Integrand size = 128, antiderivative size = 26 \[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=5+\log ^2\left (e^{6+\left (3+x-\log \left (16 e^{-2 x}\right )\right )^2}+x\right ) \] Output:
5+ln(exp(6+(x+3-ln(16/exp(x)^2))^2)+x)^2
Time = 0.39 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=\log ^2\left (e^{15+6 x+x^2-2 (3+x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right ) \] Input:
Integrate[((2 + E^(15 + 6*x + x^2 + (-6 - 2*x)*Log[16/E^(2*x)] + Log[16/E^ (2*x)]^2)*(36 + 12*x - 12*Log[16/E^(2*x)]))*Log[E^(15 + 6*x + x^2 + (-6 - 2*x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x])/(E^(15 + 6*x + x^2 + (-6 - 2*x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x),x]
Output:
Log[E^(15 + 6*x + x^2 - 2*(3 + x)*Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x ]^2
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (\left (12 x-12 \log \left (16 e^{-2 x}\right )+36\right ) \exp \left (x^2+6 x+\log ^2\left (16 e^{-2 x}\right )+(-2 x-6) \log \left (16 e^{-2 x}\right )+15\right )+2\right ) \log \left (\exp \left (x^2+6 x+\log ^2\left (16 e^{-2 x}\right )+(-2 x-6) \log \left (16 e^{-2 x}\right )+15\right )+x\right )}{\exp \left (x^2+6 x+\log ^2\left (16 e^{-2 x}\right )+(-2 x-6) \log \left (16 e^{-2 x}\right )+15\right )+x} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12 \left (x-\log \left (16 e^{-2 x}\right )+3\right ) \log \left (\exp \left (x^2+6 x+\log ^2\left (16 e^{-2 x}\right )-2 (x+3) \log \left (16 e^{-2 x}\right )+15\right )+x\right )-\frac {2^{8 x+25} \left (e^{-2 x}\right )^{2 x} \left (6 x^2+18 x-6 x \log \left (16 e^{-2 x}\right )-1\right ) \log \left (\exp \left (x^2+6 x+\log ^2\left (16 e^{-2 x}\right )-2 (x+3) \log \left (16 e^{-2 x}\right )+15\right )+x\right )}{e^{x^2+18 x+\log ^2\left (16 e^{-2 x}\right )+15}+2^{8 x+24} x \left (e^{-2 x}\right )^{2 x}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 36 \int \log \left (x+\exp \left (x^2+6 x+\log ^2\left (16 e^{-2 x}\right )-2 (x+3) \log \left (16 e^{-2 x}\right )+15\right )\right )dx+12 \int x \log \left (x+\exp \left (x^2+6 x+\log ^2\left (16 e^{-2 x}\right )-2 (x+3) \log \left (16 e^{-2 x}\right )+15\right )\right )dx+\int \frac {2^{8 x+25} \left (e^{-2 x}\right )^{2 x} \log \left (x+\exp \left (x^2+6 x+\log ^2\left (16 e^{-2 x}\right )-2 (x+3) \log \left (16 e^{-2 x}\right )+15\right )\right )}{256^{x+3} x \left (e^{-2 x}\right )^{2 x}+e^{x^2+18 x+\log ^2\left (16 e^{-2 x}\right )+15}}dx-9 \int \frac {2^{8 x+26} \left (e^{-2 x}\right )^{2 x} x \log \left (x+\exp \left (x^2+6 x+\log ^2\left (16 e^{-2 x}\right )-2 (x+3) \log \left (16 e^{-2 x}\right )+15\right )\right )}{256^{x+3} x \left (e^{-2 x}\right )^{2 x}+e^{x^2+18 x+\log ^2\left (16 e^{-2 x}\right )+15}}dx-3 \int \frac {2^{8 x+26} \left (e^{-2 x}\right )^{2 x} x^2 \log \left (x+\exp \left (x^2+6 x+\log ^2\left (16 e^{-2 x}\right )-2 (x+3) \log \left (16 e^{-2 x}\right )+15\right )\right )}{256^{x+3} x \left (e^{-2 x}\right )^{2 x}+e^{x^2+18 x+\log ^2\left (16 e^{-2 x}\right )+15}}dx-12 \int \log \left (16 e^{-2 x}\right ) \log \left (x+\exp \left (x^2+6 x+\log ^2\left (16 e^{-2 x}\right )-2 (x+3) \log \left (16 e^{-2 x}\right )+15\right )\right )dx+3 \int \frac {2^{8 x+26} \left (e^{-2 x}\right )^{2 x} x \log \left (16 e^{-2 x}\right ) \log \left (x+\exp \left (x^2+6 x+\log ^2\left (16 e^{-2 x}\right )-2 (x+3) \log \left (16 e^{-2 x}\right )+15\right )\right )}{256^{x+3} x \left (e^{-2 x}\right )^{2 x}+e^{x^2+18 x+\log ^2\left (16 e^{-2 x}\right )+15}}dx\) |
Input:
Int[((2 + E^(15 + 6*x + x^2 + (-6 - 2*x)*Log[16/E^(2*x)] + Log[16/E^(2*x)] ^2)*(36 + 12*x - 12*Log[16/E^(2*x)]))*Log[E^(15 + 6*x + x^2 + (-6 - 2*x)*L og[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x])/(E^(15 + 6*x + x^2 + (-6 - 2*x)* Log[16/E^(2*x)] + Log[16/E^(2*x)]^2) + x),x]
Output:
$Aborted
Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42
\[\ln \left ({\mathrm e}^{\ln \left (16 \,{\mathrm e}^{-2 x}\right )^{2}+\left (-2 x -6\right ) \ln \left (16 \,{\mathrm e}^{-2 x}\right )+x^{2}+6 x +15}+x \right )^{2}\]
Input:
int(((-12*ln(16/exp(x)^2)+12*x+36)*exp(ln(16/exp(x)^2)^2+(-2*x-6)*ln(16/ex p(x)^2)+x^2+6*x+15)+2)*ln(exp(ln(16/exp(x)^2)^2+(-2*x-6)*ln(16/exp(x)^2)+x ^2+6*x+15)+x)/(exp(ln(16/exp(x)^2)^2+(-2*x-6)*ln(16/exp(x)^2)+x^2+6*x+15)+ x),x)
Output:
ln(exp(ln(16/exp(x)^2)^2+(-2*x-6)*ln(16/exp(x)^2)+x^2+6*x+15)+x)^2
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=\log \left (x + e^{\left (9 \, x^{2} - 24 \, {\left (x + 1\right )} \log \left (2\right ) + 16 \, \log \left (2\right )^{2} + 18 \, x + 15\right )}\right )^{2} \] Input:
integrate(((-12*log(16/exp(x)^2)+12*x+36)*exp(log(16/exp(x)^2)^2+(-2*x-6)* log(16/exp(x)^2)+x^2+6*x+15)+2)*log(exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16 /exp(x)^2)+x^2+6*x+15)+x)/(exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)^2 )+x^2+6*x+15)+x),x, algorithm="fricas")
Output:
log(x + e^(9*x^2 - 24*(x + 1)*log(2) + 16*log(2)^2 + 18*x + 15))^2
Timed out. \[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=\text {Timed out} \] Input:
integrate(((-12*ln(16/exp(x)**2)+12*x+36)*exp(ln(16/exp(x)**2)**2+(-2*x-6) *ln(16/exp(x)**2)+x**2+6*x+15)+2)*ln(exp(ln(16/exp(x)**2)**2+(-2*x-6)*ln(1 6/exp(x)**2)+x**2+6*x+15)+x)/(exp(ln(16/exp(x)**2)**2+(-2*x-6)*ln(16/exp(x )**2)+x**2+6*x+15)+x),x)
Output:
Timed out
\[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=\int { \frac {2 \, {\left (6 \, {\left (x - \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + 3\right )} e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )} + 1\right )} \log \left (x + e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )}\right )}{x + e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )}} \,d x } \] Input:
integrate(((-12*log(16/exp(x)^2)+12*x+36)*exp(log(16/exp(x)^2)^2+(-2*x-6)* log(16/exp(x)^2)+x^2+6*x+15)+2)*log(exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16 /exp(x)^2)+x^2+6*x+15)+x)/(exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)^2 )+x^2+6*x+15)+x),x, algorithm="maxima")
Output:
2*integrate((6*(x - log(16*e^(-2*x)) + 3)*e^(x^2 - 2*(x + 3)*log(16*e^(-2* x)) + log(16*e^(-2*x))^2 + 6*x + 15) + 1)*log(x + e^(x^2 - 2*(x + 3)*log(1 6*e^(-2*x)) + log(16*e^(-2*x))^2 + 6*x + 15))/(x + e^(x^2 - 2*(x + 3)*log( 16*e^(-2*x)) + log(16*e^(-2*x))^2 + 6*x + 15)), x)
\[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=\int { \frac {2 \, {\left (6 \, {\left (x - \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + 3\right )} e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )} + 1\right )} \log \left (x + e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )}\right )}{x + e^{\left (x^{2} - 2 \, {\left (x + 3\right )} \log \left (16 \, e^{\left (-2 \, x\right )}\right ) + \log \left (16 \, e^{\left (-2 \, x\right )}\right )^{2} + 6 \, x + 15\right )}} \,d x } \] Input:
integrate(((-12*log(16/exp(x)^2)+12*x+36)*exp(log(16/exp(x)^2)^2+(-2*x-6)* log(16/exp(x)^2)+x^2+6*x+15)+2)*log(exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16 /exp(x)^2)+x^2+6*x+15)+x)/(exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)^2 )+x^2+6*x+15)+x),x, algorithm="giac")
Output:
integrate(2*(6*(x - log(16*e^(-2*x)) + 3)*e^(x^2 - 2*(x + 3)*log(16*e^(-2* x)) + log(16*e^(-2*x))^2 + 6*x + 15) + 1)*log(x + e^(x^2 - 2*(x + 3)*log(1 6*e^(-2*x)) + log(16*e^(-2*x))^2 + 6*x + 15))/(x + e^(x^2 - 2*(x + 3)*log( 16*e^(-2*x)) + log(16*e^(-2*x))^2 + 6*x + 15)), x)
Time = 7.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.54 \[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=576\,{\ln \left (2\right )}^2\,x^2-48\,\ln \left (2\right )\,x\,\ln \left (16777216\,2^{24\,x}\,x+{\mathrm {e}}^{18\,x}\,{\mathrm {e}}^{15}\,{\mathrm {e}}^{16\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{9\,x^2}\right )+1152\,{\ln \left (2\right )}^2\,x+{\ln \left (16777216\,2^{24\,x}\,x+{\mathrm {e}}^{18\,x}\,{\mathrm {e}}^{15}\,{\mathrm {e}}^{16\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{9\,x^2}\right )}^2-48\,\ln \left (2\right )\,\ln \left (16777216\,2^{24\,x}\,x+{\mathrm {e}}^{18\,x}\,{\mathrm {e}}^{15}\,{\mathrm {e}}^{16\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{9\,x^2}\right ) \] Input:
int((log(x + exp(6*x + log(16*exp(-2*x))^2 - log(16*exp(-2*x))*(2*x + 6) + x^2 + 15))*(exp(6*x + log(16*exp(-2*x))^2 - log(16*exp(-2*x))*(2*x + 6) + x^2 + 15)*(12*x - 12*log(16*exp(-2*x)) + 36) + 2))/(x + exp(6*x + log(16* exp(-2*x))^2 - log(16*exp(-2*x))*(2*x + 6) + x^2 + 15)),x)
Output:
576*x^2*log(2)^2 + log(16777216*2^(24*x)*x + exp(18*x)*exp(15)*exp(16*log( 2)^2)*exp(9*x^2))^2 + 1152*x*log(2)^2 - 48*log(16777216*2^(24*x)*x + exp(1 8*x)*exp(15)*exp(16*log(2)^2)*exp(9*x^2))*log(2) - 48*x*log(16777216*2^(24 *x)*x + exp(18*x)*exp(15)*exp(16*log(2)^2)*exp(9*x^2))*log(2)
\[ \int \frac {\left (2+e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )} \left (36+12 x-12 \log \left (16 e^{-2 x}\right )\right )\right ) \log \left (e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x\right )}{e^{15+6 x+x^2+(-6-2 x) \log \left (16 e^{-2 x}\right )+\log ^2\left (16 e^{-2 x}\right )}+x} \, dx=\int \frac {\left (\left (-12 \,\mathrm {log}\left (\frac {16}{\left ({\mathrm e}^{x}\right )^{2}}\right )+12 x +36\right ) {\mathrm e}^{{\mathrm {log}\left (\frac {16}{\left ({\mathrm e}^{x}\right )^{2}}\right )}^{2}+\left (-2 x -6\right ) \mathrm {log}\left (\frac {16}{\left ({\mathrm e}^{x}\right )^{2}}\right )+x^{2}+6 x +15}+2\right ) \mathrm {log}\left ({\mathrm e}^{{\mathrm {log}\left (\frac {16}{\left ({\mathrm e}^{x}\right )^{2}}\right )}^{2}+\left (-2 x -6\right ) \mathrm {log}\left (\frac {16}{\left ({\mathrm e}^{x}\right )^{2}}\right )+x^{2}+6 x +15}+x \right )}{{\mathrm e}^{{\mathrm {log}\left (\frac {16}{\left ({\mathrm e}^{x}\right )^{2}}\right )}^{2}+\left (-2 x -6\right ) \mathrm {log}\left (\frac {16}{\left ({\mathrm e}^{x}\right )^{2}}\right )+x^{2}+6 x +15}+x}d x \] Input:
int(((-12*log(16/exp(x)^2)+12*x+36)*exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16 /exp(x)^2)+x^2+6*x+15)+2)*log(exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x )^2)+x^2+6*x+15)+x)/(exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)^2)+x^2+ 6*x+15)+x),x)
Output:
int(((-12*log(16/exp(x)^2)+12*x+36)*exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16 /exp(x)^2)+x^2+6*x+15)+2)*log(exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x )^2)+x^2+6*x+15)+x)/(exp(log(16/exp(x)^2)^2+(-2*x-6)*log(16/exp(x)^2)+x^2+ 6*x+15)+x),x)