Optimal. Leaf size=140 \[ \frac{1}{8 x \left (3 x^4+2\right )}-\frac{5 \sqrt [4]{3} \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{64\ 2^{3/4}}+\frac{5 \sqrt [4]{3} \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{64\ 2^{3/4}}-\frac{5}{16 x}+\frac{5 \sqrt [4]{3} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32\ 2^{3/4}}-\frac{5 \sqrt [4]{3} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{32\ 2^{3/4}} \]
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Rubi [A] time = 0.0845571, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {290, 325, 297, 1162, 617, 204, 1165, 628} \[ \frac{1}{8 x \left (3 x^4+2\right )}-\frac{5 \sqrt [4]{3} \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{64\ 2^{3/4}}+\frac{5 \sqrt [4]{3} \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{64\ 2^{3/4}}-\frac{5}{16 x}+\frac{5 \sqrt [4]{3} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32\ 2^{3/4}}-\frac{5 \sqrt [4]{3} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{32\ 2^{3/4}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (2+3 x^4\right )^2} \, dx &=\frac{1}{8 x \left (2+3 x^4\right )}+\frac{5}{8} \int \frac{1}{x^2 \left (2+3 x^4\right )} \, dx\\ &=-\frac{5}{16 x}+\frac{1}{8 x \left (2+3 x^4\right )}-\frac{15}{16} \int \frac{x^2}{2+3 x^4} \, dx\\ &=-\frac{5}{16 x}+\frac{1}{8 x \left (2+3 x^4\right )}+\frac{1}{32} \left (5 \sqrt{3}\right ) \int \frac{\sqrt{2}-\sqrt{3} x^2}{2+3 x^4} \, dx-\frac{1}{32} \left (5 \sqrt{3}\right ) \int \frac{\sqrt{2}+\sqrt{3} x^2}{2+3 x^4} \, dx\\ &=-\frac{5}{16 x}+\frac{1}{8 x \left (2+3 x^4\right )}-\frac{5}{64} \int \frac{1}{\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx-\frac{5}{64} \int \frac{1}{\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx-\frac{\left (5 \sqrt [4]{3}\right ) \int \frac{\frac{2^{3/4}}{\sqrt [4]{3}}+2 x}{-\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{64\ 2^{3/4}}-\frac{\left (5 \sqrt [4]{3}\right ) \int \frac{\frac{2^{3/4}}{\sqrt [4]{3}}-2 x}{-\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{64\ 2^{3/4}}\\ &=-\frac{5}{16 x}+\frac{1}{8 x \left (2+3 x^4\right )}-\frac{5 \sqrt [4]{3} \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{64\ 2^{3/4}}+\frac{5 \sqrt [4]{3} \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{64\ 2^{3/4}}-\frac{\left (5 \sqrt [4]{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{32\ 2^{3/4}}+\frac{\left (5 \sqrt [4]{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{32\ 2^{3/4}}\\ &=-\frac{5}{16 x}+\frac{1}{8 x \left (2+3 x^4\right )}+\frac{5 \sqrt [4]{3} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{32\ 2^{3/4}}-\frac{5 \sqrt [4]{3} \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{32\ 2^{3/4}}-\frac{5 \sqrt [4]{3} \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{64\ 2^{3/4}}+\frac{5 \sqrt [4]{3} \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{64\ 2^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0775211, size = 113, normalized size = 0.81 \[ \frac{1}{128} \left (-\frac{24 x^3}{3 x^4+2}-5 \sqrt [4]{6} \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+5 \sqrt [4]{6} \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-\frac{32}{x}+10 \sqrt [4]{6} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )-10 \sqrt [4]{6} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 128, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,x}}-{\frac{{x}^{3}}{16} \left ({x}^{4}+{\frac{2}{3}} \right ) ^{-1}}-{\frac{5\,\sqrt{3}{6}^{3/4}\sqrt{2}}{384}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }-{\frac{5\,\sqrt{3}{6}^{3/4}\sqrt{2}}{384}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }-{\frac{5\,\sqrt{3}{6}^{3/4}\sqrt{2}}{768}\ln \left ({ \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48442, size = 190, normalized size = 1.36 \begin{align*} -\frac{5}{64} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) - \frac{5}{64} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{5}{128} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{5}{128} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{15 \, x^{4} + 8}{16 \,{\left (3 \, x^{5} + 2 \, x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82475, size = 647, normalized size = 4.62 \begin{align*} -\frac{120 \, x^{4} - 20 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}}{\left (3 \, x^{5} + 2 \, x\right )} \arctan \left (\frac{1}{3} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} \sqrt{3^{\frac{3}{4}} 2^{\frac{3}{4}} x + 3 \, x^{2} + \sqrt{3} \sqrt{2}} - 3^{\frac{1}{4}} 2^{\frac{1}{4}} x - 1\right ) - 20 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}}{\left (3 \, x^{5} + 2 \, x\right )} \arctan \left (\frac{1}{3} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}} \sqrt{-3^{\frac{3}{4}} 2^{\frac{3}{4}} x + 3 \, x^{2} + \sqrt{3} \sqrt{2}} - 3^{\frac{1}{4}} 2^{\frac{1}{4}} x + 1\right ) - 5 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}}{\left (3 \, x^{5} + 2 \, x\right )} \log \left (3^{\frac{3}{4}} 2^{\frac{3}{4}} x + 3 \, x^{2} + \sqrt{3} \sqrt{2}\right ) + 5 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}}{\left (3 \, x^{5} + 2 \, x\right )} \log \left (-3^{\frac{3}{4}} 2^{\frac{3}{4}} x + 3 \, x^{2} + \sqrt{3} \sqrt{2}\right ) + 64}{128 \,{\left (3 \, x^{5} + 2 \, x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.56215, size = 109, normalized size = 0.78 \begin{align*} - \frac{15 x^{4} + 8}{48 x^{5} + 32 x} - \frac{5 \sqrt [4]{6} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{128} + \frac{5 \sqrt [4]{6} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{128} - \frac{5 \sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{64} - \frac{5 \sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{64} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16147, size = 155, normalized size = 1.11 \begin{align*} -\frac{5}{64} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{5}{64} \cdot 6^{\frac{1}{4}} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{5}{128} \cdot 6^{\frac{1}{4}} \log \left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{5}{128} \cdot 6^{\frac{1}{4}} \log \left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{15 \, x^{4} + 8}{16 \,{\left (3 \, x^{5} + 2 \, x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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