Integrand size = 42, antiderivative size = 21 \[ \int \frac {-1+e^3 x+\left (1-2 e^3 x\right ) \log (x)+\left (-1+2 e^3 x\right ) \log ^2(x)}{2 e^3 \log ^2(x)} \, dx=\frac {1}{2} \left (\frac {1}{e^3}-x\right ) \left (-x+\frac {x}{\log (x)}\right ) \]
[Out]
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 6874, 2357, 2367, 2335, 2346, 2209} \[ \int \frac {-1+e^3 x+\left (1-2 e^3 x\right ) \log (x)+\left (-1+2 e^3 x\right ) \log ^2(x)}{2 e^3 \log ^2(x)} \, dx=\frac {x^2}{2}-\frac {x}{2 e^3}+\frac {\left (1-e^3 x\right ) x}{2 e^3 \log (x)} \]
[In]
[Out]
Rule 12
Rule 2209
Rule 2335
Rule 2346
Rule 2357
Rule 2367
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-1+e^3 x+\left (1-2 e^3 x\right ) \log (x)+\left (-1+2 e^3 x\right ) \log ^2(x)}{\log ^2(x)} \, dx}{2 e^3} \\ & = \frac {\int \left (-1+2 e^3 x+\frac {-1+e^3 x}{\log ^2(x)}+\frac {1-2 e^3 x}{\log (x)}\right ) \, dx}{2 e^3} \\ & = -\frac {x}{2 e^3}+\frac {x^2}{2}+\frac {\int \frac {-1+e^3 x}{\log ^2(x)} \, dx}{2 e^3}+\frac {\int \frac {1-2 e^3 x}{\log (x)} \, dx}{2 e^3} \\ & = -\frac {x}{2 e^3}+\frac {x^2}{2}+\frac {x \left (1-e^3 x\right )}{2 e^3 \log (x)}+\frac {\int \left (\frac {1}{\log (x)}-\frac {2 e^3 x}{\log (x)}\right ) \, dx}{2 e^3}+\frac {\int \frac {1}{\log (x)} \, dx}{2 e^3}+\frac {\int \frac {-1+e^3 x}{\log (x)} \, dx}{e^3} \\ & = -\frac {x}{2 e^3}+\frac {x^2}{2}+\frac {x \left (1-e^3 x\right )}{2 e^3 \log (x)}+\frac {\operatorname {LogIntegral}(x)}{2 e^3}+\frac {\int \frac {1}{\log (x)} \, dx}{2 e^3}+\frac {\int \left (-\frac {1}{\log (x)}+\frac {e^3 x}{\log (x)}\right ) \, dx}{e^3}-\int \frac {x}{\log (x)} \, dx \\ & = -\frac {x}{2 e^3}+\frac {x^2}{2}+\frac {x \left (1-e^3 x\right )}{2 e^3 \log (x)}+\frac {\operatorname {LogIntegral}(x)}{e^3}-\frac {\int \frac {1}{\log (x)} \, dx}{e^3}+\int \frac {x}{\log (x)} \, dx-\text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right ) \\ & = -\frac {x}{2 e^3}+\frac {x^2}{2}-\operatorname {ExpIntegralEi}(2 \log (x))+\frac {x \left (1-e^3 x\right )}{2 e^3 \log (x)}+\text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right ) \\ & = -\frac {x}{2 e^3}+\frac {x^2}{2}+\frac {x \left (1-e^3 x\right )}{2 e^3 \log (x)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {-1+e^3 x+\left (1-2 e^3 x\right ) \log (x)+\left (-1+2 e^3 x\right ) \log ^2(x)}{2 e^3 \log ^2(x)} \, dx=\frac {x \left (-1+e^3 x\right ) (-1+\log (x))}{2 e^3 \log (x)} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14
method | result | size |
risch | \(-\frac {x \left (x +{\mathrm e}^{-3} \ln \left (x \right )-x \ln \left (x \right )-{\mathrm e}^{-3}\right )}{2 \ln \left (x \right )}\) | \(24\) |
parallelrisch | \(\frac {{\mathrm e}^{-3} \left ({\mathrm e}^{3} x^{2} \ln \left (x \right )-x^{2} {\mathrm e}^{3}-x \ln \left (x \right )+x \right )}{2 \ln \left (x \right )}\) | \(33\) |
norman | \(\frac {-\frac {x^{2}}{2}+\frac {x \,{\mathrm e}^{-3}}{2}+\frac {x^{2} \ln \left (x \right )}{2}-\frac {x \,{\mathrm e}^{-3} \ln \left (x \right )}{2}}{\ln \left (x \right )}\) | \(35\) |
parts | \(\frac {{\mathrm e}^{-3} \left (x^{2} {\mathrm e}^{3}-x \right )}{2}-\frac {x^{2}}{2 \ln \left (x \right )}+\frac {{\mathrm e}^{-3} x}{2 \ln \left (x \right )}\) | \(38\) |
default | \(\frac {{\mathrm e}^{-3} \left (x^{2} {\mathrm e}^{3}+2 \,{\mathrm e}^{3} \operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )-x +{\mathrm e}^{3} \left (-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )\right )+\frac {x}{\ln \left (x \right )}\right )}{2}\) | \(54\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {-1+e^3 x+\left (1-2 e^3 x\right ) \log (x)+\left (-1+2 e^3 x\right ) \log ^2(x)}{2 e^3 \log ^2(x)} \, dx=-\frac {{\left (x^{2} e^{3} - {\left (x^{2} e^{3} - x\right )} \log \left (x\right ) - x\right )} e^{\left (-3\right )}}{2 \, \log \left (x\right )} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {-1+e^3 x+\left (1-2 e^3 x\right ) \log (x)+\left (-1+2 e^3 x\right ) \log ^2(x)}{2 e^3 \log ^2(x)} \, dx=\frac {x^{2}}{2} - \frac {x}{2 e^{3}} + \frac {- x^{2} e^{3} + x}{2 e^{3} \log {\left (x \right )}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.10 \[ \int \frac {-1+e^3 x+\left (1-2 e^3 x\right ) \log (x)+\left (-1+2 e^3 x\right ) \log ^2(x)}{2 e^3 \log ^2(x)} \, dx=\frac {1}{2} \, {\left (x^{2} e^{3} - 2 \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) e^{3} + 2 \, e^{3} \Gamma \left (-1, -2 \, \log \left (x\right )\right ) - x + {\rm Ei}\left (\log \left (x\right )\right ) - \Gamma \left (-1, -\log \left (x\right )\right )\right )} e^{\left (-3\right )} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {-1+e^3 x+\left (1-2 e^3 x\right ) \log (x)+\left (-1+2 e^3 x\right ) \log ^2(x)}{2 e^3 \log ^2(x)} \, dx=\frac {1}{2} \, {\left (x^{2} e^{3} - \frac {x^{2} e^{3}}{\log \left (x\right )} - x + \frac {x}{\log \left (x\right )}\right )} e^{\left (-3\right )} \]
[In]
[Out]
Time = 13.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-1+e^3 x+\left (1-2 e^3 x\right ) \log (x)+\left (-1+2 e^3 x\right ) \log ^2(x)}{2 e^3 \log ^2(x)} \, dx=\frac {x\,{\mathrm {e}}^{-3}\,\left (\ln \left (x\right )-1\right )\,\left (x\,{\mathrm {e}}^3-1\right )}{2\,\ln \left (x\right )} \]
[In]
[Out]