Integrand size = 135, antiderivative size = 24 \[ \int \frac {-12-403 x-2438 x^2-535 x^3+2360 x^4-672 x^5-128 x^6+48 x^7+e^2 \left (4 x^4+x^5\right )+e \left (-8 x^2-2 x^3-158 x^4+24 x^5+16 x^6\right )+\left (-16 x-204 x^2-48 x^3+32 x^4+8 x^5\right ) \log (4+x)+\left (-4 x^2-x^3\right ) \log ^2(4+x)}{4 x^4+x^5} \, dx=\frac {\left ((5-2 x)^2+\frac {1}{x}+e x+\log (4+x)\right )^2}{x} \]
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Leaf count is larger than twice the leaf count of optimal. \(114\) vs. \(2(24)=48\).
Time = 0.61 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.75, number of steps used = 38, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.126, Rules used = {1607, 6874, 46, 36, 29, 31, 45, 1634, 2465, 2436, 2332, 2442, 2439, 2438, 2437, 2338, 2444} \[ \int \frac {-12-403 x-2438 x^2-535 x^3+2360 x^4-672 x^5-128 x^6+48 x^7+e^2 \left (4 x^4+x^5\right )+e \left (-8 x^2-2 x^3-158 x^4+24 x^5+16 x^6\right )+\left (-16 x-204 x^2-48 x^3+32 x^4+8 x^5\right ) \log (4+x)+\left (-4 x^2-x^3\right ) \log ^2(4+x)}{4 x^4+x^5} \, dx=16 x^3+\frac {1}{x^3}+8 e x^2-160 x^2+\frac {50}{x^2}+\frac {2 \log (x+4)}{x^2}+e^2 x-40 e x+600 x+\frac {2 e}{x}+\frac {585}{x}-\frac {1}{4} \log ^2(x+4)+\frac {(x+4) \log ^2(x+4)}{4 x}+8 (x+4) \log (x+4)+2 e \log (x+4)-72 \log (x+4)+\frac {50 \log (x+4)}{x} \]
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Rule 29
Rule 31
Rule 36
Rule 45
Rule 46
Rule 1607
Rule 1634
Rule 2332
Rule 2338
Rule 2436
Rule 2437
Rule 2438
Rule 2439
Rule 2442
Rule 2444
Rule 2465
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-12-403 x-2438 x^2-535 x^3+2360 x^4-672 x^5-128 x^6+48 x^7+e^2 \left (4 x^4+x^5\right )+e \left (-8 x^2-2 x^3-158 x^4+24 x^5+16 x^6\right )+\left (-16 x-204 x^2-48 x^3+32 x^4+8 x^5\right ) \log (4+x)+\left (-4 x^2-x^3\right ) \log ^2(4+x)}{x^4 (4+x)} \, dx \\ & = \int \left (e^2+\frac {2360}{4+x}-\frac {12}{x^4 (4+x)}-\frac {403}{x^3 (4+x)}-\frac {2438}{x^2 (4+x)}-\frac {535}{x (4+x)}-\frac {672 x}{4+x}-\frac {128 x^2}{4+x}+\frac {48 x^3}{4+x}+\frac {2 e \left (-4-x-79 x^2+12 x^3+8 x^4\right )}{x^2 (4+x)}+\frac {4 \left (-4-51 x-12 x^2+8 x^3+2 x^4\right ) \log (4+x)}{x^3 (4+x)}-\frac {\log ^2(4+x)}{x^2}\right ) \, dx \\ & = e^2 x+2360 \log (4+x)+4 \int \frac {\left (-4-51 x-12 x^2+8 x^3+2 x^4\right ) \log (4+x)}{x^3 (4+x)} \, dx-12 \int \frac {1}{x^4 (4+x)} \, dx+48 \int \frac {x^3}{4+x} \, dx-128 \int \frac {x^2}{4+x} \, dx-403 \int \frac {1}{x^3 (4+x)} \, dx-535 \int \frac {1}{x (4+x)} \, dx-672 \int \frac {x}{4+x} \, dx-2438 \int \frac {1}{x^2 (4+x)} \, dx+(2 e) \int \frac {-4-x-79 x^2+12 x^3+8 x^4}{x^2 (4+x)} \, dx-\int \frac {\log ^2(4+x)}{x^2} \, dx \\ & = e^2 x+2360 \log (4+x)+\frac {(4+x) \log ^2(4+x)}{4 x}-\frac {1}{2} \int \frac {\log (4+x)}{x} \, dx+4 \int \left (2 \log (4+x)-\frac {\log (4+x)}{x^3}-\frac {25 \log (4+x)}{2 x^2}+\frac {\log (4+x)}{8 x}-\frac {\log (4+x)}{8 (4+x)}\right ) \, dx-12 \int \left (\frac {1}{4 x^4}-\frac {1}{16 x^3}+\frac {1}{64 x^2}-\frac {1}{256 x}+\frac {1}{256 (4+x)}\right ) \, dx+48 \int \left (16-4 x+x^2-\frac {64}{4+x}\right ) \, dx-128 \int \left (-4+x+\frac {16}{4+x}\right ) \, dx-\frac {535}{4} \int \frac {1}{x} \, dx+\frac {535}{4} \int \frac {1}{4+x} \, dx-403 \int \left (\frac {1}{4 x^3}-\frac {1}{16 x^2}+\frac {1}{64 x}-\frac {1}{64 (4+x)}\right ) \, dx-672 \int \left (1-\frac {4}{4+x}\right ) \, dx-2438 \int \left (\frac {1}{4 x^2}-\frac {1}{16 x}+\frac {1}{16 (4+x)}\right ) \, dx+(2 e) \int \left (-20-\frac {1}{x^2}+8 x+\frac {1}{4+x}\right ) \, dx \\ & = \frac {1}{x^3}+\frac {50}{x^2}+\frac {1169}{2 x}+\frac {2 e}{x}+608 x-40 e x+e^2 x-160 x^2+8 e x^2+16 x^3+\frac {99 \log (x)}{8}-\frac {1}{2} \log (4) \log (x)-\frac {675}{8} \log (4+x)+2 e \log (4+x)+\frac {(4+x) \log ^2(4+x)}{4 x}-\frac {1}{2} \int \frac {\log \left (1+\frac {x}{4}\right )}{x} \, dx+\frac {1}{2} \int \frac {\log (4+x)}{x} \, dx-\frac {1}{2} \int \frac {\log (4+x)}{4+x} \, dx-4 \int \frac {\log (4+x)}{x^3} \, dx+8 \int \log (4+x) \, dx-50 \int \frac {\log (4+x)}{x^2} \, dx \\ & = \frac {1}{x^3}+\frac {50}{x^2}+\frac {1169}{2 x}+\frac {2 e}{x}+608 x-40 e x+e^2 x-160 x^2+8 e x^2+16 x^3+\frac {99 \log (x)}{8}-\frac {675}{8} \log (4+x)+2 e \log (4+x)+\frac {2 \log (4+x)}{x^2}+\frac {50 \log (4+x)}{x}+\frac {(4+x) \log ^2(4+x)}{4 x}+\frac {\operatorname {PolyLog}\left (2,-\frac {x}{4}\right )}{2}+\frac {1}{2} \int \frac {\log \left (1+\frac {x}{4}\right )}{x} \, dx-\frac {1}{2} \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,4+x\right )-2 \int \frac {1}{x^2 (4+x)} \, dx+8 \text {Subst}(\int \log (x) \, dx,x,4+x)-50 \int \frac {1}{x (4+x)} \, dx \\ & = \frac {1}{x^3}+\frac {50}{x^2}+\frac {1169}{2 x}+\frac {2 e}{x}+600 x-40 e x+e^2 x-160 x^2+8 e x^2+16 x^3+\frac {99 \log (x)}{8}-\frac {675}{8} \log (4+x)+2 e \log (4+x)+\frac {2 \log (4+x)}{x^2}+\frac {50 \log (4+x)}{x}+8 (4+x) \log (4+x)-\frac {1}{4} \log ^2(4+x)+\frac {(4+x) \log ^2(4+x)}{4 x}-2 \int \left (\frac {1}{4 x^2}-\frac {1}{16 x}+\frac {1}{16 (4+x)}\right ) \, dx-\frac {25}{2} \int \frac {1}{x} \, dx+\frac {25}{2} \int \frac {1}{4+x} \, dx \\ & = \frac {1}{x^3}+\frac {50}{x^2}+\frac {585}{x}+\frac {2 e}{x}+600 x-40 e x+e^2 x-160 x^2+8 e x^2+16 x^3-72 \log (4+x)+2 e \log (4+x)+\frac {2 \log (4+x)}{x^2}+\frac {50 \log (4+x)}{x}+8 (4+x) \log (4+x)-\frac {1}{4} \log ^2(4+x)+\frac {(4+x) \log ^2(4+x)}{4 x} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(79\) vs. \(2(24)=48\).
Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.29 \[ \int \frac {-12-403 x-2438 x^2-535 x^3+2360 x^4-672 x^5-128 x^6+48 x^7+e^2 \left (4 x^4+x^5\right )+e \left (-8 x^2-2 x^3-158 x^4+24 x^5+16 x^6\right )+\left (-16 x-204 x^2-48 x^3+32 x^4+8 x^5\right ) \log (4+x)+\left (-4 x^2-x^3\right ) \log ^2(4+x)}{4 x^4+x^5} \, dx=\frac {1}{x^3}+\frac {50}{x^2}+\frac {585+2 e}{x}+\left (600-40 e+e^2\right ) x+8 (-20+e) x^2+16 x^3+2 (-20+e) \log (4+x)+\frac {2 \left (1+25 x+4 x^3\right ) \log (4+x)}{x^2}+\frac {\log ^2(4+x)}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(97\) vs. \(2(25)=50\).
Time = 0.34 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.08
method | result | size |
norman | \(\frac {1+x^{2} \ln \left (4+x \right )^{2}+\left (-160+8 \,{\mathrm e}\right ) x^{5}+\left (585+2 \,{\mathrm e}\right ) x^{2}+\left (600-40 \,{\mathrm e}+{\mathrm e}^{2}\right ) x^{4}+\left (-40+2 \,{\mathrm e}\right ) x^{3} \ln \left (4+x \right )+50 x +16 x^{6}+50 x^{2} \ln \left (4+x \right )+2 \ln \left (4+x \right ) x +8 \ln \left (4+x \right ) x^{4}}{x^{3}}\) | \(98\) |
risch | \(\frac {\ln \left (4+x \right )^{2}}{x}+\frac {2 \left (4 x^{3}+25 x +1\right ) \ln \left (4+x \right )}{x^{2}}+\frac {x^{4} {\mathrm e}^{2}+8 x^{5} {\mathrm e}+16 x^{6}+2 \ln \left (-4-x \right ) x^{3} {\mathrm e}-40 x^{4} {\mathrm e}-160 x^{5}-40 \ln \left (-4-x \right ) x^{3}+600 x^{4}+2 x^{2} {\mathrm e}+585 x^{2}+50 x +1}{x^{3}}\) | \(111\) |
parallelrisch | \(\frac {x^{4} {\mathrm e}^{2}+8 x^{5} {\mathrm e}+16 x^{6}-8 x^{3} {\mathrm e}^{2}-40 x^{4} {\mathrm e}+2 \ln \left (4+x \right ) x^{3} {\mathrm e}-160 x^{5}+8 \ln \left (4+x \right ) x^{4}+192 x^{3} {\mathrm e}+600 x^{4}-40 x^{3} \ln \left (4+x \right )+1+x^{2} \ln \left (4+x \right )^{2}+2 x^{2} {\mathrm e}-2240 x^{3}+50 x^{2} \ln \left (4+x \right )+585 x^{2}+2 \ln \left (4+x \right ) x +50 x}{x^{3}}\) | \(135\) |
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (25) = 50\).
Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.83 \[ \int \frac {-12-403 x-2438 x^2-535 x^3+2360 x^4-672 x^5-128 x^6+48 x^7+e^2 \left (4 x^4+x^5\right )+e \left (-8 x^2-2 x^3-158 x^4+24 x^5+16 x^6\right )+\left (-16 x-204 x^2-48 x^3+32 x^4+8 x^5\right ) \log (4+x)+\left (-4 x^2-x^3\right ) \log ^2(4+x)}{4 x^4+x^5} \, dx=\frac {16 \, x^{6} - 160 \, x^{5} + x^{4} e^{2} + 600 \, x^{4} + x^{2} \log \left (x + 4\right )^{2} + 585 \, x^{2} + 2 \, {\left (4 \, x^{5} - 20 \, x^{4} + x^{2}\right )} e + 2 \, {\left (4 \, x^{4} + x^{3} e - 20 \, x^{3} + 25 \, x^{2} + x\right )} \log \left (x + 4\right ) + 50 \, x + 1}{x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (22) = 44\).
Time = 0.53 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.46 \[ \int \frac {-12-403 x-2438 x^2-535 x^3+2360 x^4-672 x^5-128 x^6+48 x^7+e^2 \left (4 x^4+x^5\right )+e \left (-8 x^2-2 x^3-158 x^4+24 x^5+16 x^6\right )+\left (-16 x-204 x^2-48 x^3+32 x^4+8 x^5\right ) \log (4+x)+\left (-4 x^2-x^3\right ) \log ^2(4+x)}{4 x^4+x^5} \, dx=16 x^{3} + x^{2} \left (-160 + 8 e\right ) + x \left (- 40 e + e^{2} + 600\right ) + 2 \left (-20 + e\right ) \log {\left (x + 4 \right )} + \frac {\log {\left (x + 4 \right )}^{2}}{x} + \frac {\left (8 x^{3} + 50 x + 2\right ) \log {\left (x + 4 \right )}}{x^{2}} + \frac {x^{2} \cdot \left (2 e + 585\right ) + 50 x + 1}{x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (25) = 50\).
Time = 0.24 (sec) , antiderivative size = 179, normalized size of antiderivative = 7.46 \[ \int \frac {-12-403 x-2438 x^2-535 x^3+2360 x^4-672 x^5-128 x^6+48 x^7+e^2 \left (4 x^4+x^5\right )+e \left (-8 x^2-2 x^3-158 x^4+24 x^5+16 x^6\right )+\left (-16 x-204 x^2-48 x^3+32 x^4+8 x^5\right ) \log (4+x)+\left (-4 x^2-x^3\right ) \log ^2(4+x)}{4 x^4+x^5} \, dx=16 \, x^{3} - 160 \, x^{2} + {\left (x - 4 \, \log \left (x + 4\right )\right )} e^{2} + 8 \, {\left (x^{2} - 8 \, x + 32 \, \log \left (x + 4\right )\right )} e + 24 \, {\left (x - 4 \, \log \left (x + 4\right )\right )} e + \frac {1}{2} \, {\left (\frac {4}{x} - \log \left (x + 4\right ) + \log \left (x\right )\right )} e + \frac {1}{2} \, {\left (\log \left (x + 4\right ) - \log \left (x\right )\right )} e + 4 \, e^{2} \log \left (x + 4\right ) - 158 \, e \log \left (x + 4\right ) + 608 \, x - \frac {64 \, x^{3} - 8 \, x \log \left (x + 4\right )^{2} - {\left (64 \, x^{3} + 355 \, x^{2} + 400 \, x + 16\right )} \log \left (x + 4\right ) - 4 \, x}{8 \, x^{2}} - \frac {403 \, {\left (x - 2\right )}}{16 \, x^{2}} + \frac {1219}{2 \, x} + \frac {3 \, x^{2} - 6 \, x + 16}{16 \, x^{3}} - \frac {675}{8} \, \log \left (x + 4\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (25) = 50\).
Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.62 \[ \int \frac {-12-403 x-2438 x^2-535 x^3+2360 x^4-672 x^5-128 x^6+48 x^7+e^2 \left (4 x^4+x^5\right )+e \left (-8 x^2-2 x^3-158 x^4+24 x^5+16 x^6\right )+\left (-16 x-204 x^2-48 x^3+32 x^4+8 x^5\right ) \log (4+x)+\left (-4 x^2-x^3\right ) \log ^2(4+x)}{4 x^4+x^5} \, dx=\frac {16 \, x^{6} + 8 \, x^{5} e - 160 \, x^{5} + x^{4} e^{2} - 40 \, x^{4} e + 8 \, x^{4} \log \left (x + 4\right ) + 2 \, x^{3} e \log \left (x + 4\right ) + 600 \, x^{4} - 40 \, x^{3} \log \left (x + 4\right ) + x^{2} \log \left (x + 4\right )^{2} + 2 \, x^{2} e + 50 \, x^{2} \log \left (x + 4\right ) + 585 \, x^{2} + 2 \, x \log \left (x + 4\right ) + 50 \, x + 1}{x^{3}} \]
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Time = 14.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.67 \[ \int \frac {-12-403 x-2438 x^2-535 x^3+2360 x^4-672 x^5-128 x^6+48 x^7+e^2 \left (4 x^4+x^5\right )+e \left (-8 x^2-2 x^3-158 x^4+24 x^5+16 x^6\right )+\left (-16 x-204 x^2-48 x^3+32 x^4+8 x^5\right ) \log (4+x)+\left (-4 x^2-x^3\right ) \log ^2(4+x)}{4 x^4+x^5} \, dx=600\,x-40\,\ln \left (x+4\right )+2\,\ln \left (x+4\right )\,\mathrm {e}+8\,x\,\ln \left (x+4\right )+\frac {2\,\ln \left (x+4\right )+50}{x^2}-40\,x\,\mathrm {e}+x\,{\mathrm {e}}^2+8\,x^2\,\mathrm {e}+\frac {{\ln \left (x+4\right )}^2+50\,\ln \left (x+4\right )+2\,\mathrm {e}+585}{x}-160\,x^2+\frac {1}{x^3}+16\,x^3 \]
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