Optimal. Leaf size=97 \[ \frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.067842, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {297, 1162, 617, 204, 1165, 628} \[ \frac{\log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{\log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{\tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^2}{2+3 x^4} \, dx &=-\frac{\int \frac{\sqrt{2}-\sqrt{3} x^2}{2+3 x^4} \, dx}{2 \sqrt{3}}+\frac{\int \frac{\sqrt{2}+\sqrt{3} x^2}{2+3 x^4} \, dx}{2 \sqrt{3}}\\ &=\frac{1}{12} \int \frac{1}{\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac{1}{12} \int \frac{1}{\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\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}{4\ 6^{3/4}}+\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}{4\ 6^{3/4}}\\ &=\frac{\log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}-\frac{\log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{2\ 6^{3/4}}\\ &=-\frac{\tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{\tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{\log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}-\frac{\log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.013498, size = 77, normalized size = 0.79 \[ \frac{\log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 111, normalized size = 1.1 \begin{align*}{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{144}\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.51108, size = 163, normalized size = 1.68 \begin{align*} \frac{1}{12} \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}{12} \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}{24} \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}{24} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.83566, size = 551, normalized size = 5.68 \begin{align*} -\frac{1}{108} \cdot 54^{\frac{3}{4}} \sqrt{2} \arctan \left (-\frac{1}{18} \cdot 54^{\frac{3}{4}} \sqrt{2} x + \frac{1}{54} \cdot 54^{\frac{3}{4}} \sqrt{2} \sqrt{9 \, x^{2} + 3 \cdot 54^{\frac{1}{4}} \sqrt{2} x + 3 \, \sqrt{6}} - 1\right ) - \frac{1}{108} \cdot 54^{\frac{3}{4}} \sqrt{2} \arctan \left (-\frac{1}{18} \cdot 54^{\frac{3}{4}} \sqrt{2} x + \frac{1}{54} \cdot 54^{\frac{3}{4}} \sqrt{2} \sqrt{9 \, x^{2} - 3 \cdot 54^{\frac{1}{4}} \sqrt{2} x + 3 \, \sqrt{6}} + 1\right ) - \frac{1}{432} \cdot 54^{\frac{3}{4}} \sqrt{2} \log \left (9 \, x^{2} + 3 \cdot 54^{\frac{1}{4}} \sqrt{2} x + 3 \, \sqrt{6}\right ) + \frac{1}{432} \cdot 54^{\frac{3}{4}} \sqrt{2} \log \left (9 \, x^{2} - 3 \cdot 54^{\frac{1}{4}} \sqrt{2} x + 3 \, \sqrt{6}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.492054, size = 87, normalized size = 0.9 \begin{align*} \frac{\sqrt [4]{6} \log{\left (x^{2} - \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{24} - \frac{\sqrt [4]{6} \log{\left (x^{2} + \frac{6^{\frac{3}{4}} x}{3} + \frac{\sqrt{6}}{3} \right )}}{24} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{12} + \frac{\sqrt [4]{6} \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1714, size = 128, normalized size = 1.32 \begin{align*} \frac{1}{12} \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}{12} \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}{24} \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}{24} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]