Optimal. Leaf size=67 \[ -\frac {\left (2-3 x^2\right )^{3/4}}{6 x^3}-\frac {3 \left (2-3 x^2\right )^{3/4}}{8 x}-\frac {3 \sqrt {3} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{4\ 2^{3/4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {331, 234}
\begin {gather*} -\frac {3 \sqrt {3} E\left (\left .\frac {1}{2} \text {ArcSin}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{4\ 2^{3/4}}-\frac {3 \left (2-3 x^2\right )^{3/4}}{8 x}-\frac {\left (2-3 x^2\right )^{3/4}}{6 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 234
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt [4]{2-3 x^2}} \, dx &=-\frac {\left (2-3 x^2\right )^{3/4}}{6 x^3}+\frac {3}{4} \int \frac {1}{x^2 \sqrt [4]{2-3 x^2}} \, dx\\ &=-\frac {\left (2-3 x^2\right )^{3/4}}{6 x^3}-\frac {3 \left (2-3 x^2\right )^{3/4}}{8 x}-\frac {9}{16} \int \frac {1}{\sqrt [4]{2-3 x^2}} \, dx\\ &=-\frac {\left (2-3 x^2\right )^{3/4}}{6 x^3}-\frac {3 \left (2-3 x^2\right )^{3/4}}{8 x}-\frac {3 \sqrt {3} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{4\ 2^{3/4}}\\ \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 = 29, normalized size = 0.43 \begin {gather*} -\frac {\, _2F_1\left (-\frac {3}{2},\frac {1}{4};-\frac {1}{2};\frac {3 x^2}{2}\right )}{3 \sqrt [4]{2} x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.08, size = 20, normalized size = 0.30
method | result | size |
meijerg | \(-\frac {2^{\frac {3}{4}} \hypergeom \left (\left [-\frac {3}{2}, \frac {1}{4}\right ], \left [-\frac {1}{2}\right ], \frac {3 x^{2}}{2}\right )}{6 x^{3}}\) | \(20\) |
risch | \(\frac {27 x^{4}-6 x^{2}-8}{24 x^{3} \left (-3 x^{2}+2\right )^{\frac {1}{4}}}-\frac {9 \,2^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {3 x^{2}}{2}\right )}{32}\) | \(45\) |
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 [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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Sympy [C] Result contains complex when optimal does not.
time = 0.51, size = 34, normalized size = 0.51 \begin {gather*} - \frac {2^{\frac {3}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{2 i \pi }}{2}} \right )}}{6 x^{3}} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^4\,{\left (2-3\,x^2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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