Integrand size = 39, antiderivative size = 12 \[ \int \frac {\left (32+16 x+2 x^2\right ) \log (-2+x)+\left (-16+4 x+2 x^2\right ) \log ^2(-2+x)}{-2+x} \, dx=(4+x)^2 \log ^2(-2+x) \]
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Leaf count is larger than twice the leaf count of optimal. \(36\) vs. \(2(12)=24\).
Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 3.00, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6873, 12, 6874, 45, 2442, 2458, 2372, 2338, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int \frac {\left (32+16 x+2 x^2\right ) \log (-2+x)+\left (-16+4 x+2 x^2\right ) \log ^2(-2+x)}{-2+x} \, dx=(2-x)^2 \log ^2(x-2)-12 (2-x) \log ^2(x-2)+36 \log ^2(x-2) \]
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Rule 12
Rule 45
Rule 2332
Rule 2333
Rule 2338
Rule 2341
Rule 2342
Rule 2372
Rule 2436
Rule 2437
Rule 2442
Rule 2448
Rule 2458
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 (4+x) \log (-2+x) (-4-x+2 \log (-2+x)-x \log (-2+x))}{2-x} \, dx \\ & = 2 \int \frac {(4+x) \log (-2+x) (-4-x+2 \log (-2+x)-x \log (-2+x))}{2-x} \, dx \\ & = 2 \int \left (\frac {(4+x)^2 \log (-2+x)}{-2+x}+(4+x) \log ^2(-2+x)\right ) \, dx \\ & = 2 \int \frac {(4+x)^2 \log (-2+x)}{-2+x} \, dx+2 \int (4+x) \log ^2(-2+x) \, dx \\ & = 2 \int \left (6 \log ^2(-2+x)+(-2+x) \log ^2(-2+x)\right ) \, dx+2 \text {Subst}\left (\int \frac {(6+x)^2 \log (x)}{x} \, dx,x,-2+x\right ) \\ & = -24 (2-x) \log (-2+x)+(2-x)^2 \log (-2+x)+72 \log ^2(-2+x)+2 \int (-2+x) \log ^2(-2+x) \, dx-2 \text {Subst}\left (\int \left (12+\frac {x}{2}+\frac {36 \log (x)}{x}\right ) \, dx,x,-2+x\right )+12 \int \log ^2(-2+x) \, dx \\ & = -\frac {1}{2} (2-x)^2-24 x-24 (2-x) \log (-2+x)+(2-x)^2 \log (-2+x)+72 \log ^2(-2+x)+2 \text {Subst}\left (\int x \log ^2(x) \, dx,x,-2+x\right )+12 \text {Subst}\left (\int \log ^2(x) \, dx,x,-2+x\right )-72 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-2+x\right ) \\ & = -\frac {1}{2} (2-x)^2-24 x-24 (2-x) \log (-2+x)+(2-x)^2 \log (-2+x)+36 \log ^2(-2+x)-12 (2-x) \log ^2(-2+x)+(2-x)^2 \log ^2(-2+x)-2 \text {Subst}(\int x \log (x) \, dx,x,-2+x)-24 \text {Subst}(\int \log (x) \, dx,x,-2+x) \\ & = 36 \log ^2(-2+x)-12 (2-x) \log ^2(-2+x)+(2-x)^2 \log ^2(-2+x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(12)=24\).
Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 3.92 \[ \int \frac {\left (32+16 x+2 x^2\right ) \log (-2+x)+\left (-16+4 x+2 x^2\right ) \log ^2(-2+x)}{-2+x} \, dx=2 \left (-2 \log (2-x)+2 \log (-2+x)+8 \log ^2(-2+x)+4 x \log ^2(-2+x)+\frac {1}{2} x^2 \log ^2(-2+x)\right ) \]
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Time = 0.41 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.33
method | result | size |
risch | \(\left (x^{2}+8 x +16\right ) \ln \left (-2+x \right )^{2}\) | \(16\) |
norman | \(\ln \left (-2+x \right )^{2} x^{2}+16 \ln \left (-2+x \right )^{2}+8 \ln \left (-2+x \right )^{2} x\) | \(29\) |
parallelrisch | \(\ln \left (-2+x \right )^{2} x^{2}+16 \ln \left (-2+x \right )^{2}+8 \ln \left (-2+x \right )^{2} x\) | \(29\) |
derivativedivides | \(\left (-2+x \right )^{2} \ln \left (-2+x \right )^{2}+12 \left (-2+x \right ) \ln \left (-2+x \right )^{2}+36 \ln \left (-2+x \right )^{2}\) | \(33\) |
default | \(\left (-2+x \right )^{2} \ln \left (-2+x \right )^{2}+12 \left (-2+x \right ) \ln \left (-2+x \right )^{2}+36 \ln \left (-2+x \right )^{2}\) | \(33\) |
parts | \(\left (-2+x \right )^{2} \ln \left (-2+x \right )^{2}+12 \left (-2+x \right ) \ln \left (-2+x \right )^{2}+36 \ln \left (-2+x \right )^{2}\) | \(33\) |
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Time = 0.56 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \frac {\left (32+16 x+2 x^2\right ) \log (-2+x)+\left (-16+4 x+2 x^2\right ) \log ^2(-2+x)}{-2+x} \, dx={\left (x^{2} + 8 \, x + 16\right )} \log \left (x - 2\right )^{2} \]
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Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\left (32+16 x+2 x^2\right ) \log (-2+x)+\left (-16+4 x+2 x^2\right ) \log ^2(-2+x)}{-2+x} \, dx=\left (x^{2} + 8 x + 16\right ) \log {\left (x - 2 \right )}^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (12) = 24\).
Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 8.08 \[ \int \frac {\left (32+16 x+2 x^2\right ) \log (-2+x)+\left (-16+4 x+2 x^2\right ) \log ^2(-2+x)}{-2+x} \, dx=\frac {1}{2} \, {\left (2 \, \log \left (x - 2\right )^{2} - 2 \, \log \left (x - 2\right ) + 1\right )} {\left (x - 2\right )}^{2} + 12 \, {\left (\log \left (x - 2\right )^{2} - 2 \, \log \left (x - 2\right ) + 2\right )} {\left (x - 2\right )} - \frac {1}{2} \, x^{2} + {\left (x^{2} + 4 \, x + 8 \, \log \left (x - 2\right )\right )} \log \left (x - 2\right ) + 16 \, {\left (x + 2 \, \log \left (x - 2\right )\right )} \log \left (x - 2\right ) - 4 \, \log \left (x - 2\right )^{2} - 22 \, x - 44 \, \log \left (x - 2\right ) \]
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Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \frac {\left (32+16 x+2 x^2\right ) \log (-2+x)+\left (-16+4 x+2 x^2\right ) \log ^2(-2+x)}{-2+x} \, dx={\left (x^{2} + 8 \, x + 16\right )} \log \left (x - 2\right )^{2} \]
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Time = 11.35 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\left (32+16 x+2 x^2\right ) \log (-2+x)+\left (-16+4 x+2 x^2\right ) \log ^2(-2+x)}{-2+x} \, dx={\ln \left (x-2\right )}^2\,{\left (x+4\right )}^2 \]
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