Optimal. Leaf size=163 \[ -\frac {\left (2-3 x^2\right )^{3/4}}{16 x^2}+\frac {9 \tan ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32 \sqrt [4]{2}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32 \sqrt [4]{2}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2-3 x^2}+\sqrt {2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}} \]
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Rubi [A] time = 0.13, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {440, 266, 51, 63, 298, 203, 206, 439} \[ -\frac {\left (2-3 x^2\right )^{3/4}}{16 x^2}+\frac {9 \tan ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32 \sqrt [4]{2}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32 \sqrt [4]{2}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2-3 x^2}+\sqrt {2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 203
Rule 206
Rule 266
Rule 298
Rule 439
Rule 440
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx &=\int \left (\frac {1}{4 x^3 \sqrt [4]{2-3 x^2}}+\frac {3}{16 x \sqrt [4]{2-3 x^2}}-\frac {9 x}{16 \sqrt [4]{2-3 x^2} \left (-4+3 x^2\right )}\right ) \, dx\\ &=\frac {3}{16} \int \frac {1}{x \sqrt [4]{2-3 x^2}} \, dx+\frac {1}{4} \int \frac {1}{x^3 \sqrt [4]{2-3 x^2}} \, dx-\frac {9}{16} \int \frac {x}{\sqrt [4]{2-3 x^2} \left (-4+3 x^2\right )} \, dx\\ &=\frac {3 \tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}+\frac {3}{32} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2-3 x} x} \, dx,x,x^2\right )+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2-3 x} x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (2-3 x^2\right )^{3/4}}{16 x^2}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}+\frac {3}{64} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2-3 x} x} \, dx,x,x^2\right )-\frac {1}{8} \operatorname {Subst}\left (\int \frac {x^2}{\frac {2}{3}-\frac {x^4}{3}} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=-\frac {\left (2-3 x^2\right )^{3/4}}{16 x^2}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}-\frac {1}{16} \operatorname {Subst}\left (\int \frac {x^2}{\frac {2}{3}-\frac {x^4}{3}} \, dx,x,\sqrt [4]{2-3 x^2}\right )-\frac {3}{16} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )+\frac {3}{16} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=-\frac {\left (2-3 x^2\right )^{3/4}}{16 x^2}+\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{16 \sqrt [4]{2}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{16 \sqrt [4]{2}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}-\frac {3}{32} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )+\frac {3}{32} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=-\frac {\left (2-3 x^2\right )^{3/4}}{16 x^2}+\frac {9 \tan ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32 \sqrt [4]{2}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32 \sqrt [4]{2}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{16\ 2^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 102, normalized size = 0.63 \[ -\frac {4 \left (2-3 x^2\right )^{3/4} x^2 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {3 x^2}{2}-1\right )+4 \left (2-3 x^2\right )^{3/4}-9\ 2^{3/4} x^2 \tan ^{-1}\left (\sqrt [4]{1-\frac {3 x^2}{2}}\right )+9\ 2^{3/4} x^2 \tanh ^{-1}\left (\sqrt [4]{1-\frac {3 x^2}{2}}\right )}{64 x^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.16, size = 307, normalized size = 1.88 \[ -\frac {36 \cdot 2^{\frac {3}{4}} x^{2} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {\sqrt {2} + \sqrt {-3 \, x^{2} + 2}} - \frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + 9 \cdot 2^{\frac {3}{4}} x^{2} \log \left (2^{\frac {1}{4}} + {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - 9 \cdot 2^{\frac {3}{4}} x^{2} \log \left (-2^{\frac {1}{4}} + {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - 24 \cdot 2^{\frac {1}{4}} x^{2} \arctan \left (2^{\frac {1}{4}} \sqrt {2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}} - 2^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - 1\right ) - 24 \cdot 2^{\frac {1}{4}} x^{2} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} \sqrt {-4 \cdot 2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 4 \, \sqrt {2} + 4 \, \sqrt {-3 \, x^{2} + 2}} - 2^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 1\right ) - 6 \cdot 2^{\frac {1}{4}} x^{2} \log \left (4 \cdot 2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 4 \, \sqrt {2} + 4 \, \sqrt {-3 \, x^{2} + 2}\right ) + 6 \cdot 2^{\frac {1}{4}} x^{2} \log \left (-4 \cdot 2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 4 \, \sqrt {2} + 4 \, \sqrt {-3 \, x^{2} + 2}\right ) + 8 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}{128 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 192, normalized size = 1.18 \[ \frac {9}{64} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \frac {9}{128} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \frac {9}{128} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \frac {3}{32} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {3}{32} \cdot 2^{\frac {1}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {3}{64} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {3}{64} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {{\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}{16 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 9.18, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-3 x^{2}+2\right )^{\frac {1}{4}} \left (-3 x^{2}+4\right ) x^{3}}\, dx \]
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 \[ -\int \frac {1}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 109, normalized size = 0.67 \[ \frac {9\,2^{3/4}\,\mathrm {atan}\left (\frac {2^{3/4}\,{\left (2-3\,x^2\right )}^{1/4}}{2}\right )}{64}-\frac {{\left (2-3\,x^2\right )}^{3/4}}{16\,x^2}+\frac {2^{3/4}\,\mathrm {atan}\left (\frac {2^{3/4}\,{\left (2-3\,x^2\right )}^{1/4}\,1{}\mathrm {i}}{2}\right )\,9{}\mathrm {i}}{64}-\frac {{\left (-1\right )}^{1/4}\,2^{3/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,2^{3/4}\,{\left (2-3\,x^2\right )}^{1/4}\,1{}\mathrm {i}}{2}\right )\,3{}\mathrm {i}}{32}-\frac {{\left (-1\right )}^{3/4}\,2^{3/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{3/4}\,2^{3/4}\,{\left (2-3\,x^2\right )}^{1/4}\,1{}\mathrm {i}}{2}\right )\,3{}\mathrm {i}}{32} \]
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 \[ - \int \frac {1}{3 x^{5} \sqrt [4]{2 - 3 x^{2}} - 4 x^{3} \sqrt [4]{2 - 3 x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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