Optimal. Leaf size=101 \[ -\frac{a \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}} \]
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Rubi [A] time = 0.0960809, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {12, 211, 1165, 628, 1162, 617, 204} \[ -\frac{a \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{a}{2+3 x^4} \, dx &=a \int \frac{1}{2+3 x^4} \, dx\\ &=\frac{a \int \frac{\sqrt{2}-\sqrt{3} x^2}{2+3 x^4} \, dx}{2 \sqrt{2}}+\frac{a \int \frac{\sqrt{2}+\sqrt{3} x^2}{2+3 x^4} \, dx}{2 \sqrt{2}}\\ &=\frac{a \int \frac{1}{\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{4 \sqrt{6}}+\frac{a \int \frac{1}{\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{4 \sqrt{6}}-\frac{a \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{a \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{a \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}\\ &=-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}-\frac{a \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}\\ \end{align*}
Mathematica [A] time = 0.0290976, size = 78, normalized size = 0.77 \[ \frac{a \left (-\log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+\log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right )}{8 \sqrt [4]{6}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 114, normalized size = 1.1 \begin{align*}{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\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{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.477, size = 166, normalized size = 1.64 \begin{align*} \frac{1}{48} \,{\left (2 \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 ) + 2 \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 ) + 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 ) - 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 )\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.66575, size = 882, normalized size = 8.73 \begin{align*} -\frac{1}{48} \cdot 24^{\frac{3}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} \arctan \left (-\frac{4 \, a^{3} + 2 \cdot 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} x - 24^{\frac{1}{4}} \sqrt{2} \sqrt{\frac{1}{3}}{\left (a^{4}\right )}^{\frac{3}{4}} \sqrt{\frac{12 \, a^{2} x^{2} + 24^{\frac{3}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 4 \, \sqrt{6} \sqrt{a^{4}}}{a^{2}}}}{4 \, a^{3}}\right ) - \frac{1}{48} \cdot 24^{\frac{3}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} \arctan \left (\frac{4 \, a^{3} - 2 \cdot 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} x + 24^{\frac{1}{4}} \sqrt{2} \sqrt{\frac{1}{3}}{\left (a^{4}\right )}^{\frac{3}{4}} \sqrt{\frac{12 \, a^{2} x^{2} - 24^{\frac{3}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 4 \, \sqrt{6} \sqrt{a^{4}}}{a^{2}}}}{4 \, a^{3}}\right ) + \frac{1}{192} \cdot 24^{\frac{3}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} \log \left (12 \, a^{2} x^{2} + 24^{\frac{3}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 4 \, \sqrt{6} \sqrt{a^{4}}\right ) - \frac{1}{192} \cdot 24^{\frac{3}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} \log \left (12 \, a^{2} x^{2} - 24^{\frac{3}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 4 \, \sqrt{6} \sqrt{a^{4}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.359027, size = 88, normalized size = 0.87 \begin{align*} a \left (- \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}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11121, size = 131, normalized size = 1.3 \begin{align*} \frac{1}{48} \,{\left (2 \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 ) + 2 \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 ) + 6^{\frac{3}{4}} \log \left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - 6^{\frac{3}{4}} \log \left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right )\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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