Integrand size = 103, antiderivative size = 17 \[ \int \frac {(1296-432 x+1296 e x) \log ^3(x)+(1296-432 x+1296 e x) \log ^2(x) \log (4-x+3 e x)+(432-144 x+432 e x) \log (x) \log ^2(4-x+3 e x)+(48-16 x+48 e x) \log ^3(4-x+3 e x)}{4 x-x^2+3 e x^2} \, dx=(3 \log (x)+\log (4-x+3 e x))^4 \]
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Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {6, 1607, 6820, 12, 6818} \[ \int \frac {(1296-432 x+1296 e x) \log ^3(x)+(1296-432 x+1296 e x) \log ^2(x) \log (4-x+3 e x)+(432-144 x+432 e x) \log (x) \log ^2(4-x+3 e x)+(48-16 x+48 e x) \log ^3(4-x+3 e x)}{4 x-x^2+3 e x^2} \, dx=(3 \log (x)+\log (3 e x-x+4))^4 \]
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Rule 6
Rule 12
Rule 1607
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1296-432 x+1296 e x) \log ^3(x)+(1296-432 x+1296 e x) \log ^2(x) \log (4-x+3 e x)+(432-144 x+432 e x) \log (x) \log ^2(4-x+3 e x)+(48-16 x+48 e x) \log ^3(4-x+3 e x)}{4 x+(-1+3 e) x^2} \, dx \\ & = \int \frac {(1296-432 x+1296 e x) \log ^3(x)+(1296-432 x+1296 e x) \log ^2(x) \log (4-x+3 e x)+(432-144 x+432 e x) \log (x) \log ^2(4-x+3 e x)+(48-16 x+48 e x) \log ^3(4-x+3 e x)}{x (4+(-1+3 e) x)} \, dx \\ & = \int \frac {16 (3+(-1+3 e) x) (3 \log (x)+\log (4-x+3 e x))^3}{x (4+(-1+3 e) x)} \, dx \\ & = 16 \int \frac {(3+(-1+3 e) x) (3 \log (x)+\log (4-x+3 e x))^3}{x (4+(-1+3 e) x)} \, dx \\ & = (3 \log (x)+\log (4-x+3 e x))^4 \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {(1296-432 x+1296 e x) \log ^3(x)+(1296-432 x+1296 e x) \log ^2(x) \log (4-x+3 e x)+(432-144 x+432 e x) \log (x) \log ^2(4-x+3 e x)+(48-16 x+48 e x) \log ^3(4-x+3 e x)}{4 x-x^2+3 e x^2} \, dx=(3 \log (x)+\log (4-x+3 e x))^4 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(18)=36\).
Time = 1.88 (sec) , antiderivative size = 74, normalized size of antiderivative = 4.35
method | result | size |
risch | \(81 \ln \left (x \right )^{4}+108 \ln \left (x \right )^{3} \ln \left (3 x \,{\mathrm e}-x +4\right )+54 \ln \left (x \right )^{2} \ln \left (3 x \,{\mathrm e}-x +4\right )^{2}+12 \ln \left (x \right ) \ln \left (3 x \,{\mathrm e}-x +4\right )^{3}+\ln \left (3 x \,{\mathrm e}-x +4\right )^{4}\) | \(74\) |
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (18) = 36\).
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 4.29 \[ \int \frac {(1296-432 x+1296 e x) \log ^3(x)+(1296-432 x+1296 e x) \log ^2(x) \log (4-x+3 e x)+(432-144 x+432 e x) \log (x) \log ^2(4-x+3 e x)+(48-16 x+48 e x) \log ^3(4-x+3 e x)}{4 x-x^2+3 e x^2} \, dx=\log \left (3 \, x e - x + 4\right )^{4} + 12 \, \log \left (3 \, x e - x + 4\right )^{3} \log \left (x\right ) + 54 \, \log \left (3 \, x e - x + 4\right )^{2} \log \left (x\right )^{2} + 108 \, \log \left (3 \, x e - x + 4\right ) \log \left (x\right )^{3} + 81 \, \log \left (x\right )^{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (17) = 34\).
Time = 0.57 (sec) , antiderivative size = 76, normalized size of antiderivative = 4.47 \[ \int \frac {(1296-432 x+1296 e x) \log ^3(x)+(1296-432 x+1296 e x) \log ^2(x) \log (4-x+3 e x)+(432-144 x+432 e x) \log (x) \log ^2(4-x+3 e x)+(48-16 x+48 e x) \log ^3(4-x+3 e x)}{4 x-x^2+3 e x^2} \, dx=81 \log {\left (x \right )}^{4} + 108 \log {\left (x \right )}^{3} \log {\left (- x + 3 e x + 4 \right )} + 54 \log {\left (x \right )}^{2} \log {\left (- x + 3 e x + 4 \right )}^{2} + 12 \log {\left (x \right )} \log {\left (- x + 3 e x + 4 \right )}^{3} + \log {\left (- x + 3 e x + 4 \right )}^{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (18) = 36\).
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 4.29 \[ \int \frac {(1296-432 x+1296 e x) \log ^3(x)+(1296-432 x+1296 e x) \log ^2(x) \log (4-x+3 e x)+(432-144 x+432 e x) \log (x) \log ^2(4-x+3 e x)+(48-16 x+48 e x) \log ^3(4-x+3 e x)}{4 x-x^2+3 e x^2} \, dx=\log \left (x {\left (3 \, e - 1\right )} + 4\right )^{4} + 12 \, \log \left (x {\left (3 \, e - 1\right )} + 4\right )^{3} \log \left (x\right ) + 54 \, \log \left (x {\left (3 \, e - 1\right )} + 4\right )^{2} \log \left (x\right )^{2} + 108 \, \log \left (x {\left (3 \, e - 1\right )} + 4\right ) \log \left (x\right )^{3} + 81 \, \log \left (x\right )^{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (18) = 36\).
Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 4.41 \[ \int \frac {(1296-432 x+1296 e x) \log ^3(x)+(1296-432 x+1296 e x) \log ^2(x) \log (4-x+3 e x)+(432-144 x+432 e x) \log (x) \log ^2(4-x+3 e x)+(48-16 x+48 e x) \log ^3(4-x+3 e x)}{4 x-x^2+3 e x^2} \, dx=4 \, \log \left (3 \, x e - x + 4\right )^{4} + 12 \, \log \left (3 \, x e - x + 4\right )^{3} \log \left (x\right ) + 54 \, \log \left (3 \, x e - x + 4\right )^{2} \log \left (x\right )^{2} + 108 \, \log \left (3 \, x e - x + 4\right ) \log \left (x\right )^{3} + 81 \, \log \left (x\right )^{4} \]
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Time = 13.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 4.29 \[ \int \frac {(1296-432 x+1296 e x) \log ^3(x)+(1296-432 x+1296 e x) \log ^2(x) \log (4-x+3 e x)+(432-144 x+432 e x) \log (x) \log ^2(4-x+3 e x)+(48-16 x+48 e x) \log ^3(4-x+3 e x)}{4 x-x^2+3 e x^2} \, dx={\ln \left (3\,x\,\mathrm {e}-x+4\right )}^4+12\,{\ln \left (3\,x\,\mathrm {e}-x+4\right )}^3\,\ln \left (x\right )+54\,{\ln \left (3\,x\,\mathrm {e}-x+4\right )}^2\,{\ln \left (x\right )}^2+108\,\ln \left (3\,x\,\mathrm {e}-x+4\right )\,{\ln \left (x\right )}^3+81\,{\ln \left (x\right )}^4 \]
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