Optimal. Leaf size=59 \[ -\frac {1}{8 x}+\frac {3}{20 x \left (3-2 x^2\right )^2}+\frac {1}{8 x \left (3-2 x^2\right )}+\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{3}} x\right )}{4 \sqrt {6}} \]
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Rubi [A]
time = 0.01, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {12, 296, 331,
212} \begin {gather*} \frac {1}{8 x \left (3-2 x^2\right )}+\frac {3}{20 x \left (3-2 x^2\right )^2}-\frac {1}{8 x}+\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{3}} x\right )}{4 \sqrt {6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 296
Rule 331
Rubi steps
\begin {align*} \int \frac {9}{5 x^2 \left (3-2 x^2\right )^3} \, dx &=\frac {9}{5} \int \frac {1}{x^2 \left (3-2 x^2\right )^3} \, dx\\ &=\frac {3}{20 x \left (3-2 x^2\right )^2}+\frac {3}{4} \int \frac {1}{x^2 \left (3-2 x^2\right )^2} \, dx\\ &=\frac {3}{20 x \left (3-2 x^2\right )^2}+\frac {1}{8 x \left (3-2 x^2\right )}+\frac {3}{8} \int \frac {1}{x^2 \left (3-2 x^2\right )} \, dx\\ &=-\frac {1}{8 x}+\frac {3}{20 x \left (3-2 x^2\right )^2}+\frac {1}{8 x \left (3-2 x^2\right )}+\frac {1}{4} \int \frac {1}{3-2 x^2} \, dx\\ &=-\frac {1}{8 x}+\frac {3}{20 x \left (3-2 x^2\right )^2}+\frac {1}{8 x \left (3-2 x^2\right )}+\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{3}} x\right )}{4 \sqrt {6}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 65, normalized size = 1.10 \begin {gather*} \frac {1}{240} \left (-\frac {12 \left (12-25 x^2+10 x^4\right )}{x \left (3-2 x^2\right )^2}-5 \sqrt {6} \log \left (\sqrt {6}-2 x\right )+5 \sqrt {6} \log \left (\sqrt {6}+2 x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 39, normalized size = 0.66
method | result | size |
default | \(-\frac {1}{15 x}-\frac {8 \left (\frac {7}{16} x^{3}-\frac {27}{32} x \right )}{15 \left (2 x^{2}-3\right )^{2}}+\frac {\arctanh \left (\frac {x \sqrt {6}}{3}\right ) \sqrt {6}}{24}\) | \(39\) |
meijerg | \(\frac {i \sqrt {6}\, \left (\frac {i \sqrt {6}\, \left (\frac {20}{3} x^{4}-\frac {50}{3} x^{2}+8\right )}{4 x \left (-\frac {2 x^{2}}{3}+1\right )^{2}}-\frac {15 i \arctanh \left (\frac {x \sqrt {2}\, \sqrt {3}}{3}\right )}{2}\right )}{180}\) | \(51\) |
risch | \(\frac {-\frac {1}{2} x^{4}+\frac {5}{4} x^{2}-\frac {3}{5}}{\left (2 x^{2}-3\right )^{2} x}+\frac {\sqrt {6}\, \ln \left (2 x +\sqrt {6}\right )}{48}-\frac {\sqrt {6}\, \ln \left (2 x -\sqrt {6}\right )}{48}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.24, size = 56, normalized size = 0.95 \begin {gather*} -\frac {1}{48} \, \sqrt {6} \log \left (\frac {2 \, x - \sqrt {6}}{2 \, x + \sqrt {6}}\right ) - \frac {10 \, x^{4} - 25 \, x^{2} + 12}{20 \, {\left (4 \, x^{5} - 12 \, x^{3} + 9 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 73, normalized size = 1.24 \begin {gather*} -\frac {120 \, x^{4} - 5 \, \sqrt {6} {\left (4 \, x^{5} - 12 \, x^{3} + 9 \, x\right )} \log \left (\frac {2 \, x^{2} + 2 \, \sqrt {6} x + 3}{2 \, x^{2} - 3}\right ) - 300 \, x^{2} + 144}{240 \, {\left (4 \, x^{5} - 12 \, x^{3} + 9 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 58, normalized size = 0.98 \begin {gather*} - \frac {9 \cdot \left (10 x^{4} - 25 x^{2} + 12\right )}{720 x^{5} - 2160 x^{3} + 1620 x} - \frac {\sqrt {6} \log {\left (x - \frac {\sqrt {6}}{2} \right )}}{48} + \frac {\sqrt {6} \log {\left (x + \frac {\sqrt {6}}{2} \right )}}{48} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 55, normalized size = 0.93 \begin {gather*} -\frac {1}{48} \, \sqrt {6} \log \left (\frac {{\left | 4 \, x - 2 \, \sqrt {6} \right |}}{{\left | 4 \, x + 2 \, \sqrt {6} \right |}}\right ) - \frac {14 \, x^{3} - 27 \, x}{60 \, {\left (2 \, x^{2} - 3\right )}^{2}} - \frac {1}{15 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 41, normalized size = 0.69 \begin {gather*} \frac {\sqrt {6}\,\mathrm {atanh}\left (\frac {\sqrt {6}\,x}{3}\right )}{24}-\frac {\frac {x^4}{8}-\frac {5\,x^2}{16}+\frac {3}{20}}{x^5-3\,x^3+\frac {9\,x}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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