Optimal. Leaf size=24 \[ \log ^2\left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right ) \]
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Rubi [F]
time = 13.45, 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 {\left (e^{-3+x} \left (-2 e^5-2 x\right )-10 e^5 x-10 x^2+\left (10 e^5 x+e^{-3+x} \left (-2 x+2 e^5 x+2 x^2\right )\right ) \log (x)\right ) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{\left (5 e^5 x^2+5 x^3+e^{-3+x} \left (e^5 x+x^2\right )\right ) \log (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 {2 \left (-\left (\left (e^5+x\right ) \left (e^x+5 e^3 x\right )\right )+\left (5 e^8+e^{5+x}+e^x (-1+x)\right ) x \log (x)\right ) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{x \left (e^5+x\right ) \left (e^x+5 e^3 x\right ) \log (x)} \, dx\\ &=2 \int \frac {\left (-\left (\left (e^5+x\right ) \left (e^x+5 e^3 x\right )\right )+\left (5 e^8+e^{5+x}+e^x (-1+x)\right ) x \log (x)\right ) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{x \left (e^5+x\right ) \left (e^x+5 e^3 x\right ) \log (x)} \, dx\\ &=2 \int \left (-\frac {5 e^3 (-1+x) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{e^x+5 e^3 x}+\frac {\left (-e^5-x-\left (1-e^5\right ) x \log (x)+x^2 \log (x)\right ) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{x \left (e^5+x\right ) \log (x)}\right ) \, dx\\ &=2 \int \frac {\left (-e^5-x-\left (1-e^5\right ) x \log (x)+x^2 \log (x)\right ) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{x \left (e^5+x\right ) \log (x)} \, dx-\left (10 e^3\right ) \int \frac {(-1+x) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{e^x+5 e^3 x} \, dx\\ &=2 \int \frac {\left (-e^5-x+x \left (-1+e^5+x\right ) \log (x)\right ) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{x \left (e^5+x\right ) \log (x)} \, dx+\left (10 e^3\right ) \int \left (\frac {5 e^8+e^{5+x}+e^x (-1+x)}{\left (e^5+x\right ) \left (e^x+5 e^3 x\right )}-\frac {1}{x \log (x)}\right ) \left (-\int \frac {1}{e^x+5 e^3 x} \, dx+\int \frac {x}{e^x+5 e^3 x} \, dx\right ) \, dx+\left (10 e^3 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )\right ) \int \frac {1}{e^x+5 e^3 x} \, dx-\left (10 e^3 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )\right ) \int \frac {x}{e^x+5 e^3 x} \, dx\\ &=2 \int \left (\frac {\left (e^5+x+\left (1-e^5\right ) x \log (x)-x^2 \log (x)\right ) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{e^5 \left (e^5+x\right ) \log (x)}+\frac {\left (-e^5-x-\left (1-e^5\right ) x \log (x)+x^2 \log (x)\right ) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{e^5 x \log (x)}\right ) \, dx+\left (10 e^3\right ) \int \left (\frac {5 e^3 (-1+x) \left (\int \frac {1}{e^x+5 e^3 x} \, dx-\int \frac {x}{e^x+5 e^3 x} \, dx\right )}{e^x+5 e^3 x}+\frac {\left (e^5+x+\left (1-e^5\right ) x \log (x)-x^2 \log (x)\right ) \left (\int \frac {1}{e^x+5 e^3 x} \, dx-\int \frac {x}{e^x+5 e^3 x} \, dx\right )}{x \left (e^5+x\right ) \log (x)}\right ) \, dx+\left (10 e^3 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )\right ) \int \frac {1}{e^x+5 e^3 x} \, dx-\left (10 e^3 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )\right ) \int \frac {x}{e^x+5 e^3 x} \, dx\\ &=\frac {2 \int \frac {\left (e^5+x+\left (1-e^5\right ) x \log (x)-x^2 \log (x)\right ) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{\left (e^5+x\right ) \log (x)} \, dx}{e^5}+\frac {2 \int \frac {\left (-e^5-x-\left (1-e^5\right ) x \log (x)+x^2 \log (x)\right ) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{x \log (x)} \, dx}{e^5}+\left (10 e^3\right ) \int \frac {\left (e^5+x+\left (1-e^5\right ) x \log (x)-x^2 \log (x)\right ) \left (\int \frac {1}{e^x+5 e^3 x} \, dx-\int \frac {x}{e^x+5 e^3 x} \, dx\right )}{x \left (e^5+x\right ) \log (x)} \, dx+\left (50 e^6\right ) \int \frac {(-1+x) \left (\int \frac {1}{e^x+5 e^3 x} \, dx-\int \frac {x}{e^x+5 e^3 x} \, dx\right )}{e^x+5 e^3 x} \, dx+\left (10 e^3 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )\right ) \int \frac {1}{e^x+5 e^3 x} \, dx-\left (10 e^3 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )\right ) \int \frac {x}{e^x+5 e^3 x} \, dx\\ &=\frac {2 \int \frac {\left (e^5+x-x \left (-1+e^5+x\right ) \log (x)\right ) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{\left (e^5+x\right ) \log (x)} \, dx}{e^5}+\frac {2 \int \frac {\left (-e^5-x+x \left (-1+e^5+x\right ) \log (x)\right ) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{x \log (x)} \, dx}{e^5}+\left (10 e^3\right ) \int \frac {\left (e^5+x-x \left (-1+e^5+x\right ) \log (x)\right ) \left (\int \frac {1}{e^x+5 e^3 x} \, dx-\int \frac {x}{e^x+5 e^3 x} \, dx\right )}{x \left (e^5+x\right ) \log (x)} \, dx+\left (50 e^6\right ) \int \left (-\frac {\int \frac {1}{e^x+5 e^3 x} \, dx-\int \frac {x}{e^x+5 e^3 x} \, dx}{e^x+5 e^3 x}+\frac {x \left (\int \frac {1}{e^x+5 e^3 x} \, dx-\int \frac {x}{e^x+5 e^3 x} \, dx\right )}{e^x+5 e^3 x}\right ) \, dx+\left (10 e^3 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )\right ) \int \frac {1}{e^x+5 e^3 x} \, dx-\left (10 e^3 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )\right ) \int \frac {x}{e^x+5 e^3 x} \, dx\\ &=\frac {2 \int \left (-\left (\left (1-e^5\right ) \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )\right )+x \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )-\frac {\log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{\log (x)}-\frac {e^5 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{x \log (x)}\right ) \, dx}{e^5}+\frac {2 \int \left (\frac {\left (1-e^5\right ) x \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{e^5+x}-\frac {x^2 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{e^5+x}+\frac {e^5 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{\left (e^5+x\right ) \log (x)}+\frac {x \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{\left (e^5+x\right ) \log (x)}\right ) \, dx}{e^5}+\left (10 e^3\right ) \int \left (\frac {\left (e^5+x+\left (1-e^5\right ) x \log (x)-x^2 \log (x)\right ) \int \frac {1}{e^x+5 e^3 x} \, dx}{x \left (e^5+x\right ) \log (x)}+\frac {\left (-e^5-x-\left (1-e^5\right ) x \log (x)+x^2 \log (x)\right ) \int \frac {x}{e^x+5 e^3 x} \, dx}{x \left (e^5+x\right ) \log (x)}\right ) \, dx-\left (50 e^6\right ) \int \frac {\int \frac {1}{e^x+5 e^3 x} \, dx-\int \frac {x}{e^x+5 e^3 x} \, dx}{e^x+5 e^3 x} \, dx+\left (50 e^6\right ) \int \frac {x \left (\int \frac {1}{e^x+5 e^3 x} \, dx-\int \frac {x}{e^x+5 e^3 x} \, dx\right )}{e^x+5 e^3 x} \, dx+\left (10 e^3 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )\right ) \int \frac {1}{e^x+5 e^3 x} \, dx-\left (10 e^3 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )\right ) \int \frac {x}{e^x+5 e^3 x} \, dx\\ &=-\left (2 \int \frac {\log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{x \log (x)} \, dx\right )+2 \int \frac {\log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{\left (e^5+x\right ) \log (x)} \, dx-\left (2 \left (1-\frac {1}{e^5}\right )\right ) \int \frac {x \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{e^5+x} \, dx+\frac {2 \int x \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right ) \, dx}{e^5}-\frac {2 \int \frac {x^2 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{e^5+x} \, dx}{e^5}-\frac {2 \int \frac {\log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{\log (x)} \, dx}{e^5}+\frac {2 \int \frac {x \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )}{\left (e^5+x\right ) \log (x)} \, dx}{e^5}+\left (10 e^3\right ) \int \frac {\left (e^5+x+\left (1-e^5\right ) x \log (x)-x^2 \log (x)\right ) \int \frac {1}{e^x+5 e^3 x} \, dx}{x \left (e^5+x\right ) \log (x)} \, dx+\left (10 e^3\right ) \int \frac {\left (-e^5-x-\left (1-e^5\right ) x \log (x)+x^2 \log (x)\right ) \int \frac {x}{e^x+5 e^3 x} \, dx}{x \left (e^5+x\right ) \log (x)} \, dx-\left (50 e^6\right ) \int \left (\frac {\int \frac {1}{e^x+5 e^3 x} \, dx}{e^x+5 e^3 x}-\frac {\int \frac {x}{e^x+5 e^3 x} \, dx}{e^x+5 e^3 x}\right ) \, dx+\left (50 e^6\right ) \int \left (\frac {x \int \frac {1}{e^x+5 e^3 x} \, dx}{e^x+5 e^3 x}-\frac {x \int \frac {x}{e^x+5 e^3 x} \, dx}{e^x+5 e^3 x}\right ) \, dx+\frac {\left (2 \left (-1+e^5\right )\right ) \int \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right ) \, dx}{e^5}+\left (10 e^3 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )\right ) \int \frac {1}{e^x+5 e^3 x} \, dx-\left (10 e^3 \log \left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right )\right ) \int \frac {x}{e^x+5 e^3 x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 24, normalized size = 1.00 \begin {gather*} \log ^2\left (\frac {e^{-3+x}+5 x}{\left (e^5+x\right ) \log (x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.45, size = 1440, normalized size = 60.00
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1440\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs.
\(2 (22) = 44\).
time = 0.34, size = 104, normalized size = 4.33 \begin {gather*} 2 \, {\left (\log \left (x + e^{5}\right ) + \log \left (\log \left (x\right )\right )\right )} \log \left (5 \, x e^{3} + e^{x}\right ) - \log \left (5 \, x e^{3} + e^{x}\right )^{2} - \log \left (x + e^{5}\right )^{2} + 2 \, {\left (\log \left (5 \, x e^{3} + e^{x}\right ) - \log \left (x + e^{5}\right ) - \log \left (\log \left (x\right )\right )\right )} \log \left (\frac {5 \, x + e^{\left (x - 3\right )}}{{\left (x + e^{5}\right )} \log \left (x\right )}\right ) - 2 \, \log \left (x + e^{5}\right ) \log \left (\log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 22, normalized size = 0.92 \begin {gather*} \log \left (\frac {5 \, x + e^{\left (x - 3\right )}}{{\left (x + e^{5}\right )} \log \left (x\right )}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.86, size = 19, normalized size = 0.79 \begin {gather*} \log {\left (\frac {5 x + e^{x - 3}}{\left (x + e^{5}\right ) \log {\left (x \right )}} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.14, size = 23, normalized size = 0.96 \begin {gather*} {\ln \left (\frac {5\,x+{\mathrm {e}}^{-3}\,{\mathrm {e}}^x}{\ln \left (x\right )\,\left (x+{\mathrm {e}}^5\right )}\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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