Optimal. Leaf size=110 \[ -\frac{\log \left (4 x^2-6 x+9\right )}{17496}-\frac{\log \left (4 x^2+6 x+9\right )}{17496}-\frac{1}{2916 (2 x+3)}-\frac{\log (3-2 x)}{17496}+\frac{5 \log (2 x+3)}{17496}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{8748 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{2916 \sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.118137, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {1586, 2074, 634, 618, 204, 628} \[ -\frac{\log \left (4 x^2-6 x+9\right )}{17496}-\frac{\log \left (4 x^2+6 x+9\right )}{17496}-\frac{1}{2916 (2 x+3)}-\frac{\log (3-2 x)}{17496}+\frac{5 \log (2 x+3)}{17496}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{8748 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{2916 \sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1586
Rule 2074
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{243-162 x+108 x^2-72 x^3+48 x^4-32 x^5}{\left (729-64 x^6\right )^2} \, dx &=\int \frac{1}{(3+2 x)^2 \left (243-162 x+108 x^2-72 x^3+48 x^4-32 x^5\right )} \, dx\\ &=\int \left (-\frac{1}{8748 (-3+2 x)}+\frac{1}{1458 (3+2 x)^2}+\frac{5}{8748 (3+2 x)}+\frac{3-2 x}{4374 \left (9-6 x+4 x^2\right )}+\frac{3-2 x}{4374 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=-\frac{1}{2916 (3+2 x)}-\frac{\log (3-2 x)}{17496}+\frac{5 \log (3+2 x)}{17496}+\frac{\int \frac{3-2 x}{9-6 x+4 x^2} \, dx}{4374}+\frac{\int \frac{3-2 x}{9+6 x+4 x^2} \, dx}{4374}\\ &=-\frac{1}{2916 (3+2 x)}-\frac{\log (3-2 x)}{17496}+\frac{5 \log (3+2 x)}{17496}-\frac{\int \frac{-6+8 x}{9-6 x+4 x^2} \, dx}{17496}-\frac{\int \frac{6+8 x}{9+6 x+4 x^2} \, dx}{17496}+\frac{\int \frac{1}{9-6 x+4 x^2} \, dx}{2916}+\frac{1}{972} \int \frac{1}{9+6 x+4 x^2} \, dx\\ &=-\frac{1}{2916 (3+2 x)}-\frac{\log (3-2 x)}{17496}+\frac{5 \log (3+2 x)}{17496}-\frac{\log \left (9-6 x+4 x^2\right )}{17496}-\frac{\log \left (9+6 x+4 x^2\right )}{17496}-\frac{\operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )}{1458}-\frac{1}{486} \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,6+8 x\right )\\ &=-\frac{1}{2916 (3+2 x)}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{8748 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{3+4 x}{3 \sqrt{3}}\right )}{2916 \sqrt{3}}-\frac{\log (3-2 x)}{17496}+\frac{5 \log (3+2 x)}{17496}-\frac{\log \left (9-6 x+4 x^2\right )}{17496}-\frac{\log \left (9+6 x+4 x^2\right )}{17496}\\ \end{align*}
Mathematica [A] time = 0.0872429, size = 100, normalized size = 0.91 \[ \frac{-3 \log \left (4 x^2-6 x+9\right )-3 \log \left (4 x^2+6 x+9\right )-\frac{18}{2 x+3}-3 \log (3-2 x)+15 \log (2 x+3)+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )+6 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{52488} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 85, normalized size = 0.8 \begin{align*} -{\frac{1}{8748+5832\,x}}+{\frac{5\,\ln \left ( 3+2\,x \right ) }{17496}}-{\frac{\ln \left ( -3+2\,x \right ) }{17496}}-{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{17496}}+{\frac{\sqrt{3}}{8748}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }-{\frac{\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{17496}}+{\frac{\sqrt{3}}{26244}\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 = 1.40531, size = 113, normalized size = 1.03 \begin{align*} \frac{1}{8748} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{26244} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{1}{2916 \,{\left (2 \, x + 3\right )}} - \frac{1}{17496} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{17496} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{5}{17496} \, \log \left (2 \, x + 3\right ) - \frac{1}{17496} \, \log \left (2 \, x - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.49653, size = 342, normalized size = 3.11 \begin{align*} \frac{6 \, \sqrt{3}{\left (2 \, x + 3\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + 2 \, \sqrt{3}{\left (2 \, x + 3\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - 3 \,{\left (2 \, x + 3\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 3 \,{\left (2 \, x + 3\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 15 \,{\left (2 \, x + 3\right )} \log \left (2 \, x + 3\right ) - 3 \,{\left (2 \, x + 3\right )} \log \left (2 \, x - 3\right ) - 18}{52488 \,{\left (2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.358786, size = 105, normalized size = 0.95 \begin{align*} - \frac{\log{\left (x - \frac{3}{2} \right )}}{17496} + \frac{5 \log{\left (x + \frac{3}{2} \right )}}{17496} - \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{17496} - \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{17496} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{26244} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{8748} - \frac{1}{5832 x + 8748} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.07194, size = 116, normalized size = 1.05 \begin{align*} \frac{1}{8748} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{26244} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{1}{2916 \,{\left (2 \, x + 3\right )}} - \frac{1}{17496} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{17496} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{5}{17496} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{17496} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]