Optimal. Leaf size=124 \[ -\frac{\sqrt [4]{3} \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8\ 2^{3/4}}+\frac{\sqrt [4]{3} \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8\ 2^{3/4}}-\frac{1}{2 x}+\frac{\sqrt [4]{3} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{3} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 2^{3/4}} \]
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Rubi [A] time = 0.0760307, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {325, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\sqrt [4]{3} \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8\ 2^{3/4}}+\frac{\sqrt [4]{3} \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8\ 2^{3/4}}-\frac{1}{2 x}+\frac{\sqrt [4]{3} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{3} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 2^{3/4}} \]
Antiderivative was successfully verified.
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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 )} \, dx &=-\frac{1}{2 x}-\frac{3}{2} \int \frac{x^2}{2+3 x^4} \, dx\\ &=-\frac{1}{2 x}+\frac{1}{4} \sqrt{3} \int \frac{\sqrt{2}-\sqrt{3} x^2}{2+3 x^4} \, dx-\frac{1}{4} \sqrt{3} \int \frac{\sqrt{2}+\sqrt{3} x^2}{2+3 x^4} \, dx\\ &=-\frac{1}{2 x}-\frac{1}{8} \int \frac{1}{\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx-\frac{1}{8} \int \frac{1}{\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx-\frac{\sqrt [4]{3} \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}{8\ 2^{3/4}}-\frac{\sqrt [4]{3} \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}{8\ 2^{3/4}}\\ &=-\frac{1}{2 x}-\frac{\sqrt [4]{3} \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{8\ 2^{3/4}}+\frac{\sqrt [4]{3} \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{8\ 2^{3/4}}-\frac{\sqrt [4]{3} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{4\ 2^{3/4}}+\frac{\sqrt [4]{3} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{4\ 2^{3/4}}\\ &=-\frac{1}{2 x}+\frac{\sqrt [4]{3} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{3} \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{3} \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{8\ 2^{3/4}}+\frac{\sqrt [4]{3} \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{8\ 2^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0225613, size = 101, normalized size = 0.81 \[ -\frac{\sqrt [4]{6} x \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\sqrt [4]{6} x \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \sqrt [4]{6} x \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} x \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+8}{16 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 116, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{48}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }-{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{48}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }-{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{96}\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 ) }-{\frac{1}{2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52039, size = 170, normalized size = 1.37 \begin{align*} -\frac{1}{8} \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{1}{8} \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{1}{16} \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{1}{16} \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{1}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86949, size = 540, normalized size = 4.35 \begin{align*} \frac{4 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} x \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 ) + 4 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} x \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 ) + 3^{\frac{1}{4}} 2^{\frac{1}{4}} x \log \left (3^{\frac{3}{4}} 2^{\frac{3}{4}} x + 3 \, x^{2} + \sqrt{3} \sqrt{2}\right ) - 3^{\frac{1}{4}} 2^{\frac{1}{4}} x \log \left (-3^{\frac{3}{4}} 2^{\frac{3}{4}} x + 3 \, x^{2} + \sqrt{3} \sqrt{2}\right ) - 8}{16 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.581032, size = 92, normalized size = 0.74 \begin{align*} - \frac{\sqrt [4]{6} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{16} + \frac{\sqrt [4]{6} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{16} - \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{8} - \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{8} - \frac{1}{2 x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17495, size = 135, normalized size = 1.09 \begin{align*} -\frac{1}{8} \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{1}{8} \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{1}{16} \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{1}{16} \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{1}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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