Optimal. Leaf size=61 \[ \frac{3 (1-x)}{128 \left (-x^2+2 x+3\right )}+\frac{1-x}{16 \left (-x^2+2 x+3\right )^2}+\frac{3}{512} \log (3-x)-\frac{3}{512} \log (x+1) \]
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Rubi [A] time = 0.012479, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {614, 616, 31} \[ \frac{3 (1-x)}{128 \left (-x^2+2 x+3\right )}+\frac{1-x}{16 \left (-x^2+2 x+3\right )^2}+\frac{3}{512} \log (3-x)-\frac{3}{512} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 614
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\left (-3-2 x+x^2\right )^3} \, dx &=\frac{1-x}{16 \left (3+2 x-x^2\right )^2}-\frac{3}{16} \int \frac{1}{\left (-3-2 x+x^2\right )^2} \, dx\\ &=\frac{1-x}{16 \left (3+2 x-x^2\right )^2}+\frac{3 (1-x)}{128 \left (3+2 x-x^2\right )}+\frac{3}{128} \int \frac{1}{-3-2 x+x^2} \, dx\\ &=\frac{1-x}{16 \left (3+2 x-x^2\right )^2}+\frac{3 (1-x)}{128 \left (3+2 x-x^2\right )}+\frac{3}{512} \int \frac{1}{-3+x} \, dx-\frac{3}{512} \int \frac{1}{1+x} \, dx\\ &=\frac{1-x}{16 \left (3+2 x-x^2\right )^2}+\frac{3 (1-x)}{128 \left (3+2 x-x^2\right )}+\frac{3}{512} \log (3-x)-\frac{3}{512} \log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0224939, size = 46, normalized size = 0.75 \[ \frac{1}{512} \left (\frac{4 \left (3 x^3-9 x^2-11 x+17\right )}{\left (x^2-2 x-3\right )^2}+3 \log (3-x)-3 \log (x+1)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 42, normalized size = 0.7 \begin{align*}{\frac{1}{128\, \left ( 1+x \right ) ^{2}}}+{\frac{3}{256+256\,x}}-{\frac{3\,\ln \left ( 1+x \right ) }{512}}-{\frac{1}{128\, \left ( -3+x \right ) ^{2}}}+{\frac{3}{-768+256\,x}}+{\frac{3\,\ln \left ( -3+x \right ) }{512}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.921371, size = 68, normalized size = 1.11 \begin{align*} \frac{3 \, x^{3} - 9 \, x^{2} - 11 \, x + 17}{128 \,{\left (x^{4} - 4 \, x^{3} - 2 \, x^{2} + 12 \, x + 9\right )}} - \frac{3}{512} \, \log \left (x + 1\right ) + \frac{3}{512} \, \log \left (x - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94209, size = 224, normalized size = 3.67 \begin{align*} \frac{12 \, x^{3} - 36 \, x^{2} - 3 \,{\left (x^{4} - 4 \, x^{3} - 2 \, x^{2} + 12 \, x + 9\right )} \log \left (x + 1\right ) + 3 \,{\left (x^{4} - 4 \, x^{3} - 2 \, x^{2} + 12 \, x + 9\right )} \log \left (x - 3\right ) - 44 \, x + 68}{512 \,{\left (x^{4} - 4 \, x^{3} - 2 \, x^{2} + 12 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.145338, size = 51, normalized size = 0.84 \begin{align*} \frac{3 x^{3} - 9 x^{2} - 11 x + 17}{128 x^{4} - 512 x^{3} - 256 x^{2} + 1536 x + 1152} + \frac{3 \log{\left (x - 3 \right )}}{512} - \frac{3 \log{\left (x + 1 \right )}}{512} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06083, size = 57, normalized size = 0.93 \begin{align*} \frac{3 \, x^{3} - 9 \, x^{2} - 11 \, x + 17}{128 \,{\left (x^{2} - 2 \, x - 3\right )}^{2}} - \frac{3}{512} \, \log \left ({\left | x + 1 \right |}\right ) + \frac{3}{512} \, \log \left ({\left | x - 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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