Optimal. Leaf size=166 \[ -\frac {\sqrt [4]{2-3 x^2}}{8 x}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{16 \sqrt [4]{2}}-\frac {\sqrt {3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {2-3 x^2}+2^{3/4}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{16 \sqrt [4]{2}}+\frac {\sqrt {3} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{4 \sqrt [4]{2}} \]
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Rubi [A] time = 0.09, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {443, 325, 232, 400, 441} \[ -\frac {\sqrt [4]{2-3 x^2}}{8 x}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{16 \sqrt [4]{2}}-\frac {\sqrt {3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {2-3 x^2}+2^{3/4}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{16 \sqrt [4]{2}}+\frac {\sqrt {3} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{4 \sqrt [4]{2}} \]
Antiderivative was successfully verified.
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Rule 232
Rule 325
Rule 400
Rule 441
Rule 443
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx &=\int \left (\frac {1}{4 x^2 \left (2-3 x^2\right )^{3/4}}-\frac {3}{4 \left (2-3 x^2\right )^{3/4} \left (-4+3 x^2\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {1}{x^2 \left (2-3 x^2\right )^{3/4}} \, dx-\frac {3}{4} \int \frac {1}{\left (2-3 x^2\right )^{3/4} \left (-4+3 x^2\right )} \, dx\\ &=-\frac {\sqrt [4]{2-3 x^2}}{8 x}+2 \left (\frac {3}{16} \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx\right )-\frac {9}{16} \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (-4+3 x^2\right )} \, dx\\ &=-\frac {\sqrt [4]{2-3 x^2}}{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 )}{16 \sqrt [4]{2}}-\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 )}{16 \sqrt [4]{2}}+\frac {\sqrt {3} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{4 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 37, normalized size = 0.22 \[ -\frac {F_1\left (-\frac {1}{2};\frac {3}{4},1;\frac {1}{2};\frac {3 x^2}{2},\frac {3 x^2}{4}\right )}{4\ 2^{3/4} x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 5.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{9 \, x^{6} - 18 \, x^{4} + 8 \, x^{2}}, x\right ) \]
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 \[ \int -\frac {1}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.90, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-3 x^{2}+2\right )^{\frac {3}{4}} \left (-3 x^{2}+4\right ) x^{2}}\, dx \]
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 \[ -\int \frac {1}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} x^{2}}\,{d x} \]
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 \[ -\int \frac {1}{x^2\,{\left (2-3\,x^2\right )}^{3/4}\,\left (3\,x^2-4\right )} \,d x \]
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 \[ - \int \frac {1}{3 x^{4} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 x^{2} \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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