Optimal. Leaf size=27 \[ \frac {2}{x+\frac {x^2}{5+x}+\log \left (1+e^3+x+x^2+\log (x)\right )} \]
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Rubi [F] time = 7.45, 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 {-50-120 x-212 x^2-136 x^3-48 x^4-4 x^5+e^3 \left (-50 x-40 x^2-4 x^3\right )+\left (-50 x-40 x^2-4 x^3\right ) \log (x)}{25 x^3+45 x^4+49 x^5+24 x^6+4 x^7+e^3 \left (25 x^3+20 x^4+4 x^5\right )+\left (25 x^3+20 x^4+4 x^5\right ) \log (x)+\left (50 x^2+80 x^3+84 x^4+34 x^5+4 x^6+e^3 \left (50 x^2+30 x^3+4 x^4\right )+\left (50 x^2+30 x^3+4 x^4\right ) \log (x)\right ) \log \left (1+e^3+x+x^2+\log (x)\right )+\left (25 x+35 x^2+36 x^3+11 x^4+x^5+e^3 \left (25 x+10 x^2+x^3\right )+\left (25 x+10 x^2+x^3\right ) \log (x)\right ) \log ^2\left (1+e^3+x+x^2+\log (x)\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 {2 \left (-25-5 \left (12+5 e^3\right ) x-2 \left (53+10 e^3\right ) x^2-2 \left (34+e^3\right ) x^3-24 x^4-2 x^5-x \left (25+20 x+2 x^2\right ) \log (x)\right )}{x \left (1+e^3+x+x^2+\log (x)\right ) \left (x (5+2 x)+(5+x) \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx\\ &=2 \int \frac {-25-5 \left (12+5 e^3\right ) x-2 \left (53+10 e^3\right ) x^2-2 \left (34+e^3\right ) x^3-24 x^4-2 x^5-x \left (25+20 x+2 x^2\right ) \log (x)}{x \left (1+e^3+x+x^2+\log (x)\right ) \left (x (5+2 x)+(5+x) \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx\\ &=2 \int \left (-\frac {5 \left (12+5 e^3\right )}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2}-\frac {25}{x \left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2}-\frac {2 \left (53+10 e^3\right ) x}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2}-\frac {2 \left (34+e^3\right ) x^2}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2}-\frac {24 x^3}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2}-\frac {2 x^4}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2}-\frac {25 \log (x)}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2}-\frac {20 x \log (x)}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2}-\frac {2 x^2 \log (x)}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {x^4}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx\right )-4 \int \frac {x^2 \log (x)}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-40 \int \frac {x \log (x)}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-48 \int \frac {x^3}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-50 \int \frac {1}{x \left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-50 \int \frac {\log (x)}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-\left (4 \left (34+e^3\right )\right ) \int \frac {x^2}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-\left (10 \left (12+5 e^3\right )\right ) \int \frac {1}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-\left (4 \left (53+10 e^3\right )\right ) \int \frac {x}{\left (1+e^3+x+x^2+\log (x)\right ) \left (5 x+2 x^2+5 \log \left (1+e^3+x+x^2+\log (x)\right )+x \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx\\ &=-\left (4 \int \frac {x^4}{\left (1+e^3+x+x^2+\log (x)\right ) \left (x (5+2 x)+(5+x) \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx\right )-4 \int \frac {x^2 \log (x)}{\left (1+e^3+x+x^2+\log (x)\right ) \left (x (5+2 x)+(5+x) \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-40 \int \frac {x \log (x)}{\left (1+e^3+x+x^2+\log (x)\right ) \left (x (5+2 x)+(5+x) \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-48 \int \frac {x^3}{\left (1+e^3+x+x^2+\log (x)\right ) \left (x (5+2 x)+(5+x) \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-50 \int \frac {1}{x \left (1+e^3+x+x^2+\log (x)\right ) \left (x (5+2 x)+(5+x) \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-50 \int \frac {\log (x)}{\left (1+e^3+x+x^2+\log (x)\right ) \left (x (5+2 x)+(5+x) \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-\left (4 \left (34+e^3\right )\right ) \int \frac {x^2}{\left (1+e^3+x+x^2+\log (x)\right ) \left (x (5+2 x)+(5+x) \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-\left (10 \left (12+5 e^3\right )\right ) \int \frac {1}{\left (1+e^3+x+x^2+\log (x)\right ) \left (x (5+2 x)+(5+x) \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx-\left (4 \left (53+10 e^3\right )\right ) \int \frac {x}{\left (1+e^3+x+x^2+\log (x)\right ) \left (x (5+2 x)+(5+x) \log \left (1+e^3+x+x^2+\log (x)\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.09, size = 31, normalized size = 1.15 \begin {gather*} \frac {2 (5+x)}{x (5+2 x)+(5+x) \log \left (1+e^3+x+x^2+\log (x)\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.27, size = 31, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left (x + 5\right )}}{2 \, x^{2} + {\left (x + 5\right )} \log \left (x^{2} + x + e^{3} + \log \relax (x) + 1\right ) + 5 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 42, normalized size = 1.56 \begin {gather*} \frac {2 \, {\left (x + 5\right )}}{2 \, x^{2} + x \log \left (x^{2} + x + e^{3} + \log \relax (x) + 1\right ) + 5 \, x + 5 \, \log \left (x^{2} + x + e^{3} + \log \relax (x) + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 43, normalized size = 1.59
method | result | size |
risch | \(\frac {2 x +10}{x \ln \left (\ln \relax (x )+{\mathrm e}^{3}+x^{2}+x +1\right )+2 x^{2}+5 \ln \left (\ln \relax (x )+{\mathrm e}^{3}+x^{2}+x +1\right )+5 x}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 31, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left (x + 5\right )}}{2 \, x^{2} + {\left (x + 5\right )} \log \left (x^{2} + x + e^{3} + \log \relax (x) + 1\right ) + 5 \, x} \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 {120\,x+{\mathrm {e}}^3\,\left (4\,x^3+40\,x^2+50\,x\right )+212\,x^2+136\,x^3+48\,x^4+4\,x^5+\ln \relax (x)\,\left (4\,x^3+40\,x^2+50\,x\right )+50}{\ln \left (x+{\mathrm {e}}^3+\ln \relax (x)+x^2+1\right )\,\left (\ln \relax (x)\,\left (4\,x^4+30\,x^3+50\,x^2\right )+{\mathrm {e}}^3\,\left (4\,x^4+30\,x^3+50\,x^2\right )+50\,x^2+80\,x^3+84\,x^4+34\,x^5+4\,x^6\right )+\ln \relax (x)\,\left (4\,x^5+20\,x^4+25\,x^3\right )+{\ln \left (x+{\mathrm {e}}^3+\ln \relax (x)+x^2+1\right )}^2\,\left (25\,x+{\mathrm {e}}^3\,\left (x^3+10\,x^2+25\,x\right )+\ln \relax (x)\,\left (x^3+10\,x^2+25\,x\right )+35\,x^2+36\,x^3+11\,x^4+x^5\right )+{\mathrm {e}}^3\,\left (4\,x^5+20\,x^4+25\,x^3\right )+25\,x^3+45\,x^4+49\,x^5+24\,x^6+4\,x^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 31, normalized size = 1.15 \begin {gather*} \frac {2 x + 10}{2 x^{2} + 5 x + \left (x + 5\right ) \log {\left (x^{2} + x + \log {\relax (x )} + 1 + e^{3} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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