Optimal. Leaf size=28 \[ x^2-\frac {3}{\left (-x+\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^2} \]
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Rubi [F]
time = 3.09, 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 {-6 e^5-6 x+\left (-6 x-6 x^2-2 x^6+e^5 \left (-6 x-2 x^5\right )\right ) \log \left (\frac {2}{x}\right )+\left (6 e^5 x^4+6 x^5\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-6 e^5 x^3-6 x^4\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (2 e^5 x^2+2 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )}{\left (-e^5 x^4-x^5\right ) \log \left (\frac {2}{x}\right )+\left (3 e^5 x^3+3 x^4\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-3 e^5 x^2-3 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (e^5 x+x^2\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+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 {6 \left (e^5+x\right )+2 x \log \left (\frac {2}{x}\right ) \left (3+3 x+x^5+e^5 \left (3+x^4\right )-3 x^3 \left (e^5+x\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+3 x^2 \left (e^5+x\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )-x \left (e^5+x\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )}{x \left (e^5+x\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx\\ &=\int \left (2 x+\frac {6 \left (e^5+x+\left (1+e^5\right ) x \log \left (\frac {2}{x}\right )+x^2 \log \left (\frac {2}{x}\right )\right )}{x \left (e^5+x\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}\right ) \, dx\\ &=x^2+6 \int \frac {e^5+x+\left (1+e^5\right ) x \log \left (\frac {2}{x}\right )+x^2 \log \left (\frac {2}{x}\right )}{x \left (e^5+x\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx\\ &=x^2+6 \int \frac {e^5+x+x \left (1+e^5+x\right ) \log \left (\frac {2}{x}\right )}{x \left (e^5+x\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx\\ &=x^2+6 \int \left (\frac {-e^5-x-\left (1+e^5\right ) x \log \left (\frac {2}{x}\right )-x^2 \log \left (\frac {2}{x}\right )}{e^5 \left (e^5+x\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}+\frac {e^5+x+\left (1+e^5\right ) x \log \left (\frac {2}{x}\right )+x^2 \log \left (\frac {2}{x}\right )}{e^5 x \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}\right ) \, dx\\ &=x^2+\frac {6 \int \frac {-e^5-x-\left (1+e^5\right ) x \log \left (\frac {2}{x}\right )-x^2 \log \left (\frac {2}{x}\right )}{\left (e^5+x\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx}{e^5}+\frac {6 \int \frac {e^5+x+\left (1+e^5\right ) x \log \left (\frac {2}{x}\right )+x^2 \log \left (\frac {2}{x}\right )}{x \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx}{e^5}\\ &=x^2+\frac {6 \int \frac {-e^5-x-x \left (1+e^5+x\right ) \log \left (\frac {2}{x}\right )}{\left (e^5+x\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx}{e^5}+\frac {6 \int \frac {e^5+x+x \left (1+e^5+x\right ) \log \left (\frac {2}{x}\right )}{x \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx}{e^5}\\ &=x^2+\frac {6 \int \left (\frac {1+e^5}{\left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}+\frac {x}{\left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}+\frac {1}{\log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}+\frac {e^5}{x \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}\right ) \, dx}{e^5}+\frac {6 \int \left (-\frac {\left (1+e^5\right ) x}{\left (e^5+x\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}-\frac {x^2}{\left (e^5+x\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}-\frac {e^5}{\left (e^5+x\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}-\frac {x}{\left (e^5+x\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}\right ) \, dx}{e^5}\\ &=x^2+6 \int \frac {1}{x \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx-6 \int \frac {1}{\left (e^5+x\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx+\left (6 \left (1+\frac {1}{e^5}\right )\right ) \int \frac {1}{\left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx+\frac {6 \int \frac {x}{\left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx}{e^5}-\frac {6 \int \frac {x^2}{\left (e^5+x\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx}{e^5}+\frac {6 \int \frac {1}{\log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx}{e^5}-\frac {6 \int \frac {x}{\left (e^5+x\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx}{e^5}+\frac {\left (6 \left (-1-e^5\right )\right ) \int \frac {x}{\left (e^5+x\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx}{e^5}\\ &=x^2+6 \int \frac {1}{x \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx-6 \int \frac {1}{\left (e^5+x\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx+\left (6 \left (1+\frac {1}{e^5}\right )\right ) \int \frac {1}{\left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx-\frac {6 \int \left (-\frac {e^5}{\left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}+\frac {x}{\left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}+\frac {e^{10}}{\left (e^5+x\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}\right ) \, dx}{e^5}-\frac {6 \int \left (\frac {1}{\log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}-\frac {e^5}{\left (e^5+x\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}\right ) \, dx}{e^5}+\frac {6 \int \frac {x}{\left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx}{e^5}+\frac {6 \int \frac {1}{\log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx}{e^5}+\frac {\left (6 \left (-1-e^5\right )\right ) \int \left (\frac {1}{\left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}-\frac {e^5}{\left (e^5+x\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3}\right ) \, dx}{e^5}\\ &=x^2+6 \int \frac {1}{\left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx+6 \int \frac {1}{x \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx+\left (6 \left (1+\frac {1}{e^5}\right )\right ) \int \frac {1}{\left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx-\left (6 e^5\right ) \int \frac {1}{\left (e^5+x\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx+\frac {\left (6 \left (-1-e^5\right )\right ) \int \frac {1}{\left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx}{e^5}+\left (6 \left (1+e^5\right )\right ) \int \frac {1}{\left (e^5+x\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.19, size = 28, normalized size = 1.00 \begin {gather*} x^2-\frac {3}{\left (-x+\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 8.95, size = 0, normalized size = 0.00 \[\int \frac {\left (2 x^{2} {\mathrm e}^{5}+2 x^{3}\right ) \ln \left (\frac {2}{x}\right ) \ln \left (\frac {5 \ln \left (\frac {2}{x}\right )}{{\mathrm e}^{5}+x}\right )^{3}+\left (-6 x^{3} {\mathrm e}^{5}-6 x^{4}\right ) \ln \left (\frac {2}{x}\right ) \ln \left (\frac {5 \ln \left (\frac {2}{x}\right )}{{\mathrm e}^{5}+x}\right )^{2}+\left (6 x^{4} {\mathrm e}^{5}+6 x^{5}\right ) \ln \left (\frac {2}{x}\right ) \ln \left (\frac {5 \ln \left (\frac {2}{x}\right )}{{\mathrm e}^{5}+x}\right )+\left (\left (-2 x^{5}-6 x \right ) {\mathrm e}^{5}-2 x^{6}-6 x^{2}-6 x \right ) \ln \left (\frac {2}{x}\right )-6 \,{\mathrm e}^{5}-6 x}{\left (x \,{\mathrm e}^{5}+x^{2}\right ) \ln \left (\frac {2}{x}\right ) \ln \left (\frac {5 \ln \left (\frac {2}{x}\right )}{{\mathrm e}^{5}+x}\right )^{3}+\left (-3 x^{2} {\mathrm e}^{5}-3 x^{3}\right ) \ln \left (\frac {2}{x}\right ) \ln \left (\frac {5 \ln \left (\frac {2}{x}\right )}{{\mathrm e}^{5}+x}\right )^{2}+\left (3 x^{3} {\mathrm e}^{5}+3 x^{4}\right ) \ln \left (\frac {2}{x}\right ) \ln \left (\frac {5 \ln \left (\frac {2}{x}\right )}{{\mathrm e}^{5}+x}\right )+\left (-x^{4} {\mathrm e}^{5}-x^{5}\right ) \ln \left (\frac {2}{x}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.88, size = 215, normalized size = 7.68 \begin {gather*} \frac {2 \, {\left (i \, \pi + \log \left (5\right )\right )} x^{3} - x^{4} - x^{2} \log \left (x + e^{5}\right )^{2} - x^{2} \log \left (-\log \left (2\right ) + \log \left (x\right )\right )^{2} + {\left (\pi ^{2} - 2 i \, \pi \log \left (5\right ) - \log \left (5\right )^{2}\right )} x^{2} + 2 \, {\left ({\left (i \, \pi + \log \left (5\right )\right )} x^{2} - x^{3}\right )} \log \left (x + e^{5}\right ) + 2 \, {\left ({\left (-i \, \pi - \log \left (5\right )\right )} x^{2} + x^{3} + x^{2} \log \left (x + e^{5}\right )\right )} \log \left (-\log \left (2\right ) + \log \left (x\right )\right ) + 3}{\pi ^{2} + 2 \, {\left (i \, \pi + \log \left (5\right )\right )} x - x^{2} - 2 i \, \pi \log \left (5\right ) - \log \left (5\right )^{2} + 2 \, {\left (i \, \pi - x + \log \left (5\right )\right )} \log \left (x + e^{5}\right ) - \log \left (x + e^{5}\right )^{2} + 2 \, {\left (-i \, \pi + x - \log \left (5\right ) + \log \left (x + e^{5}\right )\right )} \log \left (-\log \left (2\right ) + \log \left (x\right )\right ) - \log \left (-\log \left (2\right ) + \log \left (x\right )\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs.
\(2 (27) = 54\).
time = 0.37, size = 88, normalized size = 3.14 \begin {gather*} \frac {x^{4} - 2 \, x^{3} \log \left (\frac {5 \, \log \left (\frac {2}{x}\right )}{x + e^{5}}\right ) + x^{2} \log \left (\frac {5 \, \log \left (\frac {2}{x}\right )}{x + e^{5}}\right )^{2} - 3}{x^{2} - 2 \, x \log \left (\frac {5 \, \log \left (\frac {2}{x}\right )}{x + e^{5}}\right ) + \log \left (\frac {5 \, \log \left (\frac {2}{x}\right )}{x + e^{5}}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.42, size = 39, normalized size = 1.39 \begin {gather*} x^{2} - \frac {3}{x^{2} - 2 x \log {\left (\frac {5 \log {\left (\frac {2}{x} \right )}}{x + e^{5}} \right )} + \log {\left (\frac {5 \log {\left (\frac {2}{x} \right )}}{x + e^{5}} \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 31752 vs.
\(2 (27) = 54\).
time = 14.25, size = 31752, normalized size = 1134.00 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.11, size = 27, normalized size = 0.96 \begin {gather*} x^2-\frac {3}{{\left (x-\ln \left (\frac {5\,\ln \left (\frac {2}{x}\right )}{x+{\mathrm {e}}^5}\right )\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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