Optimal. Leaf size=60 \[ \frac{1}{324} \log \left (4 x^2-6 x+9\right )-\frac{1}{108} \log (3-2 x)+\frac{1}{324} \log (2 x+3)-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{54 \sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0485636, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1586, 2058, 634, 618, 204, 628} \[ \frac{1}{324} \log \left (4 x^2-6 x+9\right )-\frac{1}{108} \log (3-2 x)+\frac{1}{324} \log (2 x+3)-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{54 \sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1586
Rule 2058
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{9+6 x+4 x^2}{729-64 x^6} \, dx &=\int \frac{1}{81-54 x+24 x^3-16 x^4} \, dx\\ &=\int \left (-\frac{1}{54 (-3+2 x)}+\frac{1}{162 (3+2 x)}+\frac{3+2 x}{81 \left (9-6 x+4 x^2\right )}\right ) \, dx\\ &=-\frac{1}{108} \log (3-2 x)+\frac{1}{324} \log (3+2 x)+\frac{1}{81} \int \frac{3+2 x}{9-6 x+4 x^2} \, dx\\ &=-\frac{1}{108} \log (3-2 x)+\frac{1}{324} \log (3+2 x)+\frac{1}{324} \int \frac{-6+8 x}{9-6 x+4 x^2} \, dx+\frac{1}{18} \int \frac{1}{9-6 x+4 x^2} \, dx\\ &=-\frac{1}{108} \log (3-2 x)+\frac{1}{324} \log (3+2 x)+\frac{1}{324} \log \left (9-6 x+4 x^2\right )-\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )\\ &=-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{54 \sqrt{3}}-\frac{1}{108} \log (3-2 x)+\frac{1}{324} \log (3+2 x)+\frac{1}{324} \log \left (9-6 x+4 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0108217, size = 52, normalized size = 0.87 \[ \frac{1}{324} \left (\log \left (4 x^2-6 x+9\right )-3 \log (3-2 x)+\log (2 x+3)+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 47, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 3+2\,x \right ) }{324}}-{\frac{\ln \left ( -3+2\,x \right ) }{108}}+{\frac{\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{324}}+{\frac{\sqrt{3}}{162}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.38407, size = 62, normalized size = 1.03 \begin{align*} \frac{1}{162} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{324} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{324} \, \log \left (2 \, x + 3\right ) - \frac{1}{108} \, \log \left (2 \, x - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.43353, size = 157, normalized size = 2.62 \begin{align*} \frac{1}{162} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{324} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{324} \, \log \left (2 \, x + 3\right ) - \frac{1}{108} \, \log \left (2 \, x - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.179838, size = 56, normalized size = 0.93 \begin{align*} - \frac{\log{\left (x - \frac{3}{2} \right )}}{108} + \frac{\log{\left (x + \frac{3}{2} \right )}}{324} + \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{324} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{162} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.072, size = 65, normalized size = 1.08 \begin{align*} \frac{1}{162} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{324} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{324} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{108} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]