Optimal. Leaf size=28 \[ -3+\frac {1}{2} \log \left (\frac {4}{-x+\frac {x}{\left (e^4+(5+x)^4\right )^2}}\right ) \]
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Rubi [B] time = 1.55, antiderivative size = 81, normalized size of antiderivative = 2.89, number of steps used = 6, number of rules used = 3, integrand size = 282, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {6, 2074, 1587} \begin {gather*} -\frac {1}{2} \log \left (x^4+20 x^3+150 x^2+500 x+e^4+624\right )+\log \left (x^4+20 x^3+150 x^2+500 x+e^4+625\right )-\frac {1}{2} \log \left (x^4+20 x^3+150 x^2+500 x+e^4+626\right )-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 1587
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-244140000-e^{12}-585938000 x-644531700 x^2-429687600 x^3-193359382 x^4-61875000 x^5-14437500 x^6-2475000 x^7-309375 x^8-27500 x^9-1650 x^{10}-60 x^{11}-x^{12}+e^8 \left (-1875-1500 x-450 x^2-60 x^3-3 x^4\right )+e^4 \left (-1171874-1875000 x-1312500 x^2-525000 x^3-131250 x^4-21000 x^5-2100 x^6-120 x^7-3 x^8\right )}{\left (488280000+2 e^{12}\right ) x+1171874000 x^2+1289062200 x^3+859374960 x^4+386718748 x^5+123750000 x^6+28875000 x^7+4950000 x^8+618750 x^9+55000 x^{10}+3300 x^{11}+120 x^{12}+2 x^{13}+e^8 \left (3750 x+3000 x^2+900 x^3+120 x^4+6 x^5\right )+e^4 \left (2343748 x+3750000 x^2+2625000 x^3+1050000 x^4+262500 x^5+42000 x^6+4200 x^7+240 x^8+6 x^9\right )} \, dx\\ &=\int \left (-\frac {1}{2 x}-\frac {2 (5+x)^3}{624+e^4+500 x+150 x^2+20 x^3+x^4}+\frac {4 (5+x)^3}{625+e^4+500 x+150 x^2+20 x^3+x^4}-\frac {2 (5+x)^3}{626+e^4+500 x+150 x^2+20 x^3+x^4}\right ) \, dx\\ &=-\frac {\log (x)}{2}-2 \int \frac {(5+x)^3}{624+e^4+500 x+150 x^2+20 x^3+x^4} \, dx-2 \int \frac {(5+x)^3}{626+e^4+500 x+150 x^2+20 x^3+x^4} \, dx+4 \int \frac {(5+x)^3}{625+e^4+500 x+150 x^2+20 x^3+x^4} \, dx\\ &=-\frac {\log (x)}{2}-\frac {1}{2} \log \left (624+e^4+500 x+150 x^2+20 x^3+x^4\right )+\log \left (625+e^4+500 x+150 x^2+20 x^3+x^4\right )-\frac {1}{2} \log \left (626+e^4+500 x+150 x^2+20 x^3+x^4\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.27, size = 112, normalized size = 4.00 \begin {gather*} \frac {1}{2} \left (-\log (x)+2 \log \left (625+e^4+500 x+150 x^2+20 x^3+x^4\right )-\log \left (390624+1250 e^4+e^8+625000 x+1000 e^4 x+437500 x^2+300 e^4 x^2+175000 x^3+40 e^4 x^3+43750 x^4+2 e^4 x^4+7000 x^5+700 x^6+40 x^7+x^8\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 97, normalized size = 3.46 \begin {gather*} -\frac {1}{2} \, \log \left (x^{9} + 40 \, x^{8} + 700 \, x^{7} + 7000 \, x^{6} + 43750 \, x^{5} + 175000 \, x^{4} + 437500 \, x^{3} + 625000 \, x^{2} + x e^{8} + 2 \, {\left (x^{5} + 20 \, x^{4} + 150 \, x^{3} + 500 \, x^{2} + 625 \, x\right )} e^{4} + 390624 \, x\right ) + \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 625\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.34, size = 73, normalized size = 2.61
method | result | size |
norman | \(-\frac {\ln \relax (x )}{2}-\frac {\ln \left (x^{4}+20 x^{3}+150 x^{2}+{\mathrm e}^{4}+500 x +624\right )}{2}-\frac {\ln \left (x^{4}+20 x^{3}+150 x^{2}+{\mathrm e}^{4}+500 x +626\right )}{2}+\ln \left (x^{4}+20 x^{3}+150 x^{2}+{\mathrm e}^{4}+500 x +625\right )\) | \(73\) |
risch | \(\ln \left (-x^{4}-20 x^{3}-150 x^{2}-{\mathrm e}^{4}-500 x -625\right )-\frac {\ln \left (x^{9}+40 x^{8}+700 x^{7}+7000 x^{6}+\left (2 \,{\mathrm e}^{4}+43750\right ) x^{5}+\left (40 \,{\mathrm e}^{4}+175000\right ) x^{4}+\left (300 \,{\mathrm e}^{4}+437500\right ) x^{3}+\left (1000 \,{\mathrm e}^{4}+625000\right ) x^{2}+\left ({\mathrm e}^{8}+1250 \,{\mathrm e}^{4}+390624\right ) x \right )}{2}\) | \(99\) |
default | \(-\frac {\ln \relax (x )}{2}+\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{12}+60 \textit {\_Z}^{11}+1650 \textit {\_Z}^{10}+27500 \textit {\_Z}^{9}+\left (3 \,{\mathrm e}^{4}+309375\right ) \textit {\_Z}^{8}+\left (120 \,{\mathrm e}^{4}+2475000\right ) \textit {\_Z}^{7}+\left (2100 \,{\mathrm e}^{4}+14437500\right ) \textit {\_Z}^{6}+\left (21000 \,{\mathrm e}^{4}+61875000\right ) \textit {\_Z}^{5}+\left (3 \,{\mathrm e}^{8}+131250 \,{\mathrm e}^{4}+193359374\right ) \textit {\_Z}^{4}+\left (60 \,{\mathrm e}^{8}+525000 \,{\mathrm e}^{4}+429687480\right ) \textit {\_Z}^{3}+\left (450 \,{\mathrm e}^{8}+1312500 \,{\mathrm e}^{4}+644531100\right ) \textit {\_Z}^{2}+\left (1500 \,{\mathrm e}^{8}+1875000 \,{\mathrm e}^{4}+585937000\right ) \textit {\_Z} +{\mathrm e}^{12}+1875 \,{\mathrm e}^{8}+1171874 \,{\mathrm e}^{4}+244140000\right )}{\sum }\frac {\left (-\textit {\_R}^{3}-15 \textit {\_R}^{2}-75 \textit {\_R} -125\right ) \ln \left (x -\textit {\_R} \right )}{146484250+375 \,{\mathrm e}^{8}+225 \textit {\_R} \,{\mathrm e}^{8}+468750 \,{\mathrm e}^{4}+165 \textit {\_R}^{10}+3 \textit {\_R}^{11}+4331250 \textit {\_R}^{6}+618750 \textit {\_R}^{7}+4125 \textit {\_R}^{9}+61875 \textit {\_R}^{8}+21656250 \textit {\_R}^{5}+77343750 \textit {\_R}^{4}+193359374 \textit {\_R}^{3}+322265610 \textit {\_R}^{2}+322265550 \textit {\_R} +210 \textit {\_R}^{6} {\mathrm e}^{4}+6 \textit {\_R}^{7} {\mathrm e}^{4}+393750 \textit {\_R}^{2} {\mathrm e}^{4}+656250 \textit {\_R} \,{\mathrm e}^{4}+3 \textit {\_R}^{3} {\mathrm e}^{8}+45 \textit {\_R}^{2} {\mathrm e}^{8}+131250 \textit {\_R}^{3} {\mathrm e}^{4}+26250 \textit {\_R}^{4} {\mathrm e}^{4}+3150 \textit {\_R}^{5} {\mathrm e}^{4}}\right )\) | \(287\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 72, normalized size = 2.57 \begin {gather*} -\frac {1}{2} \, \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 626\right ) + \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 625\right ) - \frac {1}{2} \, \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 624\right ) - \frac {1}{2} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.49, size = 97, normalized size = 3.46 \begin {gather*} \ln \left (x^4+20\,x^3+150\,x^2+500\,x+{\mathrm {e}}^4+625\right )-\frac {\ln \left (x\,\left (625000\,x+1250\,{\mathrm {e}}^4+{\mathrm {e}}^8+1000\,x\,{\mathrm {e}}^4+300\,x^2\,{\mathrm {e}}^4+40\,x^3\,{\mathrm {e}}^4+2\,x^4\,{\mathrm {e}}^4+437500\,x^2+175000\,x^3+43750\,x^4+7000\,x^5+700\,x^6+40\,x^7+x^8+390624\right )\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 41.52, size = 97, normalized size = 3.46 \begin {gather*} \log {\left (x^{4} + 20 x^{3} + 150 x^{2} + 500 x + e^{4} + 625 \right )} - \frac {\log {\left (x^{9} + 40 x^{8} + 700 x^{7} + 7000 x^{6} + x^{5} \left (2 e^{4} + 43750\right ) + x^{4} \left (40 e^{4} + 175000\right ) + x^{3} \left (300 e^{4} + 437500\right ) + x^{2} \left (1000 e^{4} + 625000\right ) + x \left (e^{8} + 1250 e^{4} + 390624\right ) \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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