Optimal. Leaf size=22 \[ \frac {x^5}{\left (e^x+x-x^2-\log ^2(x)\right )^4} \]
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Rubi [F] time = 11.29, 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 {-x^5-3 x^6+e^x \left (-5 x^4+4 x^5\right )-8 x^4 \log (x)+5 x^4 \log ^2(x)}{-e^{5 x}-x^5+5 x^6-10 x^7+10 x^8-5 x^9+x^{10}+e^{4 x} \left (-5 x+5 x^2\right )+e^{3 x} \left (-10 x^2+20 x^3-10 x^4\right )+e^{2 x} \left (-10 x^3+30 x^4-30 x^5+10 x^6\right )+e^x \left (-5 x^4+20 x^5-30 x^6+20 x^7-5 x^8\right )+\left (5 e^{4 x}+5 x^4-20 x^5+30 x^6-20 x^7+5 x^8+e^{3 x} \left (20 x-20 x^2\right )+e^{2 x} \left (30 x^2-60 x^3+30 x^4\right )+e^x \left (20 x^3-60 x^4+60 x^5-20 x^6\right )\right ) \log ^2(x)+\left (-10 e^{3 x}-10 x^3+30 x^4-30 x^5+10 x^6+e^{2 x} \left (-30 x+30 x^2\right )+e^x \left (-30 x^2+60 x^3-30 x^4\right )\right ) \log ^4(x)+\left (10 e^{2 x}+10 x^2-20 x^3+10 x^4+e^x \left (20 x-20 x^2\right )\right ) \log ^6(x)+\left (-5 e^x-5 x+5 x^2\right ) \log ^8(x)+\log ^{10}(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^4 \left (e^x (5-4 x)+x (1+3 x)+8 \log (x)-5 \log ^2(x)\right )}{\left (e^x-(-1+x) x-\log ^2(x)\right )^5} \, dx\\ &=\int \left (-\frac {x^4 (-5+4 x)}{\left (-e^x-x+x^2+\log ^2(x)\right )^4}+\frac {4 x^4 \left (x-3 x^2+x^3-2 \log (x)+x \log ^2(x)\right )}{\left (-e^x-x+x^2+\log ^2(x)\right )^5}\right ) \, dx\\ &=4 \int \frac {x^4 \left (x-3 x^2+x^3-2 \log (x)+x \log ^2(x)\right )}{\left (-e^x-x+x^2+\log ^2(x)\right )^5} \, dx-\int \frac {x^4 (-5+4 x)}{\left (-e^x-x+x^2+\log ^2(x)\right )^4} \, dx\\ &=4 \int \left (\frac {x^5}{\left (-e^x-x+x^2+\log ^2(x)\right )^5}-\frac {3 x^6}{\left (-e^x-x+x^2+\log ^2(x)\right )^5}+\frac {x^7}{\left (-e^x-x+x^2+\log ^2(x)\right )^5}-\frac {2 x^4 \log (x)}{\left (-e^x-x+x^2+\log ^2(x)\right )^5}+\frac {x^5 \log ^2(x)}{\left (-e^x-x+x^2+\log ^2(x)\right )^5}\right ) \, dx-\int \left (-\frac {5 x^4}{\left (-e^x-x+x^2+\log ^2(x)\right )^4}+\frac {4 x^5}{\left (-e^x-x+x^2+\log ^2(x)\right )^4}\right ) \, dx\\ &=4 \int \frac {x^5}{\left (-e^x-x+x^2+\log ^2(x)\right )^5} \, dx+4 \int \frac {x^7}{\left (-e^x-x+x^2+\log ^2(x)\right )^5} \, dx+4 \int \frac {x^5 \log ^2(x)}{\left (-e^x-x+x^2+\log ^2(x)\right )^5} \, dx-4 \int \frac {x^5}{\left (-e^x-x+x^2+\log ^2(x)\right )^4} \, dx+5 \int \frac {x^4}{\left (-e^x-x+x^2+\log ^2(x)\right )^4} \, dx-8 \int \frac {x^4 \log (x)}{\left (-e^x-x+x^2+\log ^2(x)\right )^5} \, dx-12 \int \frac {x^6}{\left (-e^x-x+x^2+\log ^2(x)\right )^5} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.79, size = 22, normalized size = 1.00 \begin {gather*} \frac {x^5}{\left (-e^x-x+x^2+\log ^2(x)\right )^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 200, normalized size = 9.09 \begin {gather*} \frac {x^{5}}{x^{8} + \log \relax (x)^{8} - 4 \, x^{7} + 4 \, {\left (x^{2} - x - e^{x}\right )} \log \relax (x)^{6} + 6 \, x^{6} - 4 \, x^{5} + 6 \, {\left (x^{4} - 2 \, x^{3} + x^{2} - 2 \, {\left (x^{2} - x\right )} e^{x} + e^{\left (2 \, x\right )}\right )} \log \relax (x)^{4} + x^{4} + 4 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3} + 3 \, {\left (x^{2} - x\right )} e^{\left (2 \, x\right )} - 3 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{x} - e^{\left (3 \, x\right )}\right )} \log \relax (x)^{2} - 4 \, {\left (x^{2} - x\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3}\right )} e^{x} + e^{\left (4 \, x\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 [A] time = 0.09, size = 22, normalized size = 1.00
method | result | size |
risch | \(\frac {x^{5}}{\left (\ln \relax (x )^{2}+x^{2}-{\mathrm e}^{x}-x \right )^{4}}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.46, size = 202, normalized size = 9.18 \begin {gather*} \frac {x^{5}}{x^{8} + \log \relax (x)^{8} - 4 \, x^{7} + 4 \, {\left (x^{2} - x\right )} \log \relax (x)^{6} + 6 \, x^{6} - 4 \, x^{5} + 6 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \relax (x)^{4} + x^{4} + 4 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3}\right )} \log \relax (x)^{2} - 4 \, {\left (x^{2} + \log \relax (x)^{2} - x\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{4} + \log \relax (x)^{4} - 2 \, x^{3} + 2 \, {\left (x^{2} - x\right )} \log \relax (x)^{2} + x^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{6} + \log \relax (x)^{6} - 3 \, x^{5} + 3 \, {\left (x^{2} - x\right )} \log \relax (x)^{4} + 3 \, x^{4} - x^{3} + 3 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \relax (x)^{2}\right )} e^{x} + e^{\left (4 \, x\right )}} \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 = -0.05 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (5\,x^4-4\,x^5\right )+8\,x^4\,\ln \relax (x)-5\,x^4\,{\ln \relax (x)}^2+x^5+3\,x^6}{{\mathrm {e}}^{5\,x}+{\mathrm {e}}^{4\,x}\,\left (5\,x-5\,x^2\right )-{\ln \relax (x)}^{10}+{\ln \relax (x)}^8\,\left (5\,x+5\,{\mathrm {e}}^x-5\,x^2\right )+{\mathrm {e}}^{3\,x}\,\left (10\,x^4-20\,x^3+10\,x^2\right )+{\mathrm {e}}^x\,\left (5\,x^8-20\,x^7+30\,x^6-20\,x^5+5\,x^4\right )+{\ln \relax (x)}^4\,\left (10\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{2\,x}\,\left (30\,x-30\,x^2\right )+{\mathrm {e}}^x\,\left (30\,x^4-60\,x^3+30\,x^2\right )+10\,x^3-30\,x^4+30\,x^5-10\,x^6\right )-{\ln \relax (x)}^2\,\left (5\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{3\,x}\,\left (20\,x-20\,x^2\right )+{\mathrm {e}}^x\,\left (-20\,x^6+60\,x^5-60\,x^4+20\,x^3\right )+{\mathrm {e}}^{2\,x}\,\left (30\,x^4-60\,x^3+30\,x^2\right )+5\,x^4-20\,x^5+30\,x^6-20\,x^7+5\,x^8\right )+{\mathrm {e}}^{2\,x}\,\left (-10\,x^6+30\,x^5-30\,x^4+10\,x^3\right )+x^5-5\,x^6+10\,x^7-10\,x^8+5\,x^9-x^{10}-{\ln \relax (x)}^6\,\left (10\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (20\,x-20\,x^2\right )+10\,x^2-20\,x^3+10\,x^4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.98, size = 270, normalized size = 12.27 \begin {gather*} \frac {x^{5}}{x^{8} - 4 x^{7} + 4 x^{6} \log {\relax (x )}^{2} + 6 x^{6} - 12 x^{5} \log {\relax (x )}^{2} - 4 x^{5} + 6 x^{4} \log {\relax (x )}^{4} + 12 x^{4} \log {\relax (x )}^{2} + x^{4} - 12 x^{3} \log {\relax (x )}^{4} - 4 x^{3} \log {\relax (x )}^{2} + 4 x^{2} \log {\relax (x )}^{6} + 6 x^{2} \log {\relax (x )}^{4} - 4 x \log {\relax (x )}^{6} + \left (- 4 x^{2} + 4 x - 4 \log {\relax (x )}^{2}\right ) e^{3 x} + \left (6 x^{4} - 12 x^{3} + 12 x^{2} \log {\relax (x )}^{2} + 6 x^{2} - 12 x \log {\relax (x )}^{2} + 6 \log {\relax (x )}^{4}\right ) e^{2 x} + \left (- 4 x^{6} + 12 x^{5} - 12 x^{4} \log {\relax (x )}^{2} - 12 x^{4} + 24 x^{3} \log {\relax (x )}^{2} + 4 x^{3} - 12 x^{2} \log {\relax (x )}^{4} - 12 x^{2} \log {\relax (x )}^{2} + 12 x \log {\relax (x )}^{4} - 4 \log {\relax (x )}^{6}\right ) e^{x} + e^{4 x} + \log {\relax (x )}^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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