Optimal. Leaf size=51 \[ \frac{\log \left (3 x^2+2 \sqrt{3} x+2\right )}{4 \sqrt{3}}-\frac{\log \left (3 x^2-2 \sqrt{3} x+2\right )}{4 \sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0213201, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1165, 628} \[ \frac{\log \left (3 x^2+2 \sqrt{3} x+2\right )}{4 \sqrt{3}}-\frac{\log \left (3 x^2-2 \sqrt{3} x+2\right )}{4 \sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{2-3 x^2}{4+9 x^4} \, dx &=-\frac{\int \frac{\frac{2}{\sqrt{3}}+2 x}{-\frac{2}{3}-\frac{2 x}{\sqrt{3}}-x^2} \, dx}{4 \sqrt{3}}-\frac{\int \frac{\frac{2}{\sqrt{3}}-2 x}{-\frac{2}{3}+\frac{2 x}{\sqrt{3}}-x^2} \, dx}{4 \sqrt{3}}\\ &=-\frac{\log \left (2-2 \sqrt{3} x+3 x^2\right )}{4 \sqrt{3}}+\frac{\log \left (2+2 \sqrt{3} x+3 x^2\right )}{4 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0127092, size = 44, normalized size = 0.86 \[ \frac{\log \left (3 x^2+2 \sqrt{3} x+2\right )-\log \left (-3 x^2+2 \sqrt{3} x-2\right )}{4 \sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.043, size = 82, normalized size = 1.6 \begin{align*}{\frac{\sqrt{6}\sqrt{2}}{48}\ln \left ({ \left ({x}^{2}+{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{6}\sqrt{2}}{48}\ln \left ({ \left ({x}^{2}-{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{6}x\sqrt{2}}{3}}+{\frac{2}{3}} \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.49291, size = 53, normalized size = 1.04 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (3 \, x^{2} + 2 \, \sqrt{3} x + 2\right ) - \frac{1}{12} \, \sqrt{3} \log \left (3 \, x^{2} - 2 \, \sqrt{3} x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.43909, size = 105, normalized size = 2.06 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (\frac{9 \, x^{4} + 24 \, x^{2} + 4 \, \sqrt{3}{\left (3 \, x^{3} + 2 \, x\right )} + 4}{9 \, x^{4} + 4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.099225, size = 49, normalized size = 0.96 \begin{align*} - \frac{\sqrt{3} \log{\left (x^{2} - \frac{2 \sqrt{3} x}{3} + \frac{2}{3} \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \frac{2 \sqrt{3} x}{3} + \frac{2}{3} \right )}}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13689, size = 54, normalized size = 1.06 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{2} \left (\frac{4}{9}\right )^{\frac{1}{4}} x + \frac{2}{3}\right ) - \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{2} \left (\frac{4}{9}\right )^{\frac{1}{4}} x + \frac{2}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]