Optimal. Leaf size=27 \[ -\frac {\sqrt {1-x^4}}{x}-E\left (\left .\sin ^{-1}(x)\right |-1\right )+F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A]
time = 0.01, 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 = {331, 313, 227,
1195, 435} \begin {gather*} F(\text {ArcSin}(x)|-1)-E(\text {ArcSin}(x)|-1)-\frac {\sqrt {1-x^4}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 313
Rule 331
Rule 435
Rule 1195
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 18, normalized size = 0.67 \begin {gather*} -\frac {\, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};x^4\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 52 vs. \(2 (25 ) = 50\).
time = 0.16, size = 53, normalized size = 1.96
method | result | size |
meijerg | \(-\frac {\hypergeom \left (\left [-\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{4}\right ], x^{4}\right )}{x}\) | \(15\) |
default | \(-\frac {\sqrt {-x^{4}+1}}{x}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{\sqrt {-x^{4}+1}}\) | \(53\) |
elliptic | \(-\frac {\sqrt {-x^{4}+1}}{x}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{\sqrt {-x^{4}+1}}\) | \(53\) |
risch | \(\frac {x^{4}-1}{x \sqrt {-x^{4}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{\sqrt {-x^{4}+1}}\) | \(57\) |
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 = 0.08, size = 28, normalized size = 1.04 \begin {gather*} -\frac {x E(\arcsin \left (x\right )\,|\,-1) - x F(\arcsin \left (x\right )\,|\,-1) + \sqrt {-x^{4} + 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 32 vs. \(2 (15) = 30\).
time = 0.34, size = 32, normalized size = 1.19 \begin {gather*} \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 )} \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 [B]
time = 1.12, size = 13, normalized size = 0.48 \begin {gather*} -\frac {{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ x^4\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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