Optimal. Leaf size=110 \[ -\frac {1}{2916 (3+2 x)}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{8748 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )}{2916 \sqrt {3}}-\frac {\log (3-2 x)}{17496}+\frac {5 \log (3+2 x)}{17496}-\frac {\log \left (9-6 x+4 x^2\right )}{17496}-\frac {\log \left (9+6 x+4 x^2\right )}{17496} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 110, normalized size of antiderivative = 1.00, 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 = {1600, 2099,
648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{8748 \sqrt {3}}+\frac {\text {ArcTan}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{2916 \sqrt {3}}-\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 632
Rule 642
Rule 648
Rule 1600
Rule 2099
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 {\text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )}{1458}-\frac {1}{486} \text {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 = 0.06, size = 100, normalized size = 0.91 \begin {gather*} \frac {-\frac {18}{3+2 x}+2 \sqrt {3} \tan ^{-1}\left (\frac {-3+4 x}{3 \sqrt {3}}\right )+6 \sqrt {3} \tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )-3 \log (3-2 x)+15 \log (3+2 x)-3 \log \left (9-6 x+4 x^2\right )-3 \log \left (9+6 x+4 x^2\right )}{52488} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.39, size = 85, normalized size = 0.77
method | result | size |
risch | \(-\frac {1}{5832 \left (x +\frac {3}{2}\right )}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (-3+4 x \right ) \sqrt {3}}{9}\right )}{26244}-\frac {\ln \left (16 x^{2}-24 x +36\right )}{17496}-\frac {\ln \left (-3+2 x \right )}{17496}+\frac {5 \ln \left (2 x +3\right )}{17496}-\frac {\ln \left (4 x^{2}+6 x +9\right )}{17496}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (2 x +\frac {3}{2}\right ) \sqrt {3}}{9}\right )}{8748}\) | \(83\) |
default | \(-\frac {\ln \left (4 x^{2}-6 x +9\right )}{17496}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{26244}-\frac {\ln \left (4 x^{2}+6 x +9\right )}{17496}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{8748}-\frac {1}{2916 \left (2 x +3\right )}+\frac {5 \ln \left (2 x +3\right )}{17496}-\frac {\ln \left (-3+2 x \right )}{17496}\) | \(85\) |
meijerg | \(-\frac {\left (-1\right )^{\frac {5}{6}} \left (\frac {4 x \left (-1\right )^{\frac {1}{6}}}{6-\frac {128 x^{6}}{243}}-\frac {5 x \left (-1\right )^{\frac {1}{6}} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}\right )}{8748}-\frac {16 x^{6}}{1594323 \left (1-\frac {64 x^{6}}{729}\right )}-\frac {\left (-1\right )^{\frac {1}{6}} \left (\frac {64 x^{5} \left (-1\right )^{\frac {5}{6}}}{81 \left (6-\frac {128 x^{6}}{243}\right )}-\frac {x^{5} \left (-1\right )^{\frac {5}{6}} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {5}{6}}}\right )}{8748}+\frac {\left (-1\right )^{\frac {1}{3}} \left (\frac {16 x^{4} \left (-1\right )^{\frac {2}{3}}}{27 \left (3-\frac {64 x^{6}}{243}\right )}-\frac {x^{4} \left (-1\right )^{\frac {2}{3}} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{3 \left (x^{6}\right )^{\frac {2}{3}}}\right )}{8748}-\frac {i \left (\frac {16 i x^{3}}{27 \left (-\frac {128 x^{6}}{729}+2\right )}+i \arctanh \left (\frac {8 x^{3}}{27}\right )\right )}{8748}+\frac {\left (-1\right )^{\frac {2}{3}} \left (\frac {4 x^{2} \left (-1\right )^{\frac {1}{3}}}{3 \left (3-\frac {64 x^{6}}{243}\right )}-\frac {2 x^{2} \left (-1\right )^{\frac {1}{3}} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{3 \left (x^{6}\right )^{\frac {1}{3}}}\right )}{8748}\) | \(525\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.52, size = 84, normalized size = 0.76 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 115, normalized size = 1.05 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.20, size = 105, normalized size = 0.95 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.83, size = 86, normalized size = 0.78 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 5.10, size = 100, normalized size = 0.91 \begin {gather*} \frac {5\,\ln \left (x+\frac {3}{2}\right )}{17496}-\frac {\ln \left (x-\frac {3}{2}\right )}{17496}-\frac {1}{5832\,\left (x+\frac {3}{2}\right )}-\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{17496}+\frac {\sqrt {3}\,1{}\mathrm {i}}{17496}\right )+\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{17496}+\frac {\sqrt {3}\,1{}\mathrm {i}}{17496}\right )-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{17496}+\frac {\sqrt {3}\,1{}\mathrm {i}}{52488}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{17496}+\frac {\sqrt {3}\,1{}\mathrm {i}}{52488}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________