Integrand size = 53, antiderivative size = 24 \[ \int \frac {(80-20 x) \log \left (\frac {1}{3} e^{-16+8 x-x^2} \left (-1-15 e^{16-8 x+x^2}\right )\right )}{3+45 e^{16-8 x+x^2}} \, dx=\frac {5}{3} \log ^2\left (-5-\frac {1}{3} e^{-(4-x)^2}\right ) \]
[Out]
Time = 0.69 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {6873, 12, 6847, 2320, 36, 29, 31, 6820, 6818} \[ \int \frac {(80-20 x) \log \left (\frac {1}{3} e^{-16+8 x-x^2} \left (-1-15 e^{16-8 x+x^2}\right )\right )}{3+45 e^{16-8 x+x^2}} \, dx=\frac {5}{3} \log ^2\left (-\frac {1}{3} e^{-(4-x)^2}-5\right ) \]
[In]
[Out]
Rule 12
Rule 29
Rule 31
Rule 36
Rule 2320
Rule 6818
Rule 6820
Rule 6847
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \int \frac {(80-20 x) \log \left (\frac {1}{3} e^{-16+8 x-x^2} \left (-1-15 e^{16-8 x+x^2}\right )\right )}{3 \left (1+15 e^{(-4+x)^2}\right )} \, dx \\ & = \frac {1}{3} \int \frac {(80-20 x) \log \left (\frac {1}{3} e^{-16+8 x-x^2} \left (-1-15 e^{16-8 x+x^2}\right )\right )}{1+15 e^{(-4+x)^2}} \, dx \\ & = \frac {1}{3} \int \frac {20 (4-x) \log \left (-5-\frac {1}{3} e^{-(-4+x)^2}\right )}{1+15 e^{(-4+x)^2}} \, dx \\ & = \frac {20}{3} \int \frac {(4-x) \log \left (-5-\frac {1}{3} e^{-(-4+x)^2}\right )}{1+15 e^{(-4+x)^2}} \, dx \\ & = \frac {5}{3} \log ^2\left (-5-\frac {1}{3} e^{-(4-x)^2}\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {(80-20 x) \log \left (\frac {1}{3} e^{-16+8 x-x^2} \left (-1-15 e^{16-8 x+x^2}\right )\right )}{3+45 e^{16-8 x+x^2}} \, dx=\frac {5}{3} \log ^2\left (-5-\frac {1}{3} e^{-(-4+x)^2}\right ) \]
[In]
[Out]
Time = 0.54 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33
method | result | size |
norman | \(\frac {5 {\ln \left (\frac {\left (-15 \,{\mathrm e}^{x^{2}-8 x +16}-1\right ) {\mathrm e}^{-x^{2}+8 x -16}}{3}\right )}^{2}}{3}\) | \(32\) |
parallelrisch | \(\frac {5 {\ln \left (-\frac {\left (15 \,{\mathrm e}^{x^{2}-8 x +16}+1\right ) {\mathrm e}^{-x^{2}+8 x -16}}{3}\right )}^{2}}{3}\) | \(32\) |
risch | \(\text {Expression too large to display}\) | \(890\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {(80-20 x) \log \left (\frac {1}{3} e^{-16+8 x-x^2} \left (-1-15 e^{16-8 x+x^2}\right )\right )}{3+45 e^{16-8 x+x^2}} \, dx=\frac {5}{3} \, \log \left (-\frac {1}{3} \, {\left (15 \, e^{\left (x^{2} - 8 \, x + 16\right )} + 1\right )} e^{\left (-x^{2} + 8 \, x - 16\right )}\right )^{2} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {(80-20 x) \log \left (\frac {1}{3} e^{-16+8 x-x^2} \left (-1-15 e^{16-8 x+x^2}\right )\right )}{3+45 e^{16-8 x+x^2}} \, dx=\frac {5 \log {\left (\left (- 5 e^{x^{2} - 8 x + 16} - \frac {1}{3}\right ) e^{- x^{2} + 8 x - 16} \right )}^{2}}{3} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (17) = 34\).
Time = 0.35 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.71 \[ \int \frac {(80-20 x) \log \left (\frac {1}{3} e^{-16+8 x-x^2} \left (-1-15 e^{16-8 x+x^2}\right )\right )}{3+45 e^{16-8 x+x^2}} \, dx=-\frac {5}{3} \, x^{4} - \frac {10}{3} \, x^{2} {\left (\log \left (5\right ) + \log \left (3\right ) + 16\right )} - \frac {10}{3} \, {\left (x^{2} - \log \left (\frac {1}{15} \, {\left (15 \, e^{\left (x^{2} + 16\right )} + e^{\left (8 \, x\right )}\right )} e^{\left (-16\right )}\right )\right )} \log \left (-\frac {1}{3} \, {\left (15 \, e^{\left (x^{2} - 8 \, x + 16\right )} + 1\right )} e^{\left (-x^{2} + 8 \, x - 16\right )}\right ) + \frac {10}{3} \, {\left (x^{2} + \log \left (5\right ) + \log \left (3\right ) + 16\right )} \log \left (15 \, e^{\left (x^{2} + 16\right )} + e^{\left (8 \, x\right )}\right ) - \frac {5}{3} \, \log \left (15 \, e^{\left (x^{2} + 16\right )} + e^{\left (8 \, x\right )}\right )^{2} \]
[In]
[Out]
\[ \int \frac {(80-20 x) \log \left (\frac {1}{3} e^{-16+8 x-x^2} \left (-1-15 e^{16-8 x+x^2}\right )\right )}{3+45 e^{16-8 x+x^2}} \, dx=\int { -\frac {20 \, {\left (x - 4\right )} \log \left (-\frac {1}{3} \, {\left (15 \, e^{\left (x^{2} - 8 \, x + 16\right )} + 1\right )} e^{\left (-x^{2} + 8 \, x - 16\right )}\right )}{3 \, {\left (15 \, e^{\left (x^{2} - 8 \, x + 16\right )} + 1\right )}} \,d x } \]
[In]
[Out]
Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {(80-20 x) \log \left (\frac {1}{3} e^{-16+8 x-x^2} \left (-1-15 e^{16-8 x+x^2}\right )\right )}{3+45 e^{16-8 x+x^2}} \, dx=\frac {5\,{\ln \left (-\frac {{\mathrm {e}}^{8\,x}\,{\mathrm {e}}^{-16}\,{\mathrm {e}}^{-x^2}}{3}-5\right )}^2}{3} \]
[In]
[Out]