Optimal. Leaf size=35 \[ x+\frac {x \log (\log (3))}{x-\frac {x^2}{\log \left (\frac {2+\frac {5}{x}+x}{5+e^3}\right )}} \]
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Rubi [F]
time = 2.74, 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 {5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5-x^2+\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )\right ) \log (\log (3))}{5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 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 x^3+x^4+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )-\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right ) (2 x-\log (\log (3)))-x^2 (-5+\log (\log (3)))+5 \log (\log (3))}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx\\ &=\int \left (\frac {2 x^3}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {x^4}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}-\frac {x^2 (-5+\log (\log (3)))}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {5 \log (\log (3))}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right ) (-2 x+\log (\log (3)))}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {x^3}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(5-\log (\log (3))) \int \frac {x^2}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(5 \log (\log (3))) \int \frac {1}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {x^4}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right ) (-2 x+\log (\log (3)))}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx\\ &=2 \int \left (-\frac {2}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {x}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {10-x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+(5-\log (\log (3))) \int \left (\frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {-5-2 x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+(5 \log (\log (3))) \int \left (\frac {i}{2 ((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {i}{2 ((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+\int \left (-\frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}-\frac {2 x}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {x^2}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {5+12 x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+\int \frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \left (-\frac {2 x \log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right ) \log (\log (3))}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {10-x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-2 \int \frac {x \log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-4 \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(5-\log (\log (3))) \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(5-\log (\log (3))) \int \frac {-5-2 x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\log (\log (3)) \int \frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-\int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {x^2}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {5+12 x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx\\ &=2 \int \left (\frac {10}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}-\frac {x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx-2 \int \frac {x \log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-4 \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(5-\log (\log (3))) \int \left (-\frac {5}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}-\frac {2 x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+(5-\log (\log (3))) \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\log (\log (3)) \int \frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \left (\frac {5}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {12 x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx-\int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {x^2}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx\\ &=-\left (2 \int \frac {x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx\right )-2 \int \frac {x \log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-4 \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+5 \int \frac {1}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+12 \int \frac {x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+20 \int \frac {1}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(5-\log (\log (3))) \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-(2 (5-\log (\log (3)))) \int \frac {x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-(5 (5-\log (\log (3)))) \int \frac {1}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\log (\log (3)) \int \frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-\int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {x^2}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [F]
time = 0.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5-x^2+\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )\right ) \log (\log (3))}{5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 5.10, size = 56, normalized size = 1.60
method | result | size |
risch | \(x -\frac {x \ln \left (\ln \left (3\right )\right )}{x -\ln \left (\frac {x^{2}+2 x +5}{x \,{\mathrm e}^{3}+5 x}\right )}\) | \(35\) |
default | \(x +\frac {-\ln \left (\ln \left (3\right )\right ) \ln \left (\frac {x^{2}+2 x +5}{x}\right )+\ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{3}+5\right )}{\ln \left ({\mathrm e}^{3}+5\right )+x -\ln \left (\frac {x^{2}+2 x +5}{x}\right )}\) | \(56\) |
norman | \(\frac {x^{2}-\ln \left (\ln \left (3\right )\right ) \ln \left (\frac {x^{2}+2 x +5}{x \,{\mathrm e}^{3}+5 x}\right )-x \ln \left (\frac {x^{2}+2 x +5}{x \,{\mathrm e}^{3}+5 x}\right )}{x -\ln \left (\frac {x^{2}+2 x +5}{x \,{\mathrm e}^{3}+5 x}\right )}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 56, normalized size = 1.60 \begin {gather*} \frac {x^{2} + x {\left (\log \left (e^{3} + 5\right ) - \log \left (\log \left (3\right )\right )\right )} - x \log \left (x^{2} + 2 \, x + 5\right ) + x \log \left (x\right )}{x - \log \left (x^{2} + 2 \, x + 5\right ) + \log \left (x\right ) + \log \left (e^{3} + 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 60, normalized size = 1.71 \begin {gather*} \frac {x^{2} - x \log \left (\frac {x^{2} + 2 \, x + 5}{x e^{3} + 5 \, x}\right ) - x \log \left (\log \left (3\right )\right )}{x - \log \left (\frac {x^{2} + 2 \, x + 5}{x e^{3} + 5 \, x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 27, normalized size = 0.77 \begin {gather*} x + \frac {x \log {\left (\log {\left (3 \right )} \right )}}{- x + \log {\left (\frac {x^{2} + 2 x + 5}{5 x + x e^{3}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 58, normalized size = 1.66 \begin {gather*} \frac {x^{2} - x \log \left (x^{2} + 2 \, x + 5\right ) + x \log \left (x e^{3} + 5 \, x\right ) - x \log \left (\log \left (3\right )\right )}{x - \log \left (x^{2} + 2 \, x + 5\right ) + \log \left (x e^{3} + 5 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.05, size = 80, normalized size = 2.29 \begin {gather*} -\frac {x\,\ln \left (\frac {x^2+2\,x+5}{5\,x+x\,{\mathrm {e}}^3}\right )-x^2+\ln \left (\frac {x^2+2\,x+5}{5\,x+x\,{\mathrm {e}}^3}\right )\,\ln \left (\ln \left (3\right )\right )}{x-\ln \left (\frac {x^2+2\,x+5}{5\,x+x\,{\mathrm {e}}^3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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