Optimal. Leaf size=101 \[ \frac{c \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0765918, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {12, 297, 1162, 617, 204, 1165, 628} \[ \frac{c \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{c x^2}{2+3 x^4} \, dx &=c \int \frac{x^2}{2+3 x^4} \, dx\\ &=-\frac{c \int \frac{\sqrt{2}-\sqrt{3} x^2}{2+3 x^4} \, dx}{2 \sqrt{3}}+\frac{c \int \frac{\sqrt{2}+\sqrt{3} x^2}{2+3 x^4} \, dx}{2 \sqrt{3}}\\ &=\frac{1}{12} c \int \frac{1}{\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac{1}{12} c \int \frac{1}{\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac{c \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{c \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{c \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}-\frac{c \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}+\frac{c \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}-\frac{c \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{2\ 6^{3/4}}\\ &=-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}-\frac{c \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0147119, size = 78, normalized size = 0.77 \[ \frac{c \left (\log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2 \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \tan ^{-1}\left (\sqrt [4]{6} x+1\right )\right )}{4\ 6^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 114, normalized size = 1.1 \begin{align*}{\frac{c\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{c\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 ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.45521, size = 166, normalized size = 1.64 \begin{align*} \frac{1}{24} \,{\left (2 \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 ) + 2 \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 ) - 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 ) + 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 )\right )} c \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.44386, size = 869, normalized size = 8.6 \begin{align*} -\frac{1}{108} \cdot 54^{\frac{3}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}} \arctan \left (-\frac{54^{\frac{3}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}} x - 54^{\frac{3}{4}} \sqrt{2} \sqrt{\frac{1}{3}}{\left (c^{4}\right )}^{\frac{1}{4}} \sqrt{\frac{3 \, c^{3} x^{2} + 54^{\frac{1}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}} x + \sqrt{6} \sqrt{c^{4}} c}{c^{3}}} + 18 \, c}{18 \, c}\right ) - \frac{1}{108} \cdot 54^{\frac{3}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}} \arctan \left (-\frac{54^{\frac{3}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}} x - 54^{\frac{3}{4}} \sqrt{2} \sqrt{\frac{1}{3}}{\left (c^{4}\right )}^{\frac{1}{4}} \sqrt{\frac{3 \, c^{3} x^{2} - 54^{\frac{1}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}} x + \sqrt{6} \sqrt{c^{4}} c}{c^{3}}} - 18 \, c}{18 \, c}\right ) - \frac{1}{432} \cdot 54^{\frac{3}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}} \log \left (9 \, c^{3} x^{2} + 3 \cdot 54^{\frac{1}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}} x + 3 \, \sqrt{6} \sqrt{c^{4}} c\right ) + \frac{1}{432} \cdot 54^{\frac{3}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{1}{4}} \log \left (9 \, c^{3} x^{2} - 3 \cdot 54^{\frac{1}{4}} \sqrt{2}{\left (c^{4}\right )}^{\frac{3}{4}} x + 3 \, \sqrt{6} \sqrt{c^{4}} c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.361851, size = 88, normalized size = 0.87 \begin{align*} c \left (\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}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.0839, size = 131, normalized size = 1.3 \begin{align*} \frac{1}{24} \,{\left (2 \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 ) + 2 \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 ) - 6^{\frac{1}{4}} \log \left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + 6^{\frac{1}{4}} \log \left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right )\right )} c \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]