Optimal. Leaf size=53 \[ \frac {1}{2 x \sqrt {1-x^4}}-\frac {3 \sqrt {1-x^4}}{2 x}-\frac {3}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {3}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {296, 331, 313,
227, 1195, 435} \begin {gather*} \frac {3}{2} F(\text {ArcSin}(x)|-1)-\frac {3}{2} E(\text {ArcSin}(x)|-1)-\frac {3 \sqrt {1-x^4}}{2 x}+\frac {1}{2 x \sqrt {1-x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 296
Rule 313
Rule 331
Rule 435
Rule 1195
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (1-x^4\right )^{3/2}} \, dx &=\frac {1}{2 x \sqrt {1-x^4}}+\frac {3}{2} \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x \sqrt {1-x^4}}-\frac {3 \sqrt {1-x^4}}{2 x}-\frac {3}{2} \int \frac {x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x \sqrt {1-x^4}}-\frac {3 \sqrt {1-x^4}}{2 x}+\frac {3}{2} \int \frac {1}{\sqrt {1-x^4}} \, dx-\frac {3}{2} \int \frac {1+x^2}{\sqrt {1-x^4}} \, dx\\ &=\frac {1}{2 x \sqrt {1-x^4}}-\frac {3 \sqrt {1-x^4}}{2 x}+\frac {3}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {3}{2} \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\\ &=\frac {1}{2 x \sqrt {1-x^4}}-\frac {3 \sqrt {1-x^4}}{2 x}-\frac {3}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {3}{2} 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 = 9.11, size = 18, normalized size = 0.34 \begin {gather*} -\frac {\, _2F_1\left (-\frac {1}{4},\frac {3}{2};\frac {3}{4};x^4\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 68, normalized size = 1.28
method | result | size |
meijerg | \(-\frac {\hypergeom \left (\left [-\frac {1}{4}, \frac {3}{2}\right ], \left [\frac {3}{4}\right ], x^{4}\right )}{x}\) | \(15\) |
risch | \(\frac {3 x^{4}-2}{2 x \sqrt {-x^{4}+1}}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}\) | \(61\) |
default | \(\frac {x^{3}}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{4}+1}}{x}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}\) | \(68\) |
elliptic | \(\frac {x^{3}}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{4}+1}}{x}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{2 \sqrt {-x^{4}+1}}\) | \(68\) |
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 = 55, normalized size = 1.04 \begin {gather*} -\frac {3 \, {\left (x^{5} - x\right )} E(\arcsin \left (x\right )\,|\,-1) - 3 \, {\left (x^{5} - x\right )} F(\arcsin \left (x\right )\,|\,-1) + {\left (3 \, x^{4} - 2\right )} \sqrt {-x^{4} + 1}}{2 \, {\left (x^{5} - x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.39, size = 32, normalized size = 0.60 \begin {gather*} \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{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 = 1.16, size = 13, normalized size = 0.25 \begin {gather*} -\frac {{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {3}{2};\ \frac {3}{4};\ x^4\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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