Integrand size = 61, antiderivative size = 22 \[ \int \frac {-8 x^2+2 x^3+3 x^4-x^5+(2-x) \log (5)+\left (-8 x^2+4 x^3+9 x^4-4 x^5-2 \log (5)\right ) \log (x)}{x^2} \, dx=(-2+x) x \left (4+x-x^2-\frac {\log (5)}{x^2}\right ) \log (x) \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(22)=44\).
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.45, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.082, Rules used = {14, 1634, 2404, 2332, 2341} \[ \int \frac {-8 x^2+2 x^3+3 x^4-x^5+(2-x) \log (5)+\left (-8 x^2+4 x^3+9 x^4-4 x^5-2 \log (5)\right ) \log (x)}{x^2} \, dx=x^4 (-\log (x))+3 x^3 \log (x)+2 x^2 \log (x)-8 x \log (x)-\log (5) \log (x)+\frac {\log (25) \log (x)}{x}+\frac {\log (25)}{x}-\frac {2 \log (5)}{x} \]
[In]
[Out]
Rule 14
Rule 1634
Rule 2332
Rule 2341
Rule 2404
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {(-2+x) \left (-4 x^2-x^3+x^4+\log (5)\right )}{x^2}-\frac {\left (8 x^2-4 x^3-9 x^4+4 x^5+\log (25)\right ) \log (x)}{x^2}\right ) \, dx \\ & = -\int \frac {(-2+x) \left (-4 x^2-x^3+x^4+\log (5)\right )}{x^2} \, dx-\int \frac {\left (8 x^2-4 x^3-9 x^4+4 x^5+\log (25)\right ) \log (x)}{x^2} \, dx \\ & = -\int \left (8-2 x-3 x^2+x^3-\frac {2 \log (5)}{x^2}+\frac {\log (5)}{x}\right ) \, dx-\int \left (8 \log (x)-4 x \log (x)-9 x^2 \log (x)+4 x^3 \log (x)+\frac {\log (25) \log (x)}{x^2}\right ) \, dx \\ & = -8 x+x^2+x^3-\frac {x^4}{4}-\frac {2 \log (5)}{x}-\log (5) \log (x)+4 \int x \log (x) \, dx-4 \int x^3 \log (x) \, dx-8 \int \log (x) \, dx+9 \int x^2 \log (x) \, dx-\log (25) \int \frac {\log (x)}{x^2} \, dx \\ & = -\frac {2 \log (5)}{x}+\frac {\log (25)}{x}-8 x \log (x)+2 x^2 \log (x)+3 x^3 \log (x)-x^4 \log (x)-\log (5) \log (x)+\frac {\log (25) \log (x)}{x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {-8 x^2+2 x^3+3 x^4-x^5+(2-x) \log (5)+\left (-8 x^2+4 x^3+9 x^4-4 x^5-2 \log (5)\right ) \log (x)}{x^2} \, dx=-8 x \log (x)+2 x^2 \log (x)+3 x^3 \log (x)-x^4 \log (x)-\log (5) \log (x)+\frac {2 \log (5) \log (x)}{x} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77
method | result | size |
risch | \(\frac {\left (-x^{5}+3 x^{4}+2 x^{3}-8 x^{2}+2 \ln \left (5\right )\right ) \ln \left (x \right )}{x}-\ln \left (5\right ) \ln \left (x \right )\) | \(39\) |
parallelrisch | \(-\frac {x^{5} \ln \left (x \right )-3 x^{4} \ln \left (x \right )-2 x^{3} \ln \left (x \right )+x \ln \left (5\right ) \ln \left (x \right )+8 x^{2} \ln \left (x \right )-2 \ln \left (5\right ) \ln \left (x \right )}{x}\) | \(46\) |
norman | \(\frac {-x \ln \left (5\right ) \ln \left (x \right )-8 x^{2} \ln \left (x \right )+2 x^{3} \ln \left (x \right )+3 x^{4} \ln \left (x \right )-x^{5} \ln \left (x \right )+2 \ln \left (5\right ) \ln \left (x \right )}{x}\) | \(47\) |
default | \(-x^{4} \ln \left (x \right )+3 x^{3} \ln \left (x \right )+2 x^{2} \ln \left (x \right )-8 x \ln \left (x \right )-2 \ln \left (5\right ) \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )-\ln \left (5\right ) \ln \left (x \right )-\frac {2 \ln \left (5\right )}{x}\) | \(58\) |
parts | \(-x^{4} \ln \left (x \right )+3 x^{3} \ln \left (x \right )+2 x^{2} \ln \left (x \right )-8 x \ln \left (x \right )-2 \ln \left (5\right ) \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )-\ln \left (5\right ) \ln \left (x \right )-\frac {2 \ln \left (5\right )}{x}\) | \(58\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {-8 x^2+2 x^3+3 x^4-x^5+(2-x) \log (5)+\left (-8 x^2+4 x^3+9 x^4-4 x^5-2 \log (5)\right ) \log (x)}{x^2} \, dx=-\frac {{\left (x^{5} - 3 \, x^{4} - 2 \, x^{3} + 8 \, x^{2} + {\left (x - 2\right )} \log \left (5\right )\right )} \log \left (x\right )}{x} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {-8 x^2+2 x^3+3 x^4-x^5+(2-x) \log (5)+\left (-8 x^2+4 x^3+9 x^4-4 x^5-2 \log (5)\right ) \log (x)}{x^2} \, dx=- \log {\left (5 \right )} \log {\left (x \right )} + \frac {\left (- x^{5} + 3 x^{4} + 2 x^{3} - 8 x^{2} + 2 \log {\left (5 \right )}\right ) \log {\left (x \right )}}{x} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).
Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.45 \[ \int \frac {-8 x^2+2 x^3+3 x^4-x^5+(2-x) \log (5)+\left (-8 x^2+4 x^3+9 x^4-4 x^5-2 \log (5)\right ) \log (x)}{x^2} \, dx=-x^{4} \log \left (x\right ) + 3 \, x^{3} \log \left (x\right ) + 2 \, x^{2} \log \left (x\right ) + 2 \, {\left (\frac {\log \left (x\right )}{x} + \frac {1}{x}\right )} \log \left (5\right ) - 8 \, x \log \left (x\right ) - \log \left (5\right ) \log \left (x\right ) - \frac {2 \, \log \left (5\right )}{x} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {-8 x^2+2 x^3+3 x^4-x^5+(2-x) \log (5)+\left (-8 x^2+4 x^3+9 x^4-4 x^5-2 \log (5)\right ) \log (x)}{x^2} \, dx=-{\left (x^{4} - 3 \, x^{3} - 2 \, x^{2} + 8 \, x - \frac {2 \, \log \left (5\right )}{x}\right )} \log \left (x\right ) - \log \left (5\right ) \log \left (x\right ) \]
[In]
[Out]
Time = 9.43 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {-8 x^2+2 x^3+3 x^4-x^5+(2-x) \log (5)+\left (-8 x^2+4 x^3+9 x^4-4 x^5-2 \log (5)\right ) \log (x)}{x^2} \, dx=-\frac {\ln \left (x\right )\,\left (x-2\right )\,\left (x^4-x^3-4\,x^2+\ln \left (5\right )\right )}{x} \]
[In]
[Out]