Integrand size = 46, antiderivative size = 21 \[ \int \frac {5 e^{5-x}+\left (5+e^{5-x} (-5-5 x)\right ) \log (x)+(5-5 \log (x)) \log (5 x)}{2 x^2} \, dx=\frac {5 \log (x) \left (e^{5-x}+\log (5 x)\right )}{2 x} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {12, 14, 2326, 2341, 2340, 2413} \[ \int \frac {5 e^{5-x}+\left (5+e^{5-x} (-5-5 x)\right ) \log (x)+(5-5 \log (x)) \log (5 x)}{2 x^2} \, dx=\frac {5 \log (5 x) \log (x)}{2 x}+\frac {5 e^{5-x} \log (x)}{2 x} \]
[In]
[Out]
Rule 12
Rule 14
Rule 2326
Rule 2340
Rule 2341
Rule 2413
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {5 e^{5-x}+\left (5+e^{5-x} (-5-5 x)\right ) \log (x)+(5-5 \log (x)) \log (5 x)}{x^2} \, dx \\ & = \frac {1}{2} \int \left (-\frac {5 e^{5-x} (-1+\log (x)+x \log (x))}{x^2}-\frac {5 (-\log (x)-\log (5 x)+\log (x) \log (5 x))}{x^2}\right ) \, dx \\ & = -\left (\frac {5}{2} \int \frac {e^{5-x} (-1+\log (x)+x \log (x))}{x^2} \, dx\right )-\frac {5}{2} \int \frac {-\log (x)-\log (5 x)+\log (x) \log (5 x)}{x^2} \, dx \\ & = \frac {5 e^{5-x} \log (x)}{2 x}-\frac {5}{2} \int \left (-\frac {\log (x)}{x^2}+\frac {(-1+\log (x)) \log (5 x)}{x^2}\right ) \, dx \\ & = \frac {5 e^{5-x} \log (x)}{2 x}+\frac {5}{2} \int \frac {\log (x)}{x^2} \, dx-\frac {5}{2} \int \frac {(-1+\log (x)) \log (5 x)}{x^2} \, dx \\ & = -\frac {5}{2 x}-\frac {5 \log (x)}{2 x}+\frac {5 e^{5-x} \log (x)}{2 x}+\frac {5 \log (x) \log (5 x)}{2 x}-\frac {5}{2} \int \frac {\log (x)}{x^2} \, dx \\ & = \frac {5 e^{5-x} \log (x)}{2 x}+\frac {5 \log (x) \log (5 x)}{2 x} \\ \end{align*}
Time = 1.55 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {5 e^{5-x}+\left (5+e^{5-x} (-5-5 x)\right ) \log (x)+(5-5 \log (x)) \log (5 x)}{2 x^2} \, dx=\frac {5 e^{-x} \log (x) \left (e^5+e^x \log (5 x)\right )}{2 x} \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19
method | result | size |
parallelrisch | \(\frac {10 \ln \left (x \right ) \ln \left (5 x \right )+10 \ln \left (x \right ) {\mathrm e}^{5-x}}{4 x}\) | \(25\) |
risch | \(\frac {5 \ln \left (x \right )^{2}}{2 x}+\frac {5 \left (2 \ln \left (5\right )+2 \,{\mathrm e}^{5-x}\right ) \ln \left (x \right )}{4 x}\) | \(31\) |
default | \(-\frac {5 \ln \left (5\right ) \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )}{2}+\frac {5 \ln \left (x \right )^{2}}{2 x}+\frac {5 \ln \left (x \right )}{2 x}+\frac {5 \ln \left (x \right ) {\mathrm e}^{5-x}}{2 x}-\frac {5 \ln \left (5 x \right )}{2 x}\) | \(57\) |
parts | \(-\frac {5 \ln \left (5\right ) \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )}{2}+\frac {5 \ln \left (x \right )^{2}}{2 x}+\frac {5 \ln \left (x \right )}{2 x}+\frac {5 \ln \left (x \right ) {\mathrm e}^{5-x}}{2 x}-\frac {5 \ln \left (5 x \right )}{2 x}\) | \(57\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {5 e^{5-x}+\left (5+e^{5-x} (-5-5 x)\right ) \log (x)+(5-5 \log (x)) \log (5 x)}{2 x^2} \, dx=\frac {5 \, {\left ({\left (e^{\left (-x + 5\right )} + \log \left (5\right )\right )} \log \left (x\right ) + \log \left (x\right )^{2}\right )}}{2 \, x} \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \frac {5 e^{5-x}+\left (5+e^{5-x} (-5-5 x)\right ) \log (x)+(5-5 \log (x)) \log (5 x)}{2 x^2} \, dx=\frac {5 e^{5 - x} \log {\left (x \right )}}{2 x} + \frac {5 \log {\left (x \right )}^{2}}{2 x} + \frac {5 \log {\left (5 \right )} \log {\left (x \right )}}{2 x} \]
[In]
[Out]
\[ \int \frac {5 e^{5-x}+\left (5+e^{5-x} (-5-5 x)\right ) \log (x)+(5-5 \log (x)) \log (5 x)}{2 x^2} \, dx=\int { -\frac {5 \, {\left ({\left (\log \left (x\right ) - 1\right )} \log \left (5 \, x\right ) + {\left ({\left (x + 1\right )} e^{\left (-x + 5\right )} - 1\right )} \log \left (x\right ) - e^{\left (-x + 5\right )}\right )}}{2 \, x^{2}} \,d x } \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {5 e^{5-x}+\left (5+e^{5-x} (-5-5 x)\right ) \log (x)+(5-5 \log (x)) \log (5 x)}{2 x^2} \, dx=\frac {5 \, {\left (e^{\left (-x + 5\right )} \log \left (x\right ) + \log \left (5\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )}}{2 \, x} \]
[In]
[Out]
Time = 13.61 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {5 e^{5-x}+\left (5+e^{5-x} (-5-5 x)\right ) \log (x)+(5-5 \log (x)) \log (5 x)}{2 x^2} \, dx=\frac {5\,{\mathrm {e}}^{-x}\,\ln \left (x\right )\,\left ({\mathrm {e}}^5+{\mathrm {e}}^x\,\ln \left (x\right )+{\mathrm {e}}^x\,\ln \left (5\right )\right )}{2\,x} \]
[In]
[Out]