Integrand size = 49, antiderivative size = 17 \[ \int \frac {352-184 x^2+4 x^4+e \left (-44 x+x^3\right )+\left (32-4 e x-16 x^2\right ) \log (e+4 x)}{4 e+16 x} \, dx=\left (11-\frac {x^2}{4}+\log (e+4 x)\right )^2 \]
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Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {6873, 6818} \[ \int \frac {352-184 x^2+4 x^4+e \left (-44 x+x^3\right )+\left (32-4 e x-16 x^2\right ) \log (e+4 x)}{4 e+16 x} \, dx=\frac {1}{16} \left (-x^2+4 \log (4 x+e)+44\right )^2 \]
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Rule 6818
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (8-e x-4 x^2\right ) \left (44-x^2+4 \log (e+4 x)\right )}{4 e+16 x} \, dx \\ & = \frac {1}{16} \left (44-x^2+4 \log (e+4 x)\right )^2 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {352-184 x^2+4 x^4+e \left (-44 x+x^3\right )+\left (32-4 e x-16 x^2\right ) \log (e+4 x)}{4 e+16 x} \, dx=\frac {1}{16} \left (-44+x^2-4 \log (e+4 x)\right )^2 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs. \(2(16)=32\).
Time = 0.62 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.47
method | result | size |
norman | \(\ln \left ({\mathrm e}+4 x \right )^{2}+22 \ln \left ({\mathrm e}+4 x \right )-\frac {11 x^{2}}{2}+\frac {x^{4}}{16}-\frac {\ln \left ({\mathrm e}+4 x \right ) x^{2}}{2}\) | \(42\) |
risch | \(\ln \left ({\mathrm e}+4 x \right )^{2}-\frac {\ln \left ({\mathrm e}+4 x \right ) x^{2}}{2}+\frac {x^{4}}{16}-\frac {11 x^{2}}{2}+121+22 \ln \left ({\mathrm e}+4 x \right )\) | \(43\) |
parallelrisch | \(\frac {x^{4}}{16}-\frac {\ln \left ({\mathrm e}+4 x \right ) x^{2}}{2}+\frac {11 \,{\mathrm e}^{2}}{32}-\frac {11 x^{2}}{2}+\ln \left ({\mathrm e}+4 x \right )^{2}+22 \ln \left ({\mathrm e}+4 x \right )\) | \(48\) |
parts | \(\frac {x^{4}}{16}-\frac {23 x^{2}}{4}+\frac {x \,{\mathrm e}}{8}+\frac {\left (-\frac {{\mathrm e}^{2}}{8}+88\right ) \ln \left ({\mathrm e}+4 x \right )}{4}+\frac {{\mathrm e} \left (\left ({\mathrm e}+4 x \right ) \ln \left ({\mathrm e}+4 x \right )-{\mathrm e}-4 x \right )}{16}-\frac {\ln \left ({\mathrm e}+4 x \right ) \left ({\mathrm e}+4 x \right )^{2}}{32}+\frac {\left ({\mathrm e}+4 x \right )^{2}}{64}+\ln \left ({\mathrm e}+4 x \right )^{2}\) | \(94\) |
derivativedivides | \(-\frac {\left ({\mathrm e}+4 x \right ) {\mathrm e}^{3}}{1024}+\frac {3 \,{\mathrm e}^{2} \left ({\mathrm e}+4 x \right )^{2}}{2048}-\frac {{\mathrm e} \left ({\mathrm e}+4 x \right )^{3}}{1024}+\frac {\left ({\mathrm e}+4 x \right )^{4}}{4096}+\frac {{\mathrm e} \left (\left ({\mathrm e}+4 x \right ) \ln \left ({\mathrm e}+4 x \right )-{\mathrm e}-4 x \right )}{16}-\frac {\ln \left ({\mathrm e}+4 x \right ) \left ({\mathrm e}+4 x \right )^{2}}{32}-\frac {11 \left ({\mathrm e}+4 x \right )^{2}}{32}-\frac {{\mathrm e}^{2} \ln \left ({\mathrm e}+4 x \right )}{32}+\frac {3 \,{\mathrm e} \left ({\mathrm e}+4 x \right )}{4}+\ln \left ({\mathrm e}+4 x \right )^{2}+22 \ln \left ({\mathrm e}+4 x \right )\) | \(144\) |
default | \(-\frac {\left ({\mathrm e}+4 x \right ) {\mathrm e}^{3}}{1024}+\frac {3 \,{\mathrm e}^{2} \left ({\mathrm e}+4 x \right )^{2}}{2048}-\frac {{\mathrm e} \left ({\mathrm e}+4 x \right )^{3}}{1024}+\frac {\left ({\mathrm e}+4 x \right )^{4}}{4096}+\frac {{\mathrm e} \left (\left ({\mathrm e}+4 x \right ) \ln \left ({\mathrm e}+4 x \right )-{\mathrm e}-4 x \right )}{16}-\frac {\ln \left ({\mathrm e}+4 x \right ) \left ({\mathrm e}+4 x \right )^{2}}{32}-\frac {11 \left ({\mathrm e}+4 x \right )^{2}}{32}-\frac {{\mathrm e}^{2} \ln \left ({\mathrm e}+4 x \right )}{32}+\frac {3 \,{\mathrm e} \left ({\mathrm e}+4 x \right )}{4}+\ln \left ({\mathrm e}+4 x \right )^{2}+22 \ln \left ({\mathrm e}+4 x \right )\) | \(144\) |
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Time = 0.49 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00 \[ \int \frac {352-184 x^2+4 x^4+e \left (-44 x+x^3\right )+\left (32-4 e x-16 x^2\right ) \log (e+4 x)}{4 e+16 x} \, dx=\frac {1}{16} \, x^{4} - \frac {11}{2} \, x^{2} - \frac {1}{2} \, {\left (x^{2} - 44\right )} \log \left (4 \, x + e\right ) + \log \left (4 \, x + e\right )^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (15) = 30\).
Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.59 \[ \int \frac {352-184 x^2+4 x^4+e \left (-44 x+x^3\right )+\left (32-4 e x-16 x^2\right ) \log (e+4 x)}{4 e+16 x} \, dx=\frac {x^{4}}{16} - \frac {x^{2} \log {\left (4 x + e \right )}}{2} - \frac {11 x^{2}}{2} + \log {\left (4 x + e \right )}^{2} + 22 \log {\left (4 x + e \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (18) = 36\).
Time = 0.20 (sec) , antiderivative size = 225, normalized size of antiderivative = 13.24 \[ \int \frac {352-184 x^2+4 x^4+e \left (-44 x+x^3\right )+\left (32-4 e x-16 x^2\right ) \log (e+4 x)}{4 e+16 x} \, dx=\frac {1}{16} \, x^{4} - \frac {1}{48} \, x^{3} e + \frac {1}{128} \, x^{2} e^{2} + \frac {1}{16} \, {\left (e \log \left (4 \, x + e\right ) - 4 \, x\right )} e \log \left (4 \, x + e\right ) + \frac {1}{32} \, e^{2} \log \left (4 \, x + e\right )^{2} - \frac {11}{2} \, x^{2} - \frac {1}{256} \, x e^{3} + \frac {1}{3072} \, {\left (64 \, x^{3} - 24 \, x^{2} e + 12 \, x e^{2} - 3 \, e^{3} \log \left (4 \, x + e\right )\right )} e - \frac {1}{32} \, {\left (e \log \left (4 \, x + e\right )^{2} + 2 \, e \log \left (4 \, x + e\right ) - 8 \, x\right )} e + \frac {11}{16} \, {\left (e \log \left (4 \, x + e\right ) - 4 \, x\right )} e + \frac {5}{2} \, x e - \frac {1}{16} \, {\left (8 \, x^{2} - 4 \, x e + e^{2} \log \left (4 \, x + e\right )\right )} \log \left (4 \, x + e\right ) + \frac {1}{1024} \, e^{4} \log \left (4 \, x + e\right ) - \frac {5}{8} \, e^{2} \log \left (4 \, x + e\right ) + \log \left (4 \, x + e\right )^{2} + 22 \, \log \left (4 \, x + e\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (18) = 36\).
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.41 \[ \int \frac {352-184 x^2+4 x^4+e \left (-44 x+x^3\right )+\left (32-4 e x-16 x^2\right ) \log (e+4 x)}{4 e+16 x} \, dx=\frac {1}{16} \, x^{4} - \frac {1}{2} \, x^{2} \log \left (4 \, x + e\right ) - \frac {11}{2} \, x^{2} + \log \left (4 \, x + e\right )^{2} + 22 \, \log \left (4 \, x + e\right ) \]
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Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.41 \[ \int \frac {352-184 x^2+4 x^4+e \left (-44 x+x^3\right )+\left (32-4 e x-16 x^2\right ) \log (e+4 x)}{4 e+16 x} \, dx=\frac {x^4}{16}-\frac {x^2\,\ln \left (4\,x+\mathrm {e}\right )}{2}-\frac {11\,x^2}{2}+{\ln \left (4\,x+\mathrm {e}\right )}^2+22\,\ln \left (4\,x+\mathrm {e}\right ) \]
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