Integrand size = 344, antiderivative size = 23 \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\frac {x^4}{\left (4 \left (x-x^2\right )+\log \left (1+e^4+x\right )\right )^4} \]
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\[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-4 x^4+\left (16+16 e^4\right ) x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx \\ & = \int \frac {4 x^3 \left (-x \left (-1+4 \left (1+e^4\right ) x+4 x^2\right )-\left (1+e^4+x\right ) \log \left (1+e^4+x\right )\right )}{\left (1+e^4+x\right ) \left (4 (-1+x) x-\log \left (1+e^4+x\right )\right )^5} \, dx \\ & = 4 \int \frac {x^3 \left (-x \left (-1+4 \left (1+e^4\right ) x+4 x^2\right )-\left (1+e^4+x\right ) \log \left (1+e^4+x\right )\right )}{\left (1+e^4+x\right ) \left (4 (-1+x) x-\log \left (1+e^4+x\right )\right )^5} \, dx \\ & = 4 \int \left (\frac {x^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^4}+\frac {x^4 \left (-5-4 e^4+4 \left (1+2 e^4\right ) x+8 x^2\right )}{\left (1+e^4+x\right ) \left (4 x-4 x^2+\log \left (1+e^4+x\right )\right )^5}\right ) \, dx \\ & = 4 \int \frac {x^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^4} \, dx+4 \int \frac {x^4 \left (-5-4 e^4+4 \left (1+2 e^4\right ) x+8 x^2\right )}{\left (1+e^4+x\right ) \left (4 x-4 x^2+\log \left (1+e^4+x\right )\right )^5} \, dx \\ & = 4 \int \left (-\frac {\left (1+e^4\right )^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}+\frac {\left (1+e^4\right )^2 x}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}-\frac {\left (1+e^4\right ) x^2}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}+\frac {x^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}+\frac {4 x^4}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}-\frac {8 x^5}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}+\frac {\left (1+e^4\right )^4}{\left (1+e^4+x\right ) \left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5}\right ) \, dx+4 \int \frac {x^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^4} \, dx \\ & = 4 \int \frac {x^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx+4 \int \frac {x^3}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^4} \, dx+16 \int \frac {x^4}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx-32 \int \frac {x^5}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx-\left (4 \left (1+e^4\right )\right ) \int \frac {x^2}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx+\left (4 \left (1+e^4\right )^2\right ) \int \frac {x}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx-\left (4 \left (1+e^4\right )^3\right ) \int \frac {1}{\left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx+\left (4 \left (1+e^4\right )^4\right ) \int \frac {1}{\left (1+e^4+x\right ) \left (-4 x+4 x^2-\log \left (1+e^4+x\right )\right )^5} \, dx \\ \end{align*}
Time = 5.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\frac {x^4}{\left (-4 (-1+x) x+\log \left (1+e^4+x\right )\right )^4} \]
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Time = 1.96 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {x^{4}}{\left (4 x^{2}-4 x -\ln \left ({\mathrm e}^{4}+x +1\right )\right )^{4}}\) | \(24\) |
parallelrisch | \(\frac {x^{4}}{256 x^{8}-1024 x^{7}-256 \ln \left ({\mathrm e}^{4}+x +1\right ) x^{6}+1536 x^{6}+768 \ln \left ({\mathrm e}^{4}+x +1\right ) x^{5}+96 \ln \left ({\mathrm e}^{4}+x +1\right )^{2} x^{4}-1024 x^{5}-768 \ln \left ({\mathrm e}^{4}+x +1\right ) x^{4}-192 \ln \left ({\mathrm e}^{4}+x +1\right )^{2} x^{3}-16 \ln \left ({\mathrm e}^{4}+x +1\right )^{3} x^{2}+256 x^{4}+256 x^{3} \ln \left ({\mathrm e}^{4}+x +1\right )+96 x^{2} \ln \left ({\mathrm e}^{4}+x +1\right )^{2}+16 x \ln \left ({\mathrm e}^{4}+x +1\right )^{3}+\ln \left ({\mathrm e}^{4}+x +1\right )^{4}}\) | \(148\) |
default | \(\frac {{\mathrm e}^{16}-4 \,{\mathrm e}^{12+\ln \left ({\mathrm e}^{4}+x +1\right )}+6 \,{\mathrm e}^{8+2 \ln \left ({\mathrm e}^{4}+x +1\right )}-4 \,{\mathrm e}^{4+3 \ln \left ({\mathrm e}^{4}+x +1\right )}+\left ({\mathrm e}^{4}+x +1\right )^{4}+4 \,{\mathrm e}^{12}-12 \,{\mathrm e}^{8+\ln \left ({\mathrm e}^{4}+x +1\right )}+12 \,{\mathrm e}^{4+2 \ln \left ({\mathrm e}^{4}+x +1\right )}-4 \left ({\mathrm e}^{4}+x +1\right )^{3}+6 \,{\mathrm e}^{8}-12 \,{\mathrm e}^{4+\ln \left ({\mathrm e}^{4}+x +1\right )}+6 \left ({\mathrm e}^{4}+x +1\right )^{2}-4 x -3}{\left (4 \,{\mathrm e}^{8}-8 \,{\mathrm e}^{4+\ln \left ({\mathrm e}^{4}+x +1\right )}+4 \left ({\mathrm e}^{4}+x +1\right )^{2}-12 x -4-\ln \left ({\mathrm e}^{4}+x +1\right )\right )^{4}}\) | \(153\) |
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Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (23) = 46\).
Time = 0.43 (sec) , antiderivative size = 106, normalized size of antiderivative = 4.61 \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\frac {x^{4}}{256 \, x^{8} - 1024 \, x^{7} + 1536 \, x^{6} - 1024 \, x^{5} + 256 \, x^{4} - 16 \, {\left (x^{2} - x\right )} \log \left (x + e^{4} + 1\right )^{3} + \log \left (x + e^{4} + 1\right )^{4} + 96 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \left (x + e^{4} + 1\right )^{2} - 256 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3}\right )} \log \left (x + e^{4} + 1\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (20) = 40\).
Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 4.78 \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\frac {x^{4}}{256 x^{8} - 1024 x^{7} + 1536 x^{6} - 1024 x^{5} + 256 x^{4} + \left (- 16 x^{2} + 16 x\right ) \log {\left (x + 1 + e^{4} \right )}^{3} + \left (96 x^{4} - 192 x^{3} + 96 x^{2}\right ) \log {\left (x + 1 + e^{4} \right )}^{2} + \left (- 256 x^{6} + 768 x^{5} - 768 x^{4} + 256 x^{3}\right ) \log {\left (x + 1 + e^{4} \right )} + \log {\left (x + 1 + e^{4} \right )}^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (23) = 46\).
Time = 0.45 (sec) , antiderivative size = 106, normalized size of antiderivative = 4.61 \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\frac {x^{4}}{256 \, x^{8} - 1024 \, x^{7} + 1536 \, x^{6} - 1024 \, x^{5} + 256 \, x^{4} - 16 \, {\left (x^{2} - x\right )} \log \left (x + e^{4} + 1\right )^{3} + \log \left (x + e^{4} + 1\right )^{4} + 96 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \left (x + e^{4} + 1\right )^{2} - 256 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3}\right )} \log \left (x + e^{4} + 1\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (23) = 46\).
Time = 1.44 (sec) , antiderivative size = 147, normalized size of antiderivative = 6.39 \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\frac {x^{4}}{256 \, x^{8} - 1024 \, x^{7} - 256 \, x^{6} \log \left (x + e^{4} + 1\right ) + 1536 \, x^{6} + 768 \, x^{5} \log \left (x + e^{4} + 1\right ) + 96 \, x^{4} \log \left (x + e^{4} + 1\right )^{2} - 1024 \, x^{5} - 768 \, x^{4} \log \left (x + e^{4} + 1\right ) - 192 \, x^{3} \log \left (x + e^{4} + 1\right )^{2} - 16 \, x^{2} \log \left (x + e^{4} + 1\right )^{3} + 256 \, x^{4} + 256 \, x^{3} \log \left (x + e^{4} + 1\right ) + 96 \, x^{2} \log \left (x + e^{4} + 1\right )^{2} + 16 \, x \log \left (x + e^{4} + 1\right )^{3} + \log \left (x + e^{4} + 1\right )^{4}} \]
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Timed out. \[ \int \frac {-4 x^4+16 x^5+16 e^4 x^5+16 x^6+\left (4 x^3+4 e^4 x^3+4 x^4\right ) \log \left (1+e^4+x\right )}{1024 x^5-4096 x^6+5120 x^7-5120 x^9+4096 x^{10}-1024 x^{11}+e^4 \left (1024 x^5-5120 x^6+10240 x^7-10240 x^8+5120 x^9-1024 x^{10}\right )+\left (1280 x^4-3840 x^5+2560 x^6+2560 x^7-3840 x^8+1280 x^9+e^4 \left (1280 x^4-5120 x^5+7680 x^6-5120 x^7+1280 x^8\right )\right ) \log \left (1+e^4+x\right )+\left (640 x^3-1280 x^4+1280 x^6-640 x^7+e^4 \left (640 x^3-1920 x^4+1920 x^5-640 x^6\right )\right ) \log ^2\left (1+e^4+x\right )+\left (160 x^2-160 x^3-160 x^4+160 x^5+e^4 \left (160 x^2-320 x^3+160 x^4\right )\right ) \log ^3\left (1+e^4+x\right )+\left (20 x-20 x^3+e^4 \left (20 x-20 x^2\right )\right ) \log ^4\left (1+e^4+x\right )+\left (1+e^4+x\right ) \log ^5\left (1+e^4+x\right )} \, dx=\text {Hanged} \]
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