Optimal. Leaf size=27 \[ -\frac {\sqrt {1-x^4}}{x}+F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {325, 307, 221, 1181, 424} \[ -\frac {\sqrt {1-x^4}}{x}+F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 221
Rule 307
Rule 325
Rule 424
Rule 1181
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx &=-\frac {\sqrt {1-x^4}}{x}-\int \frac {x^2}{\sqrt {1-x^4}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{x}+\int \frac {1}{\sqrt {1-x^4}} \, dx-\int \frac {1+x^2}{\sqrt {1-x^4}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{x}+F\left (\left .\sin ^{-1}(x)\right |-1\right )-\int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{x}-E\left (\left .\sin ^{-1}(x)\right |-1\right )+F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.00, size = 18, normalized size = 0.67 \[ -\frac {\, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};x^4\right )}{x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 1}}{x^{6} - x^{2}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-x^{4} + 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 53, normalized size = 1.96 \[ -\frac {\sqrt {-x^{4}+1}}{x}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (x , i\right )+\EllipticF \left (x , i\right )\right )}{\sqrt {-x^{4}+1}} \]
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 \[ \int \frac {1}{\sqrt {-x^{4} + 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 13, normalized size = 0.48 \[ -\frac {{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ x^4\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.13, size = 32, normalized size = 1.19 \[ \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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