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 = 0.08, antiderivative size = 140, normalized size of antiderivative = 1.00, 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 204
Rule 290
Rule 297
Rule 325
Rule 617
Rule 628
Rule 1162
Rule 1165
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*}
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Mathematica [A] time = 0.12, size = 113, normalized size = 0.81 \[ \frac {1}{128} \left (-5 \sqrt [4]{6} \log \left (\sqrt {6} x^2-2 \sqrt [4]{6} x+2\right )+5 \sqrt [4]{6} \log \left (\sqrt {6} x^2+2 \sqrt [4]{6} x+2\right )-\frac {24 x^3}{3 x^4+2}-\frac {32}{x}+10 \sqrt [4]{6} \tan ^{-1}\left (1-\sqrt [4]{6} x\right )-10 \sqrt [4]{6} \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 220, normalized size = 1.57 \[ -\frac {120 \, x^{4} - 20 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} {\left (3 \, x^{5} + 2 \, x\right )} \arctan \left (\frac {1}{3} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} \sqrt {3^{\frac {3}{4}} 2^{\frac {3}{4}} x + 3 \, x^{2} + \sqrt {3} \sqrt {2}} - 3^{\frac {1}{4}} 2^{\frac {1}{4}} x - 1\right ) - 20 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} {\left (3 \, x^{5} + 2 \, x\right )} \arctan \left (\frac {1}{3} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} \sqrt {-3^{\frac {3}{4}} 2^{\frac {3}{4}} x + 3 \, x^{2} + \sqrt {3} \sqrt {2}} - 3^{\frac {1}{4}} 2^{\frac {1}{4}} x + 1\right ) - 5 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} {\left (3 \, x^{5} + 2 \, x\right )} \log \left (3^{\frac {3}{4}} 2^{\frac {3}{4}} x + 3 \, x^{2} + \sqrt {3} \sqrt {2}\right ) + 5 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} {\left (3 \, x^{5} + 2 \, x\right )} \log \left (-3^{\frac {3}{4}} 2^{\frac {3}{4}} x + 3 \, x^{2} + \sqrt {3} \sqrt {2}\right ) + 64}{128 \, {\left (3 \, x^{5} + 2 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 115, normalized size = 0.82 \[ -\frac {5}{64} \cdot 6^{\frac {1}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) - \frac {5}{64} \cdot 6^{\frac {1}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {5}{128} \cdot 6^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {5}{128} \cdot 6^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {15 \, x^{4} + 8}{16 \, {\left (3 \, x^{5} + 2 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 128, normalized size = 0.91 \[ -\frac {x^{3}}{16 \left (x^{4}+\frac {2}{3}\right )}-\frac {5 \sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{384}-\frac {5 \sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{384}-\frac {5 \sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{768}-\frac {1}{4 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.06, size = 141, normalized size = 1.01 \[ -\frac {5}{64} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) - \frac {5}{64} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {5}{128} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) - \frac {5}{128} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) - \frac {15 \, x^{4} + 8}{16 \, {\left (3 \, x^{5} + 2 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 51, normalized size = 0.36 \[ -\frac {\frac {5\,x^4}{16}+\frac {1}{6}}{x^5+\frac {2\,x}{3}}+6^{1/4}\,\mathrm {atan}\left (6^{1/4}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {5}{64}+\frac {5}{64}{}\mathrm {i}\right )+6^{1/4}\,\mathrm {atan}\left (6^{1/4}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {5}{64}-\frac {5}{64}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 110, normalized size = 0.79 \[ \frac {- 15 x^{4} - 8}{48 x^{5} + 32 x} - \frac {5 \sqrt [4]{6} \log {\left (x^{2} - \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{128} + \frac {5 \sqrt [4]{6} \log {\left (x^{2} + \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{128} - \frac {5 \sqrt [4]{6} \operatorname {atan}{\left (\sqrt [4]{6} x - 1 \right )}}{64} - \frac {5 \sqrt [4]{6} \operatorname {atan}{\left (\sqrt [4]{6} x + 1 \right )}}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
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