Optimal. Leaf size=26 \[ e^4-x+\frac {1}{5} \log \left (x+\frac {4}{\log \left (\frac {324}{(-4+x)^2}\right )}\right ) \]
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Rubi [F] time = 1.31, 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 {8+(80-20 x) \log \left (\frac {324}{16-8 x+x^2}\right )+\left (-4+21 x-5 x^2\right ) \log ^2\left (\frac {324}{16-8 x+x^2}\right )}{(-80+20 x) \log \left (\frac {324}{16-8 x+x^2}\right )+\left (-20 x+5 x^2\right ) \log ^2\left (\frac {324}{16-8 x+x^2}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8-(80-20 x) \log \left (\frac {324}{16-8 x+x^2}\right )-\left (-4+21 x-5 x^2\right ) \log ^2\left (\frac {324}{16-8 x+x^2}\right )}{5 (4-x) \log \left (\frac {324}{(-4+x)^2}\right ) \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )} \, dx\\ &=\frac {1}{5} \int \frac {-8-(80-20 x) \log \left (\frac {324}{16-8 x+x^2}\right )-\left (-4+21 x-5 x^2\right ) \log ^2\left (\frac {324}{16-8 x+x^2}\right )}{(4-x) \log \left (\frac {324}{(-4+x)^2}\right ) \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )} \, dx\\ &=\frac {1}{5} \int \left (-\frac {20}{4+x \log \left (\frac {324}{(-4+x)^2}\right )}+\frac {8}{(-4+x) \log \left (\frac {324}{(-4+x)^2}\right ) \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )}-\frac {(-1+5 x) \log \left (\frac {324}{(-4+x)^2}\right )}{4+x \log \left (\frac {324}{(-4+x)^2}\right )}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {(-1+5 x) \log \left (\frac {324}{(-4+x)^2}\right )}{4+x \log \left (\frac {324}{(-4+x)^2}\right )} \, dx\right )+\frac {8}{5} \int \frac {1}{(-4+x) \log \left (\frac {324}{(-4+x)^2}\right ) \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )} \, dx-4 \int \frac {1}{4+x \log \left (\frac {324}{(-4+x)^2}\right )} \, dx\\ &=-\left (\frac {1}{5} \int \left (\frac {-1+5 x}{x}-\frac {4 (-1+5 x)}{x \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )}\right ) \, dx\right )+\frac {8}{5} \int \left (\frac {1}{4 (-4+x) \log \left (\frac {324}{(-4+x)^2}\right )}-\frac {x}{4 (-4+x) \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )}\right ) \, dx-4 \int \frac {1}{4+x \log \left (\frac {324}{(-4+x)^2}\right )} \, dx\\ &=-\left (\frac {1}{5} \int \frac {-1+5 x}{x} \, dx\right )+\frac {2}{5} \int \frac {1}{(-4+x) \log \left (\frac {324}{(-4+x)^2}\right )} \, dx-\frac {2}{5} \int \frac {x}{(-4+x) \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )} \, dx+\frac {4}{5} \int \frac {-1+5 x}{x \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )} \, dx-4 \int \frac {1}{4+x \log \left (\frac {324}{(-4+x)^2}\right )} \, dx\\ &=-\left (\frac {1}{5} \int \left (5-\frac {1}{x}\right ) \, dx\right )-\frac {2}{5} \int \left (\frac {1}{4+x \log \left (\frac {324}{(-4+x)^2}\right )}+\frac {4}{(-4+x) \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )}\right ) \, dx+\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{x \log \left (\frac {324}{x^2}\right )} \, dx,x,-4+x\right )+\frac {4}{5} \int \left (\frac {5}{4+x \log \left (\frac {324}{(-4+x)^2}\right )}-\frac {1}{x \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )}\right ) \, dx-4 \int \frac {1}{4+x \log \left (\frac {324}{(-4+x)^2}\right )} \, dx\\ &=-x+\frac {\log (x)}{5}-\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {324}{(-4+x)^2}\right )\right )-\frac {2}{5} \int \frac {1}{4+x \log \left (\frac {324}{(-4+x)^2}\right )} \, dx-\frac {4}{5} \int \frac {1}{x \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )} \, dx-\frac {8}{5} \int \frac {1}{(-4+x) \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )} \, dx\\ &=-x+\frac {\log (x)}{5}-\frac {1}{5} \log \left (\log \left (\frac {324}{(4-x)^2}\right )\right )-\frac {2}{5} \int \frac {1}{4+x \log \left (\frac {324}{(-4+x)^2}\right )} \, dx-\frac {4}{5} \int \frac {1}{x \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )} \, dx-\frac {8}{5} \int \frac {1}{(-4+x) \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.45, size = 33, normalized size = 1.27 \begin {gather*} \frac {1}{5} \left (20-5 x-\log \left (\log \left (\frac {324}{(-4+x)^2}\right )\right )+\log \left (4+x \log \left (\frac {324}{(-4+x)^2}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 48, normalized size = 1.85 \begin {gather*} -x + \frac {1}{5} \, \log \relax (x) + \frac {1}{5} \, \log \left (\frac {x \log \left (\frac {324}{x^{2} - 8 \, x + 16}\right ) + 4}{x}\right ) - \frac {1}{5} \, \log \left (\log \left (\frac {324}{x^{2} - 8 \, x + 16}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 40, normalized size = 1.54 \begin {gather*} -x + \frac {1}{5} \, \log \left (x \log \left (\frac {324}{x^{2} - 8 \, x + 16}\right ) + 4\right ) - \frac {1}{5} \, \log \left (\log \left (\frac {324}{x^{2} - 8 \, x + 16}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 41, normalized size = 1.58
method | result | size |
norman | \(-x -\frac {\ln \left (\ln \left (\frac {324}{x^{2}-8 x +16}\right )\right )}{5}+\frac {\ln \left (\ln \left (\frac {324}{x^{2}-8 x +16}\right ) x +4\right )}{5}\) | \(41\) |
risch | \(-x +\frac {\ln \relax (x )}{5}-\frac {\ln \left (\ln \left (\frac {324}{x^{2}-8 x +16}\right )\right )}{5}+\frac {\ln \left (\ln \left (\frac {324}{x^{2}-8 x +16}\right )+\frac {4}{x}\right )}{5}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.67, size = 50, normalized size = 1.92 \begin {gather*} -x + \frac {1}{5} \, \log \relax (x) + \frac {1}{5} \, \log \left (-\frac {x {\left (2 \, \log \relax (3) + \log \relax (2)\right )} - x \log \left (x - 4\right ) + 2}{x}\right ) - \frac {1}{5} \, \log \left (-2 \, \log \relax (3) - \log \relax (2) + \log \left (x - 4\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 104, normalized size = 4.00 \begin {gather*} \frac {\ln \left (x^2+2\,x-8\right )}{5}-\frac {\ln \left (\frac {16\,x\,\ln \left (18\right )-32\,\ln \left (\frac {1}{{\left (x-4\right )}^2}\right )-64\,\ln \left (18\right )+8\,x\,\ln \left (\frac {1}{{\left (x-4\right )}^2}\right )+8\,x^2\,\ln \left (18\right )+4\,x^2\,\ln \left (\frac {1}{{\left (x-4\right )}^2}\right )}{x^2\,\left (x-4\right )}\right )}{5}-x-\frac {\ln \relax (x)}{5}+\frac {\ln \left (\frac {8\,x\,\ln \left (18\right )+4\,x\,\ln \left (\frac {1}{{\left (x-4\right )}^2}\right )+16}{x\,\left (x-4\right )}\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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