} x + {\left (x^{2} - 1\right )} \log \left (\frac {x^{3} + \sqrt {-x^{4} + 1} \sqrt {-x^{2} + 1} - x}{x^{3} - x}\right ) - {\left (x^{2} - 1\right )} \log \left (-\frac {x^{3} - \sqrt {-x^{4} + 1} \sqrt {-x^{2} + 1} - x}{x^{3} - x}\right )}{4 \, {\left (x^{2} - 1\right )}} \]
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 \[ \int \frac {\sqrt {-x^{4} + 1}}{\sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 47, normalized size = 2.24 \[ -\frac {\sqrt {-x^{4}+1}\, \sqrt {-x^{2}+1}\, \left (\sqrt {x^{2}+1}\, x +\arcsinh \relax (x )\right )}{2 \left (x^{2}-1\right ) \sqrt {x^{2}+1}} \]
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 \[ \int \frac {\sqrt {-x^{4} + 1}}{\sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\sqrt {1-x^4}}{\sqrt {1-x^2}} \,d x \]
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 \[ \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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