Optimal. Leaf size=27 \[ \frac {1}{\left (5+\log \left (\frac {5 \left (5+e^x\right )+x-\frac {\log (3)}{5+x}}{x}\right )\right )^4} \]
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Rubi [F] time = 7.53, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {2500+1000 x+100 x^2+e^x \left (500-300 x-180 x^2-20 x^3\right )+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)+\left (1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)\right ) \log \left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (781250 x+343750 x^2+43750 x^3+1250 x^4+e^x \left (156250 x+62500 x^2+6250 x^3\right )+\left (-6250 x-1250 x^2\right ) \log (3)\right ) \log ^2\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (156250 x+68750 x^2+8750 x^3+250 x^4+e^x \left (31250 x+12500 x^2+1250 x^3\right )+\left (-1250 x-250 x^2\right ) \log (3)\right ) \log ^3\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (15625 x+6875 x^2+875 x^3+25 x^4+e^x \left (3125 x+1250 x^2+125 x^3\right )+\left (-125 x-25 x^2\right ) \log (3)\right ) \log ^4\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (625 x+275 x^2+35 x^3+x^4+e^x \left (125 x+50 x^2+5 x^3\right )+\left (-5 x-x^2\right ) \log (3)\right ) \log ^5\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (25 x^2-5 e^x (-1+x) (5+x)^2+625 \left (1-\frac {\log (3)}{125}\right )-x (-250+\log (9))\right )}{x (5+x) \left (30 x+x^2+5 e^x (5+x)+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx\\ &=4 \int \frac {25 x^2-5 e^x (-1+x) (5+x)^2+625 \left (1-\frac {\log (3)}{125}\right )-x (-250+\log (9))}{x (5+x) \left (30 x+x^2+5 e^x (5+x)+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx\\ &=4 \int \left (\frac {1-x}{x \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}+\frac {600+34 x^2+x^3+x (265-\log (3))-\log (729)}{(5+x) \left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}\right ) \, dx\\ &=4 \int \frac {1-x}{x \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx+4 \int \frac {600+34 x^2+x^3+x (265-\log (3))-\log (729)}{(5+x) \left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx\\ &=4 \int \left (-\frac {1}{\left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}+\frac {1}{x \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}\right ) \, dx+4 \int \left (\frac {29 x}{\left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}+\frac {x^2}{\left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}+\frac {120 \left (1-\frac {\log (3)}{120}\right )}{\left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}+\frac {\log (3)}{(-5-x) \left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}\right ) \, dx\\ &=-\left (4 \int \frac {1}{\left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx\right )+4 \int \frac {1}{x \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx+4 \int \frac {x^2}{\left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx+116 \int \frac {x}{\left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx+(4 (120-\log (3))) \int \frac {1}{\left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx+(4 \log (3)) \int \frac {1}{(-5-x) \left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [F] time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2500+1000 x+100 x^2+e^x \left (500-300 x-180 x^2-20 x^3\right )+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)+\left (1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)\right ) \log \left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (781250 x+343750 x^2+43750 x^3+1250 x^4+e^x \left (156250 x+62500 x^2+6250 x^3\right )+\left (-6250 x-1250 x^2\right ) \log (3)\right ) \log ^2\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (156250 x+68750 x^2+8750 x^3+250 x^4+e^x \left (31250 x+12500 x^2+1250 x^3\right )+\left (-1250 x-250 x^2\right ) \log (3)\right ) \log ^3\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (15625 x+6875 x^2+875 x^3+25 x^4+e^x \left (3125 x+1250 x^2+125 x^3\right )+\left (-125 x-25 x^2\right ) \log (3)\right ) \log ^4\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (625 x+275 x^2+35 x^3+x^4+e^x \left (125 x+50 x^2+5 x^3\right )+\left (-5 x-x^2\right ) \log (3)\right ) \log ^5\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.53, size = 136, normalized size = 5.04 \begin {gather*} \frac {1}{\log \left (\frac {x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125}{x^{2} + 5 \, x}\right )^{4} + 20 \, \log \left (\frac {x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125}{x^{2} + 5 \, x}\right )^{3} + 150 \, \log \left (\frac {x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125}{x^{2} + 5 \, x}\right )^{2} + 500 \, \log \left (\frac {x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125}{x^{2} + 5 \, x}\right ) + 625} \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 [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-20 x^{3}-180 x^{2}-300 x +500\right ) {\mathrm e}^{x}+\left (-8 x -20\right ) \ln \relax (3)+100 x^{2}+1000 x +2500}{\left (\left (5 x^{3}+50 x^{2}+125 x \right ) {\mathrm e}^{x}+\left (-x^{2}-5 x \right ) \ln \relax (3)+x^{4}+35 x^{3}+275 x^{2}+625 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \relax (3)+x^{2}+30 x +125}{x^{2}+5 x}\right )^{5}+\left (\left (125 x^{3}+1250 x^{2}+3125 x \right ) {\mathrm e}^{x}+\left (-25 x^{2}-125 x \right ) \ln \relax (3)+25 x^{4}+875 x^{3}+6875 x^{2}+15625 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \relax (3)+x^{2}+30 x +125}{x^{2}+5 x}\right )^{4}+\left (\left (1250 x^{3}+12500 x^{2}+31250 x \right ) {\mathrm e}^{x}+\left (-250 x^{2}-1250 x \right ) \ln \relax (3)+250 x^{4}+8750 x^{3}+68750 x^{2}+156250 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \relax (3)+x^{2}+30 x +125}{x^{2}+5 x}\right )^{3}+\left (\left (6250 x^{3}+62500 x^{2}+156250 x \right ) {\mathrm e}^{x}+\left (-1250 x^{2}-6250 x \right ) \ln \relax (3)+1250 x^{4}+43750 x^{3}+343750 x^{2}+781250 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \relax (3)+x^{2}+30 x +125}{x^{2}+5 x}\right )^{2}+\left (\left (15625 x^{3}+156250 x^{2}+390625 x \right ) {\mathrm e}^{x}+\left (-3125 x^{2}-15625 x \right ) \ln \relax (3)+3125 x^{4}+109375 x^{3}+859375 x^{2}+1953125 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \relax (3)+x^{2}+30 x +125}{x^{2}+5 x}\right )+\left (15625 x^{3}+156250 x^{2}+390625 x \right ) {\mathrm e}^{x}+\left (-3125 x^{2}-15625 x \right ) \ln \relax (3)+3125 x^{4}+109375 x^{3}+859375 x^{2}+1953125 x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 18.39, size = 266, normalized size = 9.85 \begin {gather*} -\frac {1}{4 \, {\left (\log \left (x + 5\right ) + \log \relax (x) - 5\right )} \log \left (x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125\right )^{3} - \log \left (x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125\right )^{4} - 4 \, {\left (\log \relax (x) - 5\right )} \log \left (x + 5\right )^{3} - \log \left (x + 5\right )^{4} - \log \relax (x)^{4} - 6 \, {\left (2 \, {\left (\log \relax (x) - 5\right )} \log \left (x + 5\right ) + \log \left (x + 5\right )^{2} + \log \relax (x)^{2} - 10 \, \log \relax (x) + 25\right )} \log \left (x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125\right )^{2} - 6 \, {\left (\log \relax (x)^{2} - 10 \, \log \relax (x) + 25\right )} \log \left (x + 5\right )^{2} + 20 \, \log \relax (x)^{3} + 4 \, {\left (3 \, {\left (\log \relax (x) - 5\right )} \log \left (x + 5\right )^{2} + \log \left (x + 5\right )^{3} + \log \relax (x)^{3} + 3 \, {\left (\log \relax (x)^{2} - 10 \, \log \relax (x) + 25\right )} \log \left (x + 5\right ) - 15 \, \log \relax (x)^{2} + 75 \, \log \relax (x) - 125\right )} \log \left (x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125\right ) - 4 \, {\left (\log \relax (x)^{3} - 15 \, \log \relax (x)^{2} + 75 \, \log \relax (x) - 125\right )} \log \left (x + 5\right ) - 150 \, \log \relax (x)^{2} + 500 \, \log \relax (x) - 625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1000\,x-\ln \relax (3)\,\left (8\,x+20\right )+100\,x^2-{\mathrm {e}}^x\,\left (20\,x^3+180\,x^2+300\,x-500\right )+2500}{1953125\,x-\ln \relax (3)\,\left (3125\,x^2+15625\,x\right )+{\ln \left (\frac {30\,x-\ln \relax (3)+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )}^4\,\left (15625\,x-\ln \relax (3)\,\left (25\,x^2+125\,x\right )+6875\,x^2+875\,x^3+25\,x^4+{\mathrm {e}}^x\,\left (125\,x^3+1250\,x^2+3125\,x\right )\right )+{\ln \left (\frac {30\,x-\ln \relax (3)+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )}^3\,\left (156250\,x-\ln \relax (3)\,\left (250\,x^2+1250\,x\right )+68750\,x^2+8750\,x^3+250\,x^4+{\mathrm {e}}^x\,\left (1250\,x^3+12500\,x^2+31250\,x\right )\right )+{\ln \left (\frac {30\,x-\ln \relax (3)+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )}^2\,\left (781250\,x-\ln \relax (3)\,\left (1250\,x^2+6250\,x\right )+343750\,x^2+43750\,x^3+1250\,x^4+{\mathrm {e}}^x\,\left (6250\,x^3+62500\,x^2+156250\,x\right )\right )+{\ln \left (\frac {30\,x-\ln \relax (3)+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )}^5\,\left (625\,x+275\,x^2+35\,x^3+x^4-\ln \relax (3)\,\left (x^2+5\,x\right )+{\mathrm {e}}^x\,\left (5\,x^3+50\,x^2+125\,x\right )\right )+859375\,x^2+109375\,x^3+3125\,x^4+\ln \left (\frac {30\,x-\ln \relax (3)+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )\,\left (1953125\,x-\ln \relax (3)\,\left (3125\,x^2+15625\,x\right )+859375\,x^2+109375\,x^3+3125\,x^4+{\mathrm {e}}^x\,\left (15625\,x^3+156250\,x^2+390625\,x\right )\right )+{\mathrm {e}}^x\,\left (15625\,x^3+156250\,x^2+390625\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.32, size = 128, normalized size = 4.74 \begin {gather*} \frac {1}{\log {\left (\frac {x^{2} + 30 x + \left (5 x + 25\right ) e^{x} - \log {\relax (3 )} + 125}{x^{2} + 5 x} \right )}^{4} + 20 \log {\left (\frac {x^{2} + 30 x + \left (5 x + 25\right ) e^{x} - \log {\relax (3 )} + 125}{x^{2} + 5 x} \right )}^{3} + 150 \log {\left (\frac {x^{2} + 30 x + \left (5 x + 25\right ) e^{x} - \log {\relax (3 )} + 125}{x^{2} + 5 x} \right )}^{2} + 500 \log {\left (\frac {x^{2} + 30 x + \left (5 x + 25\right ) e^{x} - \log {\relax (3 )} + 125}{x^{2} + 5 x} \right )} + 625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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