Optimal. Leaf size=97 \[ -\frac {\log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}+\frac {\log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}-\frac {\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac {\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}} \]
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Rubi [A] time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {211, 1165, 628, 1162, 617, 204} \[ -\frac {\log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}+\frac {\log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}-\frac {\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac {\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{2+3 x^4} \, dx &=\frac {\int \frac {\sqrt {2}-\sqrt {3} x^2}{2+3 x^4} \, dx}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2}+\sqrt {3} x^2}{2+3 x^4} \, dx}{2 \sqrt {2}}\\ &=\frac {\int \frac {1}{\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{4 \sqrt {6}}+\frac {\int \frac {1}{\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{4 \sqrt {6}}-\frac {\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 \sqrt [4]{6}}-\frac {\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 \sqrt [4]{6}}\\ &=-\frac {\log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac {\log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}\\ &=-\frac {\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac {\tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}-\frac {\log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac {\log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 77, normalized size = 0.79 \[ \frac {-\log \left (\sqrt {6} x^2-2 \sqrt [4]{6} x+2\right )+\log \left (\sqrt {6} x^2+2 \sqrt [4]{6} x+2\right )-2 \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{8 \sqrt [4]{6}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 163, normalized size = 1.68 \[ -\frac {1}{48} \cdot 24^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{12} \cdot 24^{\frac {1}{4}} \sqrt {3} \sqrt {2} \sqrt {24^{\frac {3}{4}} \sqrt {2} x + 12 \, x^{2} + 4 \, \sqrt {6}} - \frac {1}{2} \cdot 24^{\frac {1}{4}} \sqrt {2} x - 1\right ) - \frac {1}{48} \cdot 24^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{12} \cdot 24^{\frac {1}{4}} \sqrt {3} \sqrt {2} \sqrt {-24^{\frac {3}{4}} \sqrt {2} x + 12 \, x^{2} + 4 \, \sqrt {6}} - \frac {1}{2} \cdot 24^{\frac {1}{4}} \sqrt {2} x + 1\right ) + \frac {1}{192} \cdot 24^{\frac {3}{4}} \sqrt {2} \log \left (24^{\frac {3}{4}} \sqrt {2} x + 12 \, x^{2} + 4 \, \sqrt {6}\right ) - \frac {1}{192} \cdot 24^{\frac {3}{4}} \sqrt {2} \log \left (-24^{\frac {3}{4}} \sqrt {2} x + 12 \, x^{2} + 4 \, \sqrt {6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 95, normalized size = 0.98 \[ \frac {1}{24} \cdot 6^{\frac {3}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{24} \cdot 6^{\frac {3}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{48} \cdot 6^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{48} \cdot 6^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 111, normalized size = 1.14 \[ \frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{48} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 121, normalized size = 1.25 \[ \frac {1}{24} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{24} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{48} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) - \frac {1}{48} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 33, normalized size = 0.34 \[ 6^{3/4}\,\mathrm {atan}\left (6^{1/4}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{24}+\frac {1}{24}{}\mathrm {i}\right )+6^{3/4}\,\mathrm {atan}\left (6^{1/4}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{24}-\frac {1}{24}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 87, normalized size = 0.90 \[ - \frac {6^{\frac {3}{4}} \log {\left (x^{2} - \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{48} + \frac {6^{\frac {3}{4}} \log {\left (x^{2} + \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{48} + \frac {6^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac {6^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
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