Optimal. Leaf size=22 \[ \log (x)-\frac {1}{\left (x^2+\log (x) (x+x \log (5+x))\right )^2} \]
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Rubi [F] time = 4.57, 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 {10+22 x+4 x^2+5 x^5+x^6+\left (10+4 x+15 x^4+3 x^5\right ) \log (x)+\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (5 x^2+x^3\right ) \log ^3(x)+\left (10+2 x+\left (10+2 x+15 x^4+3 x^5\right ) \log (x)+\left (30 x^3+6 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^3+3 x^4\right ) \log ^2(x)+\left (15 x^2+3 x^3\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^2+x^3\right ) \log ^3(x) \log ^3(5+x)}{5 x^6+x^7+\left (15 x^5+3 x^6\right ) \log (x)+\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (5 x^3+x^4\right ) \log ^3(x)+\left (\left (15 x^5+3 x^6\right ) \log (x)+\left (30 x^4+6 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log (5+x)+\left (\left (15 x^4+3 x^5\right ) \log ^2(x)+\left (15 x^3+3 x^4\right ) \log ^3(x)\right ) \log ^2(5+x)+\left (5 x^3+x^4\right ) \log ^3(x) \log ^3(5+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 {3 x^3 (5+x) \log ^2(x) (1+\log (5+x))^2+x^2 (5+x) \log ^3(x) (1+\log (5+x))^3+(5+x) \left (2+4 x+x^5+2 \log (5+x)\right )+\log (x) \left (10+4 x+15 x^4+3 x^5+\left (10+2 x+15 x^4+3 x^5\right ) \log (5+x)\right )}{x^3 (5+x) (x+\log (x) (1+\log (5+x)))^3} \, dx\\ &=\int \left (\frac {1}{x}+\frac {2 \left (-5-x+5 \log (x)+x \log (x)+\log ^2(x)\right )}{x^2 (5+x) \log (x) (x+\log (x)+\log (x) \log (5+x))^3}+\frac {2 (1+\log (x))}{x^3 \log (x) (x+\log (x)+\log (x) \log (5+x))^2}\right ) \, dx\\ &=\log (x)+2 \int \frac {-5-x+5 \log (x)+x \log (x)+\log ^2(x)}{x^2 (5+x) \log (x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx+2 \int \frac {1+\log (x)}{x^3 \log (x) (x+\log (x)+\log (x) \log (5+x))^2} \, dx\\ &=\log (x)+2 \int \left (\frac {5+x-5 \log (x)-x \log (x)-\log ^2(x)}{25 x \log (x) (x+\log (x)+\log (x) \log (5+x))^3}+\frac {-5-x+5 \log (x)+x \log (x)+\log ^2(x)}{5 x^2 \log (x) (x+\log (x)+\log (x) \log (5+x))^3}+\frac {-5-x+5 \log (x)+x \log (x)+\log ^2(x)}{25 (5+x) \log (x) (x+\log (x)+\log (x) \log (5+x))^3}\right ) \, dx+2 \int \left (\frac {1}{x^3 (x+\log (x)+\log (x) \log (5+x))^2}+\frac {1}{x^3 \log (x) (x+\log (x)+\log (x) \log (5+x))^2}\right ) \, dx\\ &=\log (x)+\frac {2}{25} \int \frac {5+x-5 \log (x)-x \log (x)-\log ^2(x)}{x \log (x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx+\frac {2}{25} \int \frac {-5-x+5 \log (x)+x \log (x)+\log ^2(x)}{(5+x) \log (x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx+\frac {2}{5} \int \frac {-5-x+5 \log (x)+x \log (x)+\log ^2(x)}{x^2 \log (x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx+2 \int \frac {1}{x^3 (x+\log (x)+\log (x) \log (5+x))^2} \, dx+2 \int \frac {1}{x^3 \log (x) (x+\log (x)+\log (x) \log (5+x))^2} \, dx\\ &=\log (x)+\frac {2}{25} \int \left (-\frac {1}{(x+\log (x)+\log (x) \log (5+x))^3}-\frac {5}{x (x+\log (x)+\log (x) \log (5+x))^3}+\frac {1}{\log (x) (x+\log (x)+\log (x) \log (5+x))^3}+\frac {5}{x \log (x) (x+\log (x)+\log (x) \log (5+x))^3}-\frac {\log (x)}{x (x+\log (x)+\log (x) \log (5+x))^3}\right ) \, dx+\frac {2}{25} \int \left (\frac {5}{(5+x) (x+\log (x)+\log (x) \log (5+x))^3}+\frac {x}{(5+x) (x+\log (x)+\log (x) \log (5+x))^3}-\frac {5}{(5+x) \log (x) (x+\log (x)+\log (x) \log (5+x))^3}-\frac {x}{(5+x) \log (x) (x+\log (x)+\log (x) \log (5+x))^3}+\frac {\log (x)}{(5+x) (x+\log (x)+\log (x) \log (5+x))^3}\right ) \, dx+\frac {2}{5} \int \left (\frac {5}{x^2 (x+\log (x)+\log (x) \log (5+x))^3}+\frac {1}{x (x+\log (x)+\log (x) \log (5+x))^3}-\frac {5}{x^2 \log (x) (x+\log (x)+\log (x) \log (5+x))^3}-\frac {1}{x \log (x) (x+\log (x)+\log (x) \log (5+x))^3}+\frac {\log (x)}{x^2 (x+\log (x)+\log (x) \log (5+x))^3}\right ) \, dx+2 \int \frac {1}{x^3 (x+\log (x)+\log (x) \log (5+x))^2} \, dx+2 \int \frac {1}{x^3 \log (x) (x+\log (x)+\log (x) \log (5+x))^2} \, dx\\ &=\log (x)-\frac {2}{25} \int \frac {1}{(x+\log (x)+\log (x) \log (5+x))^3} \, dx+\frac {2}{25} \int \frac {x}{(5+x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx+\frac {2}{25} \int \frac {1}{\log (x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx-\frac {2}{25} \int \frac {x}{(5+x) \log (x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx-\frac {2}{25} \int \frac {\log (x)}{x (x+\log (x)+\log (x) \log (5+x))^3} \, dx+\frac {2}{25} \int \frac {\log (x)}{(5+x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx+\frac {2}{5} \int \frac {1}{(5+x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx-\frac {2}{5} \int \frac {1}{(5+x) \log (x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx+\frac {2}{5} \int \frac {\log (x)}{x^2 (x+\log (x)+\log (x) \log (5+x))^3} \, dx+2 \int \frac {1}{x^2 (x+\log (x)+\log (x) \log (5+x))^3} \, dx-2 \int \frac {1}{x^2 \log (x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx+2 \int \frac {1}{x^3 (x+\log (x)+\log (x) \log (5+x))^2} \, dx+2 \int \frac {1}{x^3 \log (x) (x+\log (x)+\log (x) \log (5+x))^2} \, dx\\ &=\log (x)-\frac {2}{25} \int \frac {1}{(x+\log (x)+\log (x) \log (5+x))^3} \, dx+\frac {2}{25} \int \frac {1}{\log (x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx-\frac {2}{25} \int \frac {\log (x)}{x (x+\log (x)+\log (x) \log (5+x))^3} \, dx+\frac {2}{25} \int \frac {\log (x)}{(5+x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx+\frac {2}{25} \int \left (\frac {1}{(x+\log (x)+\log (x) \log (5+x))^3}-\frac {5}{(5+x) (x+\log (x)+\log (x) \log (5+x))^3}\right ) \, dx-\frac {2}{25} \int \left (\frac {1}{\log (x) (x+\log (x)+\log (x) \log (5+x))^3}-\frac {5}{(5+x) \log (x) (x+\log (x)+\log (x) \log (5+x))^3}\right ) \, dx+\frac {2}{5} \int \frac {1}{(5+x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx-\frac {2}{5} \int \frac {1}{(5+x) \log (x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx+\frac {2}{5} \int \frac {\log (x)}{x^2 (x+\log (x)+\log (x) \log (5+x))^3} \, dx+2 \int \frac {1}{x^2 (x+\log (x)+\log (x) \log (5+x))^3} \, dx-2 \int \frac {1}{x^2 \log (x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx+2 \int \frac {1}{x^3 (x+\log (x)+\log (x) \log (5+x))^2} \, dx+2 \int \frac {1}{x^3 \log (x) (x+\log (x)+\log (x) \log (5+x))^2} \, dx\\ &=\log (x)-\frac {2}{25} \int \frac {\log (x)}{x (x+\log (x)+\log (x) \log (5+x))^3} \, dx+\frac {2}{25} \int \frac {\log (x)}{(5+x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx+\frac {2}{5} \int \frac {\log (x)}{x^2 (x+\log (x)+\log (x) \log (5+x))^3} \, dx+2 \int \frac {1}{x^2 (x+\log (x)+\log (x) \log (5+x))^3} \, dx-2 \int \frac {1}{x^2 \log (x) (x+\log (x)+\log (x) \log (5+x))^3} \, dx+2 \int \frac {1}{x^3 (x+\log (x)+\log (x) \log (5+x))^2} \, dx+2 \int \frac {1}{x^3 \log (x) (x+\log (x)+\log (x) \log (5+x))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 21, normalized size = 0.95 \begin {gather*} \log (x)-\frac {1}{x^2 (x+\log (x)+\log (x) \log (5+x))^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 119, normalized size = 5.41 \begin {gather*} \frac {x^{2} \log \left (x + 5\right )^{2} \log \relax (x)^{3} + x^{4} \log \relax (x) + 2 \, x^{3} \log \relax (x)^{2} + x^{2} \log \relax (x)^{3} + 2 \, {\left (x^{3} \log \relax (x)^{2} + x^{2} \log \relax (x)^{3}\right )} \log \left (x + 5\right ) - 1}{x^{2} \log \left (x + 5\right )^{2} \log \relax (x)^{2} + x^{4} + 2 \, x^{3} \log \relax (x) + x^{2} \log \relax (x)^{2} + 2 \, {\left (x^{3} \log \relax (x) + x^{2} \log \relax (x)^{2}\right )} \log \left (x + 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.64, size = 281, normalized size = 12.77 \begin {gather*} -\frac {x \log \relax (x) + \log \relax (x)^{2} - x + 5 \, \log \relax (x) - 5}{x^{3} \log \left (x + 5\right )^{2} \log \relax (x)^{3} + x^{2} \log \left (x + 5\right )^{2} \log \relax (x)^{4} + 2 \, x^{4} \log \left (x + 5\right ) \log \relax (x)^{2} - x^{3} \log \left (x + 5\right )^{2} \log \relax (x)^{2} + 4 \, x^{3} \log \left (x + 5\right ) \log \relax (x)^{3} + 5 \, x^{2} \log \left (x + 5\right )^{2} \log \relax (x)^{3} + 2 \, x^{2} \log \left (x + 5\right ) \log \relax (x)^{4} + x^{5} \log \relax (x) - 2 \, x^{4} \log \left (x + 5\right ) \log \relax (x) + 3 \, x^{4} \log \relax (x)^{2} + 8 \, x^{3} \log \left (x + 5\right ) \log \relax (x)^{2} - 5 \, x^{2} \log \left (x + 5\right )^{2} \log \relax (x)^{2} + 3 \, x^{3} \log \relax (x)^{3} + 10 \, x^{2} \log \left (x + 5\right ) \log \relax (x)^{3} + x^{2} \log \relax (x)^{4} - x^{5} + 3 \, x^{4} \log \relax (x) - 10 \, x^{3} \log \left (x + 5\right ) \log \relax (x) + 9 \, x^{3} \log \relax (x)^{2} - 10 \, x^{2} \log \left (x + 5\right ) \log \relax (x)^{2} + 5 \, x^{2} \log \relax (x)^{3} - 5 \, x^{4} - 10 \, x^{3} \log \relax (x) - 5 \, x^{2} \log \relax (x)^{2}} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 22, normalized size = 1.00
method | result | size |
risch | \(\ln \relax (x )-\frac {1}{x^{2} \left (\ln \relax (x ) \ln \left (5+x \right )+\ln \relax (x )+x \right )^{2}}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 61, normalized size = 2.77 \begin {gather*} -\frac {1}{x^{2} \log \left (x + 5\right )^{2} \log \relax (x)^{2} + x^{4} + 2 \, x^{3} \log \relax (x) + x^{2} \log \relax (x)^{2} + 2 \, {\left (x^{3} \log \relax (x) + x^{2} \log \relax (x)^{2}\right )} \log \left (x + 5\right )} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.09, size = 82, normalized size = 3.73 \begin {gather*} \frac {x^4\,\ln \relax (x)+2\,x^3\,\ln \left (x+5\right )\,{\ln \relax (x)}^2+2\,x^3\,{\ln \relax (x)}^2+x^2\,{\ln \left (x+5\right )}^2\,{\ln \relax (x)}^3+2\,x^2\,\ln \left (x+5\right )\,{\ln \relax (x)}^3+x^2\,{\ln \relax (x)}^3-1}{x^2\,{\left (x+\ln \relax (x)+\ln \left (x+5\right )\,\ln \relax (x)\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.52, size = 63, normalized size = 2.86 \begin {gather*} \log {\relax (x )} - \frac {1}{x^{4} + 2 x^{3} \log {\relax (x )} + x^{2} \log {\relax (x )}^{2} \log {\left (x + 5 \right )}^{2} + x^{2} \log {\relax (x )}^{2} + \left (2 x^{3} \log {\relax (x )} + 2 x^{2} \log {\relax (x )}^{2}\right ) \log {\left (x + 5 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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