Optimal. Leaf size=43 \[ -\frac {18-7 x}{20 \left (-3 x^2+4 x+2\right )}-\frac {7 \tanh ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{20 \sqrt {10}} \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {638, 618, 206} \[ -\frac {18-7 x}{20 \left (-3 x^2+4 x+2\right )}-\frac {7 \tanh ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{20 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 638
Rubi steps
\begin {align*} \int \frac {5-4 x}{\left (-2-4 x+3 x^2\right )^2} \, dx &=-\frac {18-7 x}{20 \left (2+4 x-3 x^2\right )}-\frac {7}{20} \int \frac {1}{-2-4 x+3 x^2} \, dx\\ &=-\frac {18-7 x}{20 \left (2+4 x-3 x^2\right )}+\frac {7}{10} \operatorname {Subst}\left (\int \frac {1}{40-x^2} \, dx,x,-4+6 x\right )\\ &=-\frac {18-7 x}{20 \left (2+4 x-3 x^2\right )}-\frac {7 \tanh ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{20 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 62, normalized size = 1.44 \[ \frac {18-7 x}{20 \left (3 x^2-4 x-2\right )}-\frac {7 \log \left (-3 x+\sqrt {10}+2\right )}{40 \sqrt {10}}+\frac {7 \log \left (3 x+\sqrt {10}-2\right )}{40 \sqrt {10}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 48, normalized size = 1.12 \[ \frac {18-7 x}{20 \left (3 x^2-4 x-2\right )}-\frac {7 \tanh ^{-1}\left (\sqrt {\frac {2}{5}}-\frac {3 x}{\sqrt {10}}\right )}{20 \sqrt {10}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 68, normalized size = 1.58 \[ \frac {7 \, \sqrt {10} {\left (3 \, x^{2} - 4 \, x - 2\right )} \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {10} {\left (3 \, x - 2\right )} - 12 \, x + 14}{3 \, x^{2} - 4 \, x - 2}\right ) - 140 \, x + 360}{400 \, {\left (3 \, x^{2} - 4 \, x - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 51, normalized size = 1.19 \[ -\frac {7}{400} \, \sqrt {10} \log \left (\frac {{\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}}{{\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}}\right ) - \frac {7 \, x - 18}{20 \, {\left (3 \, x^{2} - 4 \, x - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 37, normalized size = 0.86
method | result | size |
default | \(-\frac {14 x -36}{40 \left (3 x^{2}-4 x -2\right )}+\frac {7 \sqrt {10}\, \arctanh \left (\frac {\left (6 x -4\right ) \sqrt {10}}{20}\right )}{200}\) | \(37\) |
risch | \(\frac {-\frac {7 x}{60}+\frac {3}{10}}{x^{2}-\frac {4}{3} x -\frac {2}{3}}+\frac {7 \sqrt {10}\, \ln \left (3 x -2+\sqrt {10}\right )}{400}-\frac {7 \sqrt {10}\, \ln \left (3 x -2-\sqrt {10}\right )}{400}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 47, normalized size = 1.09 \[ -\frac {7}{400} \, \sqrt {10} \log \left (\frac {3 \, x - \sqrt {10} - 2}{3 \, x + \sqrt {10} - 2}\right ) - \frac {7 \, x - 18}{20 \, {\left (3 \, x^{2} - 4 \, x - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 34, normalized size = 0.79 \[ \frac {7\,\sqrt {10}\,\mathrm {atanh}\left (\sqrt {10}\,\left (\frac {3\,x}{10}-\frac {1}{5}\right )\right )}{200}+\frac {\frac {7\,x}{60}-\frac {3}{10}}{-x^2+\frac {4\,x}{3}+\frac {2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 58, normalized size = 1.35 \[ - \frac {7 x - 18}{60 x^{2} - 80 x - 40} + \frac {7 \sqrt {10} \log {\left (x - \frac {2}{3} + \frac {\sqrt {10}}{3} \right )}}{400} - \frac {7 \sqrt {10} \log {\left (x - \frac {\sqrt {10}}{3} - \frac {2}{3} \right )}}{400} \]
Verification of antiderivative is not currently implemented for this CAS.
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