Optimal. Leaf size=148 \[ -\frac {1}{708588 (3+2 x)}+\frac {3-x}{708588 \left (9-6 x+4 x^2\right )}+\frac {x}{236196 \left (9+6 x+4 x^2\right )}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{1417176 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}-\frac {\log (3-2 x)}{4251528}+\frac {\log (3+2 x)}{472392}-\frac {\log \left (9-6 x+4 x^2\right )}{944784}+\frac {\log \left (9+6 x+4 x^2\right )}{8503056} \]
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Rubi [A]
time = 0.12, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {1600, 2099,
652, 632, 210, 648, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{1417176 \sqrt {3}}+\frac {\text {ArcTan}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}+\frac {3-x}{708588 \left (4 x^2-6 x+9\right )}+\frac {x}{236196 \left (4 x^2+6 x+9\right )}-\frac {\log \left (4 x^2-6 x+9\right )}{944784}+\frac {\log \left (4 x^2+6 x+9\right )}{8503056}-\frac {1}{708588 (2 x+3)}-\frac {\log (3-2 x)}{4251528}+\frac {\log (2 x+3)}{472392} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 652
Rule 1600
Rule 2099
Rubi steps
\begin {align*} \int \frac {3-2 x}{\left (729-64 x^6\right )^2} \, dx &=\int \frac {1}{(3-2 x) \left (243+162 x+108 x^2+72 x^3+48 x^4+32 x^5\right )^2} \, dx\\ &=\int \left (-\frac {1}{2125764 (-3+2 x)}+\frac {1}{354294 (3+2 x)^2}+\frac {1}{236196 (3+2 x)}-\frac {x}{39366 \left (9-6 x+4 x^2\right )^2}+\frac {7-6 x}{708588 \left (9-6 x+4 x^2\right )}+\frac {3+x}{39366 \left (9+6 x+4 x^2\right )^2}+\frac {33+2 x}{2125764 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=-\frac {1}{708588 (3+2 x)}-\frac {\log (3-2 x)}{4251528}+\frac {\log (3+2 x)}{472392}+\frac {\int \frac {33+2 x}{9+6 x+4 x^2} \, dx}{2125764}+\frac {\int \frac {7-6 x}{9-6 x+4 x^2} \, dx}{708588}-\frac {\int \frac {x}{\left (9-6 x+4 x^2\right )^2} \, dx}{39366}+\frac {\int \frac {3+x}{\left (9+6 x+4 x^2\right )^2} \, dx}{39366}\\ &=-\frac {1}{708588 (3+2 x)}+\frac {3-x}{708588 \left (9-6 x+4 x^2\right )}+\frac {x}{236196 \left (9+6 x+4 x^2\right )}-\frac {\log (3-2 x)}{4251528}+\frac {\log (3+2 x)}{472392}+\frac {\int \frac {6+8 x}{9+6 x+4 x^2} \, dx}{8503056}-\frac {\int \frac {-6+8 x}{9-6 x+4 x^2} \, dx}{944784}-\frac {\int \frac {1}{9-6 x+4 x^2} \, dx}{708588}+\frac {5 \int \frac {1}{9-6 x+4 x^2} \, dx}{1417176}+\frac {\int \frac {1}{9+6 x+4 x^2} \, dx}{236196}+\frac {7 \int \frac {1}{9+6 x+4 x^2} \, dx}{472392}\\ &=-\frac {1}{708588 (3+2 x)}+\frac {3-x}{708588 \left (9-6 x+4 x^2\right )}+\frac {x}{236196 \left (9+6 x+4 x^2\right )}-\frac {\log (3-2 x)}{4251528}+\frac {\log (3+2 x)}{472392}-\frac {\log \left (9-6 x+4 x^2\right )}{944784}+\frac {\log \left (9+6 x+4 x^2\right )}{8503056}+\frac {\text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )}{354294}-\frac {5 \text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )}{708588}-\frac {\text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,6+8 x\right )}{118098}-\frac {7 \text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,6+8 x\right )}{236196}\\ &=-\frac {1}{708588 (3+2 x)}+\frac {3-x}{708588 \left (9-6 x+4 x^2\right )}+\frac {x}{236196 \left (9+6 x+4 x^2\right )}-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{1417176 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )}{157464 \sqrt {3}}-\frac {\log (3-2 x)}{4251528}+\frac {\log (3+2 x)}{472392}-\frac {\log \left (9-6 x+4 x^2\right )}{944784}+\frac {\log \left (9+6 x+4 x^2\right )}{8503056}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 119, normalized size = 0.80 \begin {gather*} \frac {\frac {1944 x}{243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}+2 \sqrt {3} \tan ^{-1}\left (\frac {-3+4 x}{3 \sqrt {3}}\right )+18 \sqrt {3} \tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )-2 \log (3-2 x)+18 \log (3+2 x)-9 \log \left (9-6 x+4 x^2\right )+\log \left (9+6 x+4 x^2\right )}{8503056} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 115, normalized size = 0.78
method | result | size |
risch | \(\frac {x}{139968 x^{5}+209952 x^{4}+314928 x^{3}+472392 x^{2}+708588 x +1062882}+\frac {\ln \left (2 x +3\right )}{472392}+\frac {\ln \left (36 x^{2}+54 x +81\right )}{8503056}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (6 x +\frac {9}{2}\right ) \sqrt {3}}{27}\right )}{472392}-\frac {\ln \left (-3+2 x \right )}{4251528}-\frac {\ln \left (4 x^{2}-6 x +9\right )}{944784}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (2 x -\frac {3}{2}\right ) \sqrt {3}}{9}\right )}{4251528}\) | \(104\) |
default | \(-\frac {\frac {x}{4}-\frac {3}{4}}{708588 \left (x^{2}-\frac {3}{2} x +\frac {9}{4}\right )}-\frac {\ln \left (4 x^{2}-6 x +9\right )}{944784}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{4251528}+\frac {x}{944784 x^{2}+1417176 x +2125764}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{8503056}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{472392}-\frac {1}{708588 \left (2 x +3\right )}+\frac {\ln \left (2 x +3\right )}{472392}-\frac {\ln \left (-3+2 x \right )}{4251528}\) | \(115\) |
meijerg | \(-\frac {\left (-1\right )^{\frac {5}{6}} \left (\frac {4 x \left (-1\right )^{\frac {1}{6}}}{6-\frac {128 x^{6}}{243}}-\frac {5 x \left (-1\right )^{\frac {1}{6}} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}\right )}{708588}+\frac {\left (-1\right )^{\frac {2}{3}} \left (\frac {4 x^{2} \left (-1\right )^{\frac {1}{3}}}{3 \left (3-\frac {64 x^{6}}{243}\right )}-\frac {2 x^{2} \left (-1\right )^{\frac {1}{3}} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{3 \left (x^{6}\right )^{\frac {1}{3}}}\right )}{708588}\) | \(242\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 105, normalized size = 0.71 \begin {gather*} \frac {1}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{4251528} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {x}{4374 \, {\left (32 \, x^{5} + 48 \, x^{4} + 72 \, x^{3} + 108 \, x^{2} + 162 \, x + 243\right )}} + \frac {1}{8503056} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{472392} \, \log \left (2 \, x + 3\right ) - \frac {1}{4251528} \, \log \left (2 \, x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 256 vs.
\(2 (116) = 232\).
time = 0.41, size = 256, normalized size = 1.73 \begin {gather*} \frac {18 \, \sqrt {3} {\left (32 \, x^{5} + 48 \, x^{4} + 72 \, x^{3} + 108 \, x^{2} + 162 \, x + 243\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + 2 \, \sqrt {3} {\left (32 \, x^{5} + 48 \, x^{4} + 72 \, x^{3} + 108 \, x^{2} + 162 \, x + 243\right )} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + {\left (32 \, x^{5} + 48 \, x^{4} + 72 \, x^{3} + 108 \, x^{2} + 162 \, x + 243\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 9 \, {\left (32 \, x^{5} + 48 \, x^{4} + 72 \, x^{3} + 108 \, x^{2} + 162 \, x + 243\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 18 \, {\left (32 \, x^{5} + 48 \, x^{4} + 72 \, x^{3} + 108 \, x^{2} + 162 \, x + 243\right )} \log \left (2 \, x + 3\right ) - 2 \, {\left (32 \, x^{5} + 48 \, x^{4} + 72 \, x^{3} + 108 \, x^{2} + 162 \, x + 243\right )} \log \left (2 \, x - 3\right ) + 1944 \, x}{8503056 \, {\left (32 \, x^{5} + 48 \, x^{4} + 72 \, x^{3} + 108 \, x^{2} + 162 \, x + 243\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.32, size = 124, normalized size = 0.84 \begin {gather*} \frac {x}{139968 x^{5} + 209952 x^{4} + 314928 x^{3} + 472392 x^{2} + 708588 x + 1062882} - \frac {\log {\left (x - \frac {3}{2} \right )}}{4251528} + \frac {\log {\left (x + \frac {3}{2} \right )}}{472392} - \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {9}{4} \right )}}{944784} + \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {9}{4} \right )}}{8503056} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{4251528} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} + \frac {\sqrt {3}}{3} \right )}}{472392} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.47, size = 111, normalized size = 0.75 \begin {gather*} \frac {1}{472392} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x + 3\right )}\right ) + \frac {1}{4251528} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {x}{4374 \, {\left (4 \, x^{2} + 6 \, x + 9\right )} {\left (4 \, x^{2} - 6 \, x + 9\right )} {\left (2 \, x + 3\right )}} + \frac {1}{8503056} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{944784} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac {1}{472392} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {1}{4251528} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 120, normalized size = 0.81 \begin {gather*} \frac {\ln \left (x+\frac {3}{2}\right )}{472392}-\frac {\ln \left (x-\frac {3}{2}\right )}{4251528}-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8503056}\right )-\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{8503056}+\frac {\sqrt {3}\,1{}\mathrm {i}}{944784}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{944784}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8503056}\right )+\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{8503056}+\frac {\sqrt {3}\,1{}\mathrm {i}}{944784}\right )+\frac {x}{139968\,\left (x^5+\frac {3\,x^4}{2}+\frac {9\,x^3}{4}+\frac {27\,x^2}{8}+\frac {81\,x}{16}+\frac {243}{32}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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