Optimal. Leaf size=215 \[ -\frac{\sqrt [4]{2-3 x^2}}{16 x^2}+\frac{3 \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{64 \sqrt [4]{2}}-\frac{3 \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{64 \sqrt [4]{2}}-\frac{15 \tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}-\frac{3 \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{32 \sqrt [4]{2}}+\frac{3 \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{32 \sqrt [4]{2}}-\frac{15 \tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}} \]
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Rubi [A] time = 0.240897, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 14, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {443, 266, 51, 63, 212, 206, 203, 444, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt [4]{2-3 x^2}}{16 x^2}+\frac{3 \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{64 \sqrt [4]{2}}-\frac{3 \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{64 \sqrt [4]{2}}-\frac{15 \tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}-\frac{3 \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{32 \sqrt [4]{2}}+\frac{3 \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{32 \sqrt [4]{2}}-\frac{15 \tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}} \]
Antiderivative was successfully verified.
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Rule 443
Rule 266
Rule 51
Rule 63
Rule 212
Rule 206
Rule 203
Rule 444
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx &=\int \left (\frac{1}{4 x^3 \left (2-3 x^2\right )^{3/4}}+\frac{3}{16 x \left (2-3 x^2\right )^{3/4}}-\frac{9 x}{16 \left (2-3 x^2\right )^{3/4} \left (-4+3 x^2\right )}\right ) \, dx\\ &=\frac{3}{16} \int \frac{1}{x \left (2-3 x^2\right )^{3/4}} \, dx+\frac{1}{4} \int \frac{1}{x^3 \left (2-3 x^2\right )^{3/4}} \, dx-\frac{9}{16} \int \frac{x}{\left (2-3 x^2\right )^{3/4} \left (-4+3 x^2\right )} \, dx\\ &=\frac{3}{32} \operatorname{Subst}\left (\int \frac{1}{(2-3 x)^{3/4} x} \, dx,x,x^2\right )+\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{(2-3 x)^{3/4} x^2} \, dx,x,x^2\right )-\frac{9}{32} \operatorname{Subst}\left (\int \frac{1}{(2-3 x)^{3/4} (-4+3 x)} \, dx,x,x^2\right )\\ &=-\frac{\sqrt [4]{2-3 x^2}}{16 x^2}-\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{\frac{2}{3}-\frac{x^4}{3}} \, dx,x,\sqrt [4]{2-3 x^2}\right )+\frac{9}{64} \operatorname{Subst}\left (\int \frac{1}{(2-3 x)^{3/4} x} \, dx,x,x^2\right )+\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{-2-x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=-\frac{\sqrt [4]{2-3 x^2}}{16 x^2}-\frac{3}{16} \operatorname{Subst}\left (\int \frac{1}{\frac{2}{3}-\frac{x^4}{3}} \, dx,x,\sqrt [4]{2-3 x^2}\right )-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{16 \sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{16 \sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}-x^2}{-2-x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{16 \sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}+x^2}{-2-x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{16 \sqrt{2}}\\ &=-\frac{\sqrt [4]{2-3 x^2}}{16 x^2}-\frac{3 \tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{16\ 2^{3/4}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{16\ 2^{3/4}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}-2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{32 \sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}+2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{32 \sqrt{2}}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{32 \sqrt{2}}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{32 \sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{2^{3/4}+2 x}{-\sqrt{2}-2^{3/4} x-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{64 \sqrt [4]{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{2^{3/4}-2 x}{-\sqrt{2}+2^{3/4} x-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{64 \sqrt [4]{2}}\\ &=-\frac{\sqrt [4]{2-3 x^2}}{16 x^2}-\frac{15 \tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}-\frac{15 \tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}+\frac{3 \log \left (\sqrt{2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )}{64 \sqrt [4]{2}}-\frac{3 \log \left (\sqrt{2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )}{64 \sqrt [4]{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{4-6 x^2}\right )}{32 \sqrt [4]{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{4-6 x^2}\right )}{32 \sqrt [4]{2}}\\ &=-\frac{\sqrt [4]{2-3 x^2}}{16 x^2}-\frac{15 \tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}-\frac{3 \tan ^{-1}\left (1+\sqrt [4]{4-6 x^2}\right )}{32 \sqrt [4]{2}}+\frac{3 \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{32 \sqrt [4]{2}}-\frac{15 \tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}+\frac{3 \log \left (\sqrt{2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )}{64 \sqrt [4]{2}}-\frac{3 \log \left (\sqrt{2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )}{64 \sqrt [4]{2}}\\ \end{align*}
Mathematica [A] time = 0.0780583, size = 210, normalized size = 0.98 \[ -\frac{8 \sqrt [4]{2-3 x^2}-3\ 2^{3/4} x^2 \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )+3\ 2^{3/4} x^2 \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )+30 \sqrt [4]{2} x^2 \tan ^{-1}\left (\sqrt [4]{1-\frac{3 x^2}{2}}\right )-6\ 2^{3/4} x^2 \tan ^{-1}\left (1-\sqrt [4]{4-6 x^2}\right )+6\ 2^{3/4} x^2 \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+30 \sqrt [4]{2} x^2 \tanh ^{-1}\left (\sqrt [4]{1-\frac{3 x^2}{2}}\right )}{128 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( -3\,{x}^{2}+4 \right ) } \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, 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{3}{4}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81365, size = 1081, normalized size = 5.03 \begin{align*} \frac{12 \cdot 8^{\frac{3}{4}} \sqrt{2} x^{2} \arctan \left (\frac{1}{4} \cdot 8^{\frac{1}{4}} \sqrt{2} \sqrt{8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} + 4 \, \sqrt{-3 \, x^{2} + 2}} - \frac{1}{2} \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} - 1\right ) + 12 \cdot 8^{\frac{3}{4}} \sqrt{2} x^{2} \arctan \left (\frac{1}{16} \cdot 8^{\frac{1}{4}} \sqrt{2} \sqrt{-16 \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 64 \, \sqrt{2} + 64 \, \sqrt{-3 \, x^{2} + 2}} - \frac{1}{2} \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 1\right ) - 3 \cdot 8^{\frac{3}{4}} \sqrt{2} x^{2} \log \left (16 \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 64 \, \sqrt{2} + 64 \, \sqrt{-3 \, x^{2} + 2}\right ) + 3 \cdot 8^{\frac{3}{4}} \sqrt{2} x^{2} \log \left (-16 \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 64 \, \sqrt{2} + 64 \, \sqrt{-3 \, x^{2} + 2}\right ) + 60 \cdot 8^{\frac{3}{4}} x^{2} \arctan \left (\frac{1}{2} \cdot 8^{\frac{1}{4}} \sqrt{\sqrt{2} + \sqrt{-3 \, x^{2} + 2}} - \frac{1}{2} \cdot 8^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) - 15 \cdot 8^{\frac{3}{4}} x^{2} \log \left (8^{\frac{3}{4}} + 4 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) + 15 \cdot 8^{\frac{3}{4}} x^{2} \log \left (-8^{\frac{3}{4}} + 4 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) - 32 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}{512 \, 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} \left (2 - 3 x^{2}\right )^{\frac{3}{4}} - 4 x^{3} \left (2 - 3 x^{2}\right )^{\frac{3}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26565, size = 259, normalized size = 1.2 \begin{align*} -\frac{3}{64} \cdot 2^{\frac{3}{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{3}{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}{128} \cdot 2^{\frac{3}{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}{128} \cdot 2^{\frac{3}{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{15}{64} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) - \frac{15}{128} \cdot 2^{\frac{1}{4}} \log \left (2^{\frac{1}{4}} +{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) + \frac{15}{128} \cdot 2^{\frac{1}{4}} \log \left (2^{\frac{1}{4}} -{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) - \frac{{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}{16 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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