Integrand size = 29, antiderivative size = 21 \[ \int \frac {-2+x^2+3 x^4-4 x^3 \log (3)+x^2 \log ^2(3)}{x^2} \, dx=6+x+\frac {2 (1+x)}{x}+x (-x+\log (3))^2 \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {6, 14} \[ \int \frac {-2+x^2+3 x^4-4 x^3 \log (3)+x^2 \log ^2(3)}{x^2} \, dx=x^3-2 x^2 \log (3)+\frac {2}{x}+x \left (1+\log ^2(3)\right ) \]
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Rule 6
Rule 14
Rubi steps \begin{align*} \text {integral}& = \int \frac {-2+3 x^4-4 x^3 \log (3)+x^2 \left (1+\log ^2(3)\right )}{x^2} \, dx \\ & = \int \left (1-\frac {2}{x^2}+3 x^2-4 x \log (3)+\log ^2(3)\right ) \, dx \\ & = \frac {2}{x}+x^3-2 x^2 \log (3)+x \left (1+\log ^2(3)\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {-2+x^2+3 x^4-4 x^3 \log (3)+x^2 \log ^2(3)}{x^2} \, dx=\frac {2}{x}+x+x^3-2 x^2 \log (3)+x \log ^2(3) \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14
method | result | size |
default | \(x^{3}-2 x^{2} \ln \left (3\right )+x \ln \left (3\right )^{2}+x +\frac {2}{x}\) | \(24\) |
risch | \(x^{3}-2 x^{2} \ln \left (3\right )+x \ln \left (3\right )^{2}+x +\frac {2}{x}\) | \(24\) |
norman | \(\frac {x^{4}-2 x^{3} \ln \left (3\right )+\left (\ln \left (3\right )^{2}+1\right ) x^{2}+2}{x}\) | \(27\) |
gosper | \(\frac {x^{2} \ln \left (3\right )^{2}-2 x^{3} \ln \left (3\right )+x^{4}+x^{2}+2}{x}\) | \(28\) |
parallelrisch | \(\frac {x^{2} \ln \left (3\right )^{2}-2 x^{3} \ln \left (3\right )+x^{4}+x^{2}+2}{x}\) | \(28\) |
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none
Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {-2+x^2+3 x^4-4 x^3 \log (3)+x^2 \log ^2(3)}{x^2} \, dx=\frac {x^{4} - 2 \, x^{3} \log \left (3\right ) + x^{2} \log \left (3\right )^{2} + x^{2} + 2}{x} \]
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Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {-2+x^2+3 x^4-4 x^3 \log (3)+x^2 \log ^2(3)}{x^2} \, dx=x^{3} - 2 x^{2} \log {\left (3 \right )} + x \left (1 + \log {\left (3 \right )}^{2}\right ) + \frac {2}{x} \]
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none
Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {-2+x^2+3 x^4-4 x^3 \log (3)+x^2 \log ^2(3)}{x^2} \, dx=x^{3} - 2 \, x^{2} \log \left (3\right ) + {\left (\log \left (3\right )^{2} + 1\right )} x + \frac {2}{x} \]
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {-2+x^2+3 x^4-4 x^3 \log (3)+x^2 \log ^2(3)}{x^2} \, dx=x^{3} - 2 \, x^{2} \log \left (3\right ) + x \log \left (3\right )^{2} + x + \frac {2}{x} \]
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Time = 8.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {-2+x^2+3 x^4-4 x^3 \log (3)+x^2 \log ^2(3)}{x^2} \, dx=x\,\left ({\ln \left (3\right )}^2+1\right )-2\,x^2\,\ln \left (3\right )+\frac {2}{x}+x^3 \]
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