Integrand size = 72, antiderivative size = 23 \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=\left (-1+e^3-e^4-x+\log \left (3+\frac {3}{x}\right )\right )^2 \]
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Time = 0.18 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1607, 6820, 12, 6818} \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=\left (x-\log \left (\frac {3}{x}+3\right )+e^4-e^3+1\right )^2 \]
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Rule 12
Rule 1607
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x (1+x)} \, dx \\ & = \int \frac {2 \left (1+x+x^2\right ) \left (1+(-1+e) e^3+x-\log \left (3+\frac {3}{x}\right )\right )}{x (1+x)} \, dx \\ & = 2 \int \frac {\left (1+x+x^2\right ) \left (1+(-1+e) e^3+x-\log \left (3+\frac {3}{x}\right )\right )}{x (1+x)} \, dx \\ & = \left (1-e^3+e^4+x-\log \left (3+\frac {3}{x}\right )\right )^2 \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=\left (1-e^3+e^4+x-\log \left (3+\frac {3}{x}\right )\right )^2 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(21)=42\).
Time = 0.66 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74
method | result | size |
derivativedivides | \(-2 \left ({\mathrm e}^{4}-{\mathrm e}^{3}\right ) \ln \left (\frac {3}{x}+3\right )+\frac {2 \left (3 \,{\mathrm e}^{4}-3 \,{\mathrm e}^{3}+3\right ) x}{3}+x^{2}+\ln \left (\frac {3}{x}+3\right )^{2}-\frac {2 \ln \left (\frac {3}{x}+3\right ) \left (\frac {3}{x}+3\right ) x}{3}\) | \(63\) |
default | \(-2 \left ({\mathrm e}^{4}-{\mathrm e}^{3}\right ) \ln \left (\frac {3}{x}+3\right )+\frac {2 \left (3 \,{\mathrm e}^{4}-3 \,{\mathrm e}^{3}+3\right ) x}{3}+x^{2}+\ln \left (\frac {3}{x}+3\right )^{2}-\frac {2 \ln \left (\frac {3}{x}+3\right ) \left (\frac {3}{x}+3\right ) x}{3}\) | \(63\) |
norman | \(x^{2}+\ln \left (\frac {3 x +3}{x}\right )^{2}+\left (2+2 \,{\mathrm e}^{4}-2 \,{\mathrm e}^{3}\right ) x +\left (-2 \,{\mathrm e}^{4}+2 \,{\mathrm e}^{3}-2\right ) \ln \left (\frac {3 x +3}{x}\right )-2 x \ln \left (\frac {3 x +3}{x}\right )\) | \(63\) |
parts | \(x^{2}+2 x \,{\mathrm e}^{4}-2 x \,{\mathrm e}^{3}+2 x +2 \left ({\mathrm e}^{4}-{\mathrm e}^{3}+1\right ) \ln \left (x \right )+2 \left (-{\mathrm e}^{4}+{\mathrm e}^{3}\right ) \ln \left (1+x \right )+\ln \left (\frac {3}{x}+3\right )^{2}+2 \ln \left (\frac {3}{x}\right )-\frac {2 \ln \left (\frac {3}{x}+3\right ) \left (\frac {3}{x}+3\right ) x}{3}\) | \(79\) |
parallelrisch | \(2 x \,{\mathrm e}^{4}-2 \,{\mathrm e}^{4} \ln \left (\frac {3 x +3}{x}\right )-2 x \,{\mathrm e}^{3}+2 \ln \left (\frac {3 x +3}{x}\right ) {\mathrm e}^{3}-5+x^{2}-2 x \ln \left (\frac {3 x +3}{x}\right )+\ln \left (\frac {3 x +3}{x}\right )^{2}-4 \,{\mathrm e}^{4}+4 \,{\mathrm e}^{3}+2 x -2 \ln \left (\frac {3 x +3}{x}\right )\) | \(87\) |
risch | \(2 x \,{\mathrm e}^{4}-2 x \,{\mathrm e}^{3}+x^{2}+2 x +2 \,{\mathrm e}^{4} \ln \left (x \right )-2 \ln \left (x \right ) {\mathrm e}^{3}+2 \ln \left (x \right )-2 \,{\mathrm e}^{3} \ln \left (1+x \right ) {\mathrm e}+2 \,{\mathrm e}^{3} \ln \left (1+x \right )+\ln \left (\frac {3}{x}+3\right )^{2}+2 \ln \left (\frac {3}{x}\right )-\frac {2 \ln \left (\frac {3}{x}+3\right ) \left (\frac {3}{x}+3\right ) x}{3}\) | \(88\) |
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=x^{2} + 2 \, x e^{4} - 2 \, x e^{3} - 2 \, {\left (x + e^{4} - e^{3} + 1\right )} \log \left (\frac {3 \, {\left (x + 1\right )}}{x}\right ) + \log \left (\frac {3 \, {\left (x + 1\right )}}{x}\right )^{2} + 2 \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (17) = 34\).
Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.04 \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=x^{2} - 2 x \log {\left (\frac {3 x + 3}{x} \right )} + x \left (- 2 e^{3} + 2 + 2 e^{4}\right ) + \left (- 2 e^{3} + 2 + 2 e^{4}\right ) \log {\left (x \right )} + \log {\left (\frac {3 x + 3}{x} \right )}^{2} + \left (- 2 e^{4} - 2 + 2 e^{3}\right ) \log {\left (x + 1 \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (21) = 42\).
Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 5.74 \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=x^{2} + 2 \, x {\left (e^{4} - e^{3} - \log \left (3\right ) + 1\right )} - 2 \, {\left (\log \left (x + 1\right ) - \log \left (x\right )\right )} e^{4} + 2 \, {\left (\log \left (x + 1\right ) - \log \left (x\right )\right )} e^{3} - 2 \, {\left (x + e^{4} - e^{3} + 2\right )} \log \left (x + 1\right ) + 2 \, e^{4} \log \left (x + 1\right ) - 2 \, e^{3} \log \left (x + 1\right ) - \log \left (x + 1\right )^{2} + 2 \, x \log \left (x\right ) + 2 \, \log \left (x + 1\right ) \log \left (x\right ) - \log \left (x\right )^{2} + 2 \, {\left (\log \left (x + 1\right ) - \log \left (x\right )\right )} \log \left (\frac {3}{x} + 3\right ) + 2 \, \log \left (x + 1\right ) + 2 \, \log \left (x\right ) \]
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\[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=\int { \frac {2 \, {\left (x^{3} + 2 \, x^{2} + {\left (x^{2} + x + 1\right )} e^{4} - {\left (x^{2} + x + 1\right )} e^{3} - {\left (x^{2} + x + 1\right )} \log \left (\frac {3 \, {\left (x + 1\right )}}{x}\right ) + 2 \, x + 1\right )}}{x^{2} + x} \,d x } \]
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Time = 13.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=\left (x-\ln \left (\frac {3\,\left (x+1\right )}{x}\right )\right )\,\left (x-2\,{\mathrm {e}}^3+2\,{\mathrm {e}}^4-\ln \left (\frac {3\,\left (x+1\right )}{x}\right )+2\right ) \]
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