Optimal. Leaf size=166 \[ -\frac {\left (2-3 x^2\right )^{3/4}}{8 x}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{8\ 2^{3/4}}+\frac {\sqrt {3} \tanh ^{-1}\left (\frac {2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{8\ 2^{3/4}}-\frac {\sqrt {3} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{4\ 2^{3/4}} \]
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Rubi [A]
time = 0.05, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {451, 331, 234,
406} \begin {gather*} -\frac {\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 {\sqrt {3} \text {ArcTan}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{8\ 2^{3/4}}-\frac {\left (2-3 x^2\right )^{3/4}}{8 x}+\frac {\sqrt {3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {2-3 x^2}+2^{3/4}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{8\ 2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 234
Rule 331
Rule 406
Rule 451
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx &=\int \left (\frac {1}{4 x^2 \sqrt [4]{2-3 x^2}}-\frac {3}{4 \sqrt [4]{2-3 x^2} \left (-4+3 x^2\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {1}{x^2 \sqrt [4]{2-3 x^2}} \, dx-\frac {3}{4} \int \frac {1}{\sqrt [4]{2-3 x^2} \left (-4+3 x^2\right )} \, dx\\ &=-\frac {\left (2-3 x^2\right )^{3/4}}{8 x}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{8\ 2^{3/4}}+\frac {\sqrt {3} \tanh ^{-1}\left (\frac {2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{8\ 2^{3/4}}-\frac {3}{16} \int \frac {1}{\sqrt [4]{2-3 x^2}} \, dx\\ &=-\frac {\left (2-3 x^2\right )^{3/4}}{8 x}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{8\ 2^{3/4}}+\frac {\sqrt {3} \tanh ^{-1}\left (\frac {2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{8\ 2^{3/4}}-\frac {\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 6 vs. order 4 in
optimal.
time = 20.05, size = 56, normalized size = 0.34 \begin {gather*} -\frac {\left (2-3 x^2\right )^{3/4}}{8 x}+\frac {3 x^3 F_1\left (\frac {3}{2};\frac {1}{4},1;\frac {5}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )}{64 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}} \left (-3 x^{2}+4\right )}\, dx\]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{3 x^{4} \sqrt [4]{2 - 3 x^{2}} - 4 x^{2} \sqrt [4]{2 - 3 x^{2}}}\, dx \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^2\,{\left (2-3\,x^2\right )}^{1/4}\,\left (3\,x^2-4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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