Integrand size = 48, antiderivative size = 21 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=\log \left (\frac {9-\log ^2(36)+\frac {\log ^2(x)}{x^2}}{x}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6, 2641, 6874, 2621} \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=\log \left (x^2 \left (9-\log ^2(36)\right )+\log ^2(x)\right )-3 \log (x) \]
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Rule 6
Rule 2621
Rule 2641
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 \left (-9+\log ^2(36)\right )+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx \\ & = \int \frac {x^2 \left (-9+\log ^2(36)\right )+2 \log (x)-3 \log ^2(x)}{x^3 \left (9-\log ^2(36)\right )+x \log ^2(x)} \, dx \\ & = \int \frac {x^2 \left (-9+\log ^2(36)\right )+2 \log (x)-3 \log ^2(x)}{x \left (x^2 \left (9-\log ^2(36)\right )+\log ^2(x)\right )} \, dx \\ & = \int \left (-\frac {3}{x}+\frac {2 \left (9 x^2 \left (1-\frac {\log ^2(36)}{9}\right )+\log (x)\right )}{x \left (9 x^2 \left (1-\frac {\log ^2(36)}{9}\right )+\log ^2(x)\right )}\right ) \, dx \\ & = -3 \log (x)+2 \int \frac {9 x^2 \left (1-\frac {\log ^2(36)}{9}\right )+\log (x)}{x \left (9 x^2 \left (1-\frac {\log ^2(36)}{9}\right )+\log ^2(x)\right )} \, dx \\ & = -3 \log (x)+\log \left (x^2 \left (9-\log ^2(36)\right )+\log ^2(x)\right ) \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=-3 \log (x)+\log \left (x^2 \left (-9+\log ^2(36)\right )-\log ^2(x)\right ) \]
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Time = 3.92 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33
method | result | size |
default | \(-3 \ln \left (x \right )+\ln \left (4 x^{2} \ln \left (6\right )^{2}-9 x^{2}-\ln \left (x \right )^{2}\right )\) | \(28\) |
norman | \(-3 \ln \left (x \right )+\ln \left (4 x^{2} \ln \left (6\right )^{2}-9 x^{2}-\ln \left (x \right )^{2}\right )\) | \(28\) |
parallelrisch | \(\ln \left (\frac {4 x^{2} \ln \left (6\right )^{2}-9 x^{2}-\ln \left (x \right )^{2}}{4 \ln \left (6\right )^{2}-9}\right )-3 \ln \left (x \right )\) | \(39\) |
risch | \(-3 \ln \left (x \right )+\ln \left (-4 x^{2} \ln \left (2\right )^{2}-8 \ln \left (2\right ) \ln \left (3\right ) x^{2}-4 x^{2} \ln \left (3\right )^{2}+9 x^{2}+\ln \left (x \right )^{2}\right )\) | \(44\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=\log \left (-4 \, x^{2} \log \left (6\right )^{2} + 9 \, x^{2} + \log \left (x\right )^{2}\right ) - 3 \, \log \left (x\right ) \]
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Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=- 3 \log {\left (x \right )} + \log {\left (- 4 x^{2} \log {\left (6 \right )}^{2} + 9 x^{2} + \log {\left (x \right )}^{2} \right )} \]
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Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=\log \left (-{\left (4 \, \log \left (3\right )^{2} + 8 \, \log \left (3\right ) \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 9\right )} x^{2} + \log \left (x\right )^{2}\right ) - 3 \, \log \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=\log \left (4 \, x^{2} \log \left (6\right )^{2} - 9 \, x^{2} - \log \left (x\right )^{2}\right ) - 3 \, \log \left (x\right ) \]
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Time = 8.66 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-9 x^2+x^2 \log ^2(36)+2 \log (x)-3 \log ^2(x)}{9 x^3-x^3 \log ^2(36)+x \log ^2(x)} \, dx=\ln \left ({\ln \left (x\right )}^2-4\,x^2\,{\ln \left (6\right )}^2+9\,x^2\right )-3\,\ln \left (x\right ) \]
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