Integrand size = 28, antiderivative size = 17 \[ \int \frac {4 x^2+4 x \log (x)+2 x \log ^2(x)+2 \log ^3(x)}{x} \, dx=\frac {1}{2} \left (\log (4)+\left (2 x+\log ^2(x)\right )^2\right ) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {14, 2332, 2333, 2339, 30} \[ \int \frac {4 x^2+4 x \log (x)+2 x \log ^2(x)+2 \log ^3(x)}{x} \, dx=2 x^2+\frac {\log ^4(x)}{2}+2 x \log ^2(x) \]
[In]
[Out]
Rule 14
Rule 30
Rule 2332
Rule 2333
Rule 2339
Rubi steps \begin{align*} \text {integral}& = \int \left (4 x+4 \log (x)+2 \log ^2(x)+\frac {2 \log ^3(x)}{x}\right ) \, dx \\ & = 2 x^2+2 \int \log ^2(x) \, dx+2 \int \frac {\log ^3(x)}{x} \, dx+4 \int \log (x) \, dx \\ & = -4 x+2 x^2+4 x \log (x)+2 x \log ^2(x)+2 \text {Subst}\left (\int x^3 \, dx,x,\log (x)\right )-4 \int \log (x) \, dx \\ & = 2 x^2+2 x \log ^2(x)+\frac {\log ^4(x)}{2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int \frac {4 x^2+4 x \log (x)+2 x \log ^2(x)+2 \log ^3(x)}{x} \, dx=2 x^2+2 x \log ^2(x)+\frac {\log ^4(x)}{2} \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18
method | result | size |
default | \(\frac {\ln \left (x \right )^{4}}{2}+2 x \ln \left (x \right )^{2}+2 x^{2}\) | \(20\) |
norman | \(\frac {\ln \left (x \right )^{4}}{2}+2 x \ln \left (x \right )^{2}+2 x^{2}\) | \(20\) |
risch | \(\frac {\ln \left (x \right )^{4}}{2}+2 x \ln \left (x \right )^{2}+2 x^{2}\) | \(20\) |
parts | \(\frac {\ln \left (x \right )^{4}}{2}+2 x \ln \left (x \right )^{2}+2 x^{2}\) | \(20\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {4 x^2+4 x \log (x)+2 x \log ^2(x)+2 \log ^3(x)}{x} \, dx=\frac {1}{2} \, \log \left (x\right )^{4} + 2 \, x \log \left (x\right )^{2} + 2 \, x^{2} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {4 x^2+4 x \log (x)+2 x \log ^2(x)+2 \log ^3(x)}{x} \, dx=2 x^{2} + 2 x \log {\left (x \right )}^{2} + \frac {\log {\left (x \right )}^{4}}{2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).
Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {4 x^2+4 x \log (x)+2 x \log ^2(x)+2 \log ^3(x)}{x} \, dx=\frac {1}{2} \, \log \left (x\right )^{4} + 2 \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x + 2 \, x^{2} + 4 \, x \log \left (x\right ) - 4 \, x \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {4 x^2+4 x \log (x)+2 x \log ^2(x)+2 \log ^3(x)}{x} \, dx=\frac {1}{2} \, \log \left (x\right )^{4} + 2 \, x \log \left (x\right )^{2} + 2 \, x^{2} \]
[In]
[Out]
Time = 8.15 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \frac {4 x^2+4 x \log (x)+2 x \log ^2(x)+2 \log ^3(x)}{x} \, dx=\frac {{\left ({\ln \left (x\right )}^2+2\,x\right )}^2}{2} \]
[In]
[Out]