Integrand size = 106, antiderivative size = 19 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=x+\frac {5}{x \left (13+\log ^2(4)-\log (x)\right )} \]
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Time = 0.33 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6, 6820, 6874, 2343, 2346, 2209} \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=x+\frac {5}{x \left (-\log (x)+13+\log ^2(4)\right )} \]
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Rule 6
Rule 2209
Rule 2343
Rule 2346
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{x^2 \log ^4(4)+x^2 \left (169+26 \log ^2(4)\right )+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx \\ & = \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{x^2 \left (169+26 \log ^2(4)+\log ^4(4)\right )+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx \\ & = \int \frac {-60+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \left (169+\log ^4(4)\right )+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{x^2 \left (169+26 \log ^2(4)+\log ^4(4)\right )+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx \\ & = \int \frac {-5 \left (12+\log ^2(4)\right )+x^2 \left (13+\log ^2(4)\right )^2+\left (5-2 x^2 \left (13+\log ^2(4)\right )\right ) \log (x)+x^2 \log ^2(x)}{x^2 \left (13 \left (1+\frac {\log ^2(4)}{13}\right )-\log (x)\right )^2} \, dx \\ & = \int \left (1+\frac {5}{x^2 \left (13 \left (1+\frac {\log ^2(4)}{13}\right )-\log (x)\right )^2}+\frac {5}{x^2 \left (-13 \left (1+\frac {\log ^2(4)}{13}\right )+\log (x)\right )}\right ) \, dx \\ & = x+5 \int \frac {1}{x^2 \left (13 \left (1+\frac {\log ^2(4)}{13}\right )-\log (x)\right )^2} \, dx+5 \int \frac {1}{x^2 \left (-13 \left (1+\frac {\log ^2(4)}{13}\right )+\log (x)\right )} \, dx \\ & = x+\frac {5}{x \left (13+\log ^2(4)-\log (x)\right )}+5 \int \frac {1}{x^2 \left (13 \left (1+\frac {\log ^2(4)}{13}\right )-\log (x)\right )} \, dx+5 \text {Subst}\left (\int \frac {e^{-x}}{x-13 \left (1+\frac {\log ^2(4)}{13}\right )} \, dx,x,\log (x)\right ) \\ & = x+5 e^{-13-\log ^2(4)} \text {Ei}\left (13+\log ^2(4)-\log (x)\right )+\frac {5}{x \left (13+\log ^2(4)-\log (x)\right )}+5 \text {Subst}\left (\int \frac {e^{-x}}{-x+13 \left (1+\frac {\log ^2(4)}{13}\right )} \, dx,x,\log (x)\right ) \\ & = x+\frac {5}{x \left (13+\log ^2(4)-\log (x)\right )} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=x-\frac {5}{x \left (-13-\log ^2(4)+\log (x)\right )} \]
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Time = 0.52 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16
method | result | size |
risch | \(x +\frac {5}{\left (13-\ln \left (x \right )+4 \ln \left (2\right )^{2}\right ) x}\) | \(22\) |
norman | \(\frac {5+\left (4 \ln \left (2\right )^{2}+13\right ) x^{2}-x^{2} \ln \left (x \right )}{x \left (13-\ln \left (x \right )+4 \ln \left (2\right )^{2}\right )}\) | \(40\) |
parallelrisch | \(-\frac {-5-4 x^{2} \ln \left (2\right )^{2}+x^{2} \ln \left (x \right )-13 x^{2}}{x \left (13-\ln \left (x \right )+4 \ln \left (2\right )^{2}\right )}\) | \(42\) |
default | \(-\frac {20 \ln \left (2\right )^{2}}{x \left (13-\ln \left (x \right )+4 \ln \left (2\right )^{2}\right )}+\frac {20 \ln \left (2\right )^{2}+65}{x \left (13-\ln \left (x \right )+4 \ln \left (2\right )^{2}\right )}-\frac {60}{\left (13-\ln \left (x \right )+4 \ln \left (2\right )^{2}\right ) x}+x\) | \(72\) |
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Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=\frac {4 \, x^{2} \log \left (2\right )^{2} - x^{2} \log \left (x\right ) + 13 \, x^{2} + 5}{4 \, x \log \left (2\right )^{2} - x \log \left (x\right ) + 13 \, x} \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=x - \frac {5}{x \log {\left (x \right )} - 13 x - 4 x \log {\left (2 \right )}^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=\frac {{\left (4 \, \log \left (2\right )^{2} + 13\right )} x^{2} - x^{2} \log \left (x\right ) + 5}{{\left (4 \, \log \left (2\right )^{2} + 13\right )} x - x \log \left (x\right )} \]
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Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=x + \frac {5}{4 \, x \log \left (2\right )^{2} - x \log \left (x\right ) + 13 \, x} \]
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Time = 11.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=x+\frac {5}{x\,\left (4\,{\ln \left (2\right )}^2-\ln \left (x\right )+13\right )} \]
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