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 = 0.132789, antiderivative size = 163, normalized size of antiderivative = 1., 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 440
Rule 266
Rule 51
Rule 63
Rule 298
Rule 203
Rule 206
Rule 439
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*}
Mathematica [C] time = 0.0506956, 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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Maple [F] time = 0.077, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( -3\,{x}^{2}+4 \right ) }{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (3 \, x^{2} - 4\right )}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.45525, size = 909, normalized size = 5.58 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{3 x^{5} \sqrt [4]{2 - 3 x^{2}} - 4 x^{3} \sqrt [4]{2 - 3 x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33623, size = 259, normalized size = 1.59 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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