Optimal. Leaf size=197 \[ \frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{16 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{16 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac{\tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{8 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{8 \sqrt [4]{2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}} \]
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Rubi [A] time = 0.189395, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542, Rules used = {443, 266, 63, 212, 206, 203, 444, 211, 1165, 628, 1162, 617, 204} \[ \frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{16 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{16 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac{\tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{8 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{8 \sqrt [4]{2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}} \]
Antiderivative was successfully verified.
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Rule 443
Rule 266
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 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx &=\int \left (\frac{1}{4 x \left (2-3 x^2\right )^{3/4}}-\frac{3 x}{4 \left (2-3 x^2\right )^{3/4} \left (-4+3 x^2\right )}\right ) \, dx\\ &=\frac{1}{4} \int \frac{1}{x \left (2-3 x^2\right )^{3/4}} \, dx-\frac{3}{4} \int \frac{x}{\left (2-3 x^2\right )^{3/4} \left (-4+3 x^2\right )} \, dx\\ &=\frac{1}{8} \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}{(2-3 x)^{3/4} (-4+3 x)} \, dx,x,x^2\right )\\ &=-\left (\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\frac{2}{3}-\frac{x^4}{3}} \, dx,x,\sqrt [4]{2-3 x^2}\right )\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-2-x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{2}-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{4 \sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{2}+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{4 \sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-x^2}{-2-x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{4 \sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+x^2}{-2-x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{4 \sqrt{2}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{2}-2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{8 \sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{2}+2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{8 \sqrt{2}}+\frac{\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 )}{16 \sqrt [4]{2}}+\frac{\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 )}{16 \sqrt [4]{2}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}+\frac{\log \left (\sqrt{2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )}{16 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )}{16 \sqrt [4]{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{4-6 x^2}\right )}{8 \sqrt [4]{2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{4-6 x^2}\right )}{8 \sqrt [4]{2}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac{\tan ^{-1}\left (1+\sqrt [4]{4-6 x^2}\right )}{8 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{8 \sqrt [4]{2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}+\frac{\log \left (\sqrt{2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )}{16 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )}{16 \sqrt [4]{2}}\\ \end{align*}
Mathematica [A] time = 0.0567033, size = 156, normalized size = 0.79 \[ \frac{-4 \tan ^{-1}\left (\sqrt [4]{1-\frac{3 x^2}{2}}\right )-4 \tanh ^{-1}\left (\sqrt [4]{1-\frac{3 x^2}{2}}\right )+\sqrt{2} \left (\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )+2 \tan ^{-1}\left (1-\sqrt [4]{4-6 x^2}\right )-2 \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )\right )}{16\ 2^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73771, size = 1019, normalized size = 5.17 \begin{align*} \frac{1}{32} \cdot 8^{\frac{3}{4}} \sqrt{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 ) + \frac{1}{32} \cdot 8^{\frac{3}{4}} \sqrt{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 ) - \frac{1}{128} \cdot 8^{\frac{3}{4}} \sqrt{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 ) + \frac{1}{128} \cdot 8^{\frac{3}{4}} \sqrt{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 ) + \frac{1}{16} \cdot 8^{\frac{3}{4}} \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 ) - \frac{1}{64} \cdot 8^{\frac{3}{4}} \log \left (8^{\frac{3}{4}} + 4 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) + \frac{1}{64} \cdot 8^{\frac{3}{4}} \log \left (-8^{\frac{3}{4}} + 4 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) \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^{3} \left (2 - 3 x^{2}\right )^{\frac{3}{4}} - 4 x \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 = 1.25155, size = 284, normalized size = 1.44 \begin{align*} -\frac{1}{16} \cdot 4^{\frac{1}{8}} \sqrt{2} \arctan \left (\frac{1}{8} \cdot 4^{\frac{7}{8}} \sqrt{2}{\left (4^{\frac{1}{8}} \sqrt{2} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{16} \cdot 4^{\frac{1}{8}} \sqrt{2} \arctan \left (-\frac{1}{8} \cdot 4^{\frac{7}{8}} \sqrt{2}{\left (4^{\frac{1}{8}} \sqrt{2} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{32} \cdot 4^{\frac{1}{8}} \sqrt{2} \log \left (4^{\frac{1}{8}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-3 \, x^{2} + 2} + 4^{\frac{1}{4}}\right ) + \frac{1}{32} \cdot 4^{\frac{1}{8}} \sqrt{2} \log \left (-4^{\frac{1}{8}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-3 \, x^{2} + 2} + 4^{\frac{1}{4}}\right ) - \frac{1}{8} \cdot 4^{\frac{1}{8}} \arctan \left (\frac{1}{4} \cdot 4^{\frac{7}{8}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) - \frac{1}{16} \cdot 4^{\frac{1}{8}} \log \left ({\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4^{\frac{1}{8}}\right ) + \frac{1}{16} \cdot 4^{\frac{1}{8}} \log \left (-{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4^{\frac{1}{8}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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