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 = 0.0230899, antiderivative size = 43, normalized size of antiderivative = 1., 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 638
Rule 618
Rule 206
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*}
Mathematica [A] time = 0.0374513, 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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Maple [A] time = 0.003, size = 37, normalized size = 0.9 \begin{align*} -{\frac{14\,x-36}{120\,{x}^{2}-160\,x-80}}+{\frac{7\,\sqrt{10}}{200}{\it Artanh} \left ({\frac{ \left ( 6\,x-4 \right ) \sqrt{10}}{20}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47678, size = 63, normalized size = 1.47 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98568, size = 184, normalized size = 4.28 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.124447, size = 58, normalized size = 1.35 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06143, size = 69, normalized size = 1.6 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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