Optimal. Leaf size=26 \[ 2-\frac {x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \]
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Rubi [F]
time = 5.93, 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 {4 x^2+e^x \left (1-4 x+4 x^2\right )+\left (-6 x^2+12 x^3\right ) \log \left (\frac {-5+10 x}{2 x}\right )}{e^{3 x} (-1+2 x)+e^{2 x} \left (-6 x^2+12 x^3\right ) \log \left (\frac {-5+10 x}{2 x}\right )+e^x \left (-12 x^4+24 x^5\right ) \log ^2\left (\frac {-5+10 x}{2 x}\right )+\left (-8 x^6+16 x^7\right ) \log ^3\left (\frac {-5+10 x}{2 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 {-e^x (1-2 x)^2-4 x^2-6 x^2 (-1+2 x) \log \left (5-\frac {5}{2 x}\right )}{(1-2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx\\ &=\int \left (\frac {-1+2 x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2}-\frac {4 x^2 \left (-1+2 \log \left (5-\frac {5}{2 x}\right )-5 x \log \left (5-\frac {5}{2 x}\right )+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}\right ) \, dx\\ &=-\left (4 \int \frac {x^2 \left (-1+2 \log \left (5-\frac {5}{2 x}\right )-5 x \log \left (5-\frac {5}{2 x}\right )+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx\right )+\int \frac {-1+2 x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx\\ &=-\left (4 \int \frac {x^2 \left (1-\left (2-5 x+2 x^2\right ) \log \left (5-\frac {5}{2 x}\right )\right )}{(1-2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx\right )+\int \left (-\frac {1}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2}+\frac {2 x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx-4 \int \left (\frac {-1+2 \log \left (5-\frac {5}{2 x}\right )-5 x \log \left (5-\frac {5}{2 x}\right )+2 x^2 \log \left (5-\frac {5}{2 x}\right )}{4 \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}+\frac {x \left (-1+2 \log \left (5-\frac {5}{2 x}\right )-5 x \log \left (5-\frac {5}{2 x}\right )+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )}{2 \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}+\frac {-1+2 \log \left (5-\frac {5}{2 x}\right )-5 x \log \left (5-\frac {5}{2 x}\right )+2 x^2 \log \left (5-\frac {5}{2 x}\right )}{4 (-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}\right ) \, dx-\int \frac {1}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx\\ &=2 \int \frac {x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx-2 \int \frac {x \left (-1+2 \log \left (5-\frac {5}{2 x}\right )-5 x \log \left (5-\frac {5}{2 x}\right )+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-\int \frac {1}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx-\int \frac {-1+2 \log \left (5-\frac {5}{2 x}\right )-5 x \log \left (5-\frac {5}{2 x}\right )+2 x^2 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-\int \frac {-1+2 \log \left (5-\frac {5}{2 x}\right )-5 x \log \left (5-\frac {5}{2 x}\right )+2 x^2 \log \left (5-\frac {5}{2 x}\right )}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx\\ &=2 \int \frac {x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx-2 \int \frac {x \left (-1+\left (2-5 x+2 x^2\right ) \log \left (5-\frac {5}{2 x}\right )\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-\int \frac {1}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx-\int \frac {1-\left (2-5 x+2 x^2\right ) \log \left (5-\frac {5}{2 x}\right )}{(1-2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-\int \frac {-1+\left (2-5 x+2 x^2\right ) \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx\\ &=2 \int \frac {x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx-2 \int \left (-\frac {x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}+\frac {2 x \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}-\frac {5 x^2 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}+\frac {2 x^3 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}\right ) \, dx-\int \frac {1}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx-\int \left (-\frac {1}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}+\frac {2 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}-\frac {5 x \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}+\frac {2 x^2 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}\right ) \, dx-\int \left (-\frac {1}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}+\frac {2 \log \left (5-\frac {5}{2 x}\right )}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}-\frac {5 x \log \left (5-\frac {5}{2 x}\right )}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}+\frac {2 x^2 \log \left (5-\frac {5}{2 x}\right )}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}\right ) \, dx\\ &=2 \int \frac {x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-2 \int \frac {\log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-2 \int \frac {x^2 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-2 \int \frac {\log \left (5-\frac {5}{2 x}\right )}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-2 \int \frac {x^2 \log \left (5-\frac {5}{2 x}\right )}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+2 \int \frac {x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx-4 \int \frac {x \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-4 \int \frac {x^3 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+5 \int \frac {x \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+5 \int \frac {x \log \left (5-\frac {5}{2 x}\right )}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+10 \int \frac {x^2 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+\int \frac {1}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+\int \frac {1}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-\int \frac {1}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx\\ &=2 \int \frac {x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-2 \int \frac {\log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-2 \int \frac {x^2 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-2 \int \frac {\log \left (5-\frac {5}{2 x}\right )}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+2 \int \frac {x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx-2 \int \left (\frac {\log \left (5-\frac {5}{2 x}\right )}{4 \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}+\frac {x \log \left (5-\frac {5}{2 x}\right )}{2 \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}+\frac {\log \left (5-\frac {5}{2 x}\right )}{4 (-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}\right ) \, dx-4 \int \frac {x \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-4 \int \frac {x^3 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+5 \int \frac {x \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+5 \int \left (\frac {\log \left (5-\frac {5}{2 x}\right )}{2 \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}+\frac {\log \left (5-\frac {5}{2 x}\right )}{2 (-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3}\right ) \, dx+10 \int \frac {x^2 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+\int \frac {1}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+\int \frac {1}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-\int \frac {1}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx\right )-\frac {1}{2} \int \frac {\log \left (5-\frac {5}{2 x}\right )}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+2 \int \frac {x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-2 \int \frac {\log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-2 \int \frac {x^2 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-2 \int \frac {\log \left (5-\frac {5}{2 x}\right )}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+2 \int \frac {x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx+\frac {5}{2} \int \frac {\log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+\frac {5}{2} \int \frac {\log \left (5-\frac {5}{2 x}\right )}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-4 \int \frac {x \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-4 \int \frac {x^3 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+5 \int \frac {x \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+10 \int \frac {x^2 \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+\int \frac {1}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx+\int \frac {1}{(-1+2 x) \left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-\int \frac {x \log \left (5-\frac {5}{2 x}\right )}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^3} \, dx-\int \frac {1}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.15, size = 24, normalized size = 0.92 \begin {gather*} -\frac {x}{\left (e^x+2 x^2 \log \left (5-\frac {5}{2 x}\right )\right )^2} \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 = 0.79, size = 134, normalized size = 5.15
method | result | size |
risch | \(-\frac {x}{\left (2 x^{2} \ln \left (x -\frac {1}{2}\right )-i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (x -\frac {1}{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{2}\right )}{x}\right )+i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{2}\right )}{x}\right )^{2}+i \pi \,x^{2} \mathrm {csgn}\left (i \left (x -\frac {1}{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{2}\right )}{x}\right )^{2}-i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (x -\frac {1}{2}\right )}{x}\right )^{3}+2 x^{2} \ln \left (5\right )-2 x^{2} \ln \left (x \right )+{\mathrm e}^{x}\right )^{2}}\) | \(134\) |
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. 123 vs.
\(2 (23) = 46\).
time = 0.61, size = 123, normalized size = 4.73 \begin {gather*} -\frac {x}{4 \, x^{4} \log \left (2 \, x - 1\right )^{2} - 8 \, x^{4} {\left (\log \left (5\right ) - \log \left (2\right )\right )} \log \left (x\right ) + 4 \, x^{4} \log \left (x\right )^{2} + 4 \, {\left (\log \left (5\right )^{2} - 2 \, \log \left (5\right ) \log \left (2\right ) + \log \left (2\right )^{2}\right )} x^{4} + 4 \, {\left (x^{2} {\left (\log \left (5\right ) - \log \left (2\right )\right )} - x^{2} \log \left (x\right )\right )} e^{x} + 4 \, {\left (2 \, x^{4} {\left (\log \left (5\right ) - \log \left (2\right )\right )} - 2 \, x^{4} \log \left (x\right ) + x^{2} e^{x}\right )} \log \left (2 \, x - 1\right ) + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 46, normalized size = 1.77 \begin {gather*} -\frac {x}{4 \, x^{4} \log \left (\frac {5 \, {\left (2 \, x - 1\right )}}{2 \, x}\right )^{2} + 4 \, x^{2} e^{x} \log \left (\frac {5 \, {\left (2 \, x - 1\right )}}{2 \, x}\right ) + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs.
\(2 (20) = 40\).
time = 0.11, size = 42, normalized size = 1.62 \begin {gather*} - \frac {x}{4 x^{4} \log {\left (\frac {5 x - \frac {5}{2}}{x} \right )}^{2} + 4 x^{2} e^{x} \log {\left (\frac {5 x - \frac {5}{2}}{x} \right )} + e^{2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 46, normalized size = 1.77 \begin {gather*} -\frac {x}{4 \, x^{4} \log \left (\frac {5 \, {\left (2 \, x - 1\right )}}{2 \, x}\right )^{2} + 4 \, x^{2} e^{x} \log \left (\frac {5 \, {\left (2 \, x - 1\right )}}{2 \, x}\right ) + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.15, size = 116, normalized size = 4.46 \begin {gather*} \frac {{\mathrm {e}}^x\,\left (-4\,x^5+12\,x^4-9\,x^3+2\,x^2\right )+2\,x^4-4\,x^5}{\left ({\mathrm {e}}^{2\,x}+4\,x^4\,{\ln \left (\frac {5\,x-\frac {5}{2}}{x}\right )}^2+4\,x^2\,{\mathrm {e}}^x\,\ln \left (\frac {5\,x-\frac {5}{2}}{x}\right )\right )\,\left (9\,x^2\,{\mathrm {e}}^x-12\,x^3\,{\mathrm {e}}^x+4\,x^4\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^x-2\,x^3+4\,x^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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