Integrand size = 34, antiderivative size = 20 \[ \int \frac {3-6 e^{-11 x} \log (5)+e^{-11 x} (-6+66 x) \log (5) \log (x)}{2 \log (5)} \, dx=x \left (\frac {3}{2 \log (5)}-3 e^{-11 x} \log (x)\right ) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 2225, 2207, 2634} \[ \int \frac {3-6 e^{-11 x} \log (5)+e^{-11 x} (-6+66 x) \log (5) \log (x)}{2 \log (5)} \, dx=\frac {3 x}{2 \log (5)}-\frac {3}{11} e^{-11 x} \log (x)+\frac {3}{11} e^{-11 x} (1-11 x) \log (x) \]
[In]
[Out]
Rule 12
Rule 2207
Rule 2225
Rule 2634
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (3-6 e^{-11 x} \log (5)+e^{-11 x} (-6+66 x) \log (5) \log (x)\right ) \, dx}{2 \log (5)} \\ & = \frac {3 x}{2 \log (5)}+\frac {1}{2} \int e^{-11 x} (-6+66 x) \log (x) \, dx-3 \int e^{-11 x} \, dx \\ & = \frac {3 e^{-11 x}}{11}+\frac {3 x}{2 \log (5)}-\frac {3}{11} e^{-11 x} \log (x)+\frac {3}{11} e^{-11 x} (1-11 x) \log (x)-\frac {1}{2} \int -6 e^{-11 x} \, dx \\ & = \frac {3 e^{-11 x}}{11}+\frac {3 x}{2 \log (5)}-\frac {3}{11} e^{-11 x} \log (x)+\frac {3}{11} e^{-11 x} (1-11 x) \log (x)+3 \int e^{-11 x} \, dx \\ & = \frac {3 x}{2 \log (5)}-\frac {3}{11} e^{-11 x} \log (x)+\frac {3}{11} e^{-11 x} (1-11 x) \log (x) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {3-6 e^{-11 x} \log (5)+e^{-11 x} (-6+66 x) \log (5) \log (x)}{2 \log (5)} \, dx=\frac {3 \left (x-2 e^{-11 x} x \log (5) \log (x)\right )}{\log (25)} \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
method | result | size |
norman | \(\frac {3 x}{2 \ln \left (5\right )}-3 x \,{\mathrm e}^{-11 x} \ln \left (x \right )\) | \(18\) |
risch | \(\frac {3 x}{2 \ln \left (5\right )}-3 x \,{\mathrm e}^{-11 x} \ln \left (x \right )\) | \(18\) |
parts | \(\frac {3 x}{2 \ln \left (5\right )}-3 x \,{\mathrm e}^{-11 x} \ln \left (x \right )\) | \(18\) |
default | \(\frac {3 x -6 \ln \left (5\right ) {\mathrm e}^{-11 x} \ln \left (x \right ) x}{2 \ln \left (5\right )}\) | \(22\) |
parallelrisch | \(\frac {3 x -6 \ln \left (5\right ) {\mathrm e}^{-11 x} \ln \left (x \right ) x}{2 \ln \left (5\right )}\) | \(22\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {3-6 e^{-11 x} \log (5)+e^{-11 x} (-6+66 x) \log (5) \log (x)}{2 \log (5)} \, dx=-\frac {3 \, {\left (2 \, x e^{\left (-11 \, x\right )} \log \left (5\right ) \log \left (x\right ) - x\right )}}{2 \, \log \left (5\right )} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {3-6 e^{-11 x} \log (5)+e^{-11 x} (-6+66 x) \log (5) \log (x)}{2 \log (5)} \, dx=\frac {3 x}{2 \log {\left (5 \right )}} - 3 x e^{- 11 x} \log {\left (x \right )} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {3-6 e^{-11 x} \log (5)+e^{-11 x} (-6+66 x) \log (5) \log (x)}{2 \log (5)} \, dx=-\frac {3 \, {\left (2 \, x e^{\left (-11 \, x\right )} \log \left (5\right ) \log \left (x\right ) - x\right )}}{2 \, \log \left (5\right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {3-6 e^{-11 x} \log (5)+e^{-11 x} (-6+66 x) \log (5) \log (x)}{2 \log (5)} \, dx=-\frac {3 \, {\left (2 \, {\left (11 \, x e^{\left (-11 \, x\right )} \log \left (x\right ) + e^{\left (-11 \, x\right )}\right )} \log \left (5\right ) - 2 \, e^{\left (-11 \, x\right )} \log \left (5\right ) - 11 \, x\right )}}{22 \, \log \left (5\right )} \]
[In]
[Out]
Time = 13.58 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {3-6 e^{-11 x} \log (5)+e^{-11 x} (-6+66 x) \log (5) \log (x)}{2 \log (5)} \, dx=\frac {3\,x}{2\,\ln \left (5\right )}-3\,x\,{\mathrm {e}}^{-11\,x}\,\ln \left (x\right ) \]
[In]
[Out]