Integrand size = 200, antiderivative size = 26 \[ \int \frac {-e^{3 x}+e^{2 x} (-1-2 \log (4))-3 x \log (4)-\log ^2(4)+e^x \left (-3 x-2 \log (4)-\log ^2(4)\right )+\left (-e^{3 x} x-3 x \log (4)-2 e^{2 x} x \log (4)+e^x \left (-3 x+3 x^2-x \log ^2(4)\right )\right ) \log (x)}{3 \left (e^{4 x} x+9 x^3+6 x^2 \log (4)+x \log ^2(4)+e^{3 x} (2 x+2 x \log (4))+e^{2 x} \left (x+6 x^2+4 x \log (4)+x \log ^2(4)\right )+e^x \left (6 x^2+\left (2 x+6 x^2\right ) \log (4)+2 x \log ^2(4)\right )\right ) \log ^2(x)} \, dx=\frac {1}{3 \left (1+e^x+\frac {3 x}{e^x+\log (4)}\right ) \log (x)} \] Output:
1/3/ln(x)/(1+3*x/(exp(x)+2*ln(2))+exp(x))
\[ \int \frac {-e^{3 x}+e^{2 x} (-1-2 \log (4))-3 x \log (4)-\log ^2(4)+e^x \left (-3 x-2 \log (4)-\log ^2(4)\right )+\left (-e^{3 x} x-3 x \log (4)-2 e^{2 x} x \log (4)+e^x \left (-3 x+3 x^2-x \log ^2(4)\right )\right ) \log (x)}{3 \left (e^{4 x} x+9 x^3+6 x^2 \log (4)+x \log ^2(4)+e^{3 x} (2 x+2 x \log (4))+e^{2 x} \left (x+6 x^2+4 x \log (4)+x \log ^2(4)\right )+e^x \left (6 x^2+\left (2 x+6 x^2\right ) \log (4)+2 x \log ^2(4)\right )\right ) \log ^2(x)} \, dx=\int \frac {-e^{3 x}+e^{2 x} (-1-2 \log (4))-3 x \log (4)-\log ^2(4)+e^x \left (-3 x-2 \log (4)-\log ^2(4)\right )+\left (-e^{3 x} x-3 x \log (4)-2 e^{2 x} x \log (4)+e^x \left (-3 x+3 x^2-x \log ^2(4)\right )\right ) \log (x)}{3 \left (e^{4 x} x+9 x^3+6 x^2 \log (4)+x \log ^2(4)+e^{3 x} (2 x+2 x \log (4))+e^{2 x} \left (x+6 x^2+4 x \log (4)+x \log ^2(4)\right )+e^x \left (6 x^2+\left (2 x+6 x^2\right ) \log (4)+2 x \log ^2(4)\right )\right ) \log ^2(x)} \, dx \] Input:
Integrate[(-E^(3*x) + E^(2*x)*(-1 - 2*Log[4]) - 3*x*Log[4] - Log[4]^2 + E^ x*(-3*x - 2*Log[4] - Log[4]^2) + (-(E^(3*x)*x) - 3*x*Log[4] - 2*E^(2*x)*x* Log[4] + E^x*(-3*x + 3*x^2 - x*Log[4]^2))*Log[x])/(3*(E^(4*x)*x + 9*x^3 + 6*x^2*Log[4] + x*Log[4]^2 + E^(3*x)*(2*x + 2*x*Log[4]) + E^(2*x)*(x + 6*x^ 2 + 4*x*Log[4] + x*Log[4]^2) + E^x*(6*x^2 + (2*x + 6*x^2)*Log[4] + 2*x*Log [4]^2))*Log[x]^2),x]
Output:
Integrate[(-E^(3*x) + E^(2*x)*(-1 - 2*Log[4]) - 3*x*Log[4] - Log[4]^2 + E^ x*(-3*x - 2*Log[4] - Log[4]^2) + (-(E^(3*x)*x) - 3*x*Log[4] - 2*E^(2*x)*x* Log[4] + E^x*(-3*x + 3*x^2 - x*Log[4]^2))*Log[x])/((E^(4*x)*x + 9*x^3 + 6* x^2*Log[4] + x*Log[4]^2 + E^(3*x)*(2*x + 2*x*Log[4]) + E^(2*x)*(x + 6*x^2 + 4*x*Log[4] + x*Log[4]^2) + E^x*(6*x^2 + (2*x + 6*x^2)*Log[4] + 2*x*Log[4 ]^2))*Log[x]^2), x]/3
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.
| \(\displaystyle \int \frac {\left (e^x \left (3 x^2-3 x-x \log ^2(4)\right )-e^{3 x} x-2 e^{2 x} x \log (4)-3 x \log (4)\right ) \log (x)-e^{3 x}+e^x \left (-3 x-\log ^2(4)-2 \log (4)\right )-3 x \log (4)+e^{2 x} (-1-2 \log (4))-\log ^2(4)}{3 \left (9 x^3+e^{2 x} \left (6 x^2+x+x \log ^2(4)+4 x \log (4)\right )+e^x \left (6 x^2+\left (6 x^2+2 x\right ) \log (4)+2 x \log ^2(4)\right )+6 x^2 \log (4)+e^{4 x} x+x \log ^2(4)+e^{3 x} (2 x+2 x \log (4))\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int -\frac {3 \log (4) x+e^{3 x}+e^x (3 x+\log (4) (2+\log (4)))+\left (e^{3 x} x+2 e^{2 x} \log (4) x+3 \log (4) x+e^x \left (-3 x^2+\log ^2(4) x+3 x\right )\right ) \log (x)+e^{2 x} (1+\log (16))+\log ^2(4)}{\left (9 x^3+6 \log (4) x^2+e^{4 x} x+e^{3 x} (2+\log (16)) x+\log ^2(4) x+e^{2 x} \left (6 x^2+\log ^2(4) x+4 \log (4) x+x\right )+2 e^x \left (3 x^2+\log ^2(4) x+\left (3 x^2+x\right ) \log (4)\right )\right ) \log ^2(x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{3} \int \frac {3 \log (4) x+e^{3 x}+e^x (3 x+\log (4) (2+\log (4)))+\left (e^{3 x} x+2 e^{2 x} \log (4) x+3 \log (4) x+e^x \left (-3 x^2+\log ^2(4) x+3 x\right )\right ) \log (x)+e^{2 x} (1+\log (16))+\log ^2(4)}{\left (9 x^3+6 \log (4) x^2+e^{4 x} x+e^{3 x} (2+\log (16)) x+\log ^2(4) x+e^{2 x} \left (6 x^2+\log ^2(4) x+4 \log (4) x+x\right )+2 e^x \left (3 x^2+\log ^2(4) x+\left (3 x^2+x\right ) \log (4)\right )\right ) \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{3} \int \left (\frac {3 e^x x}{\left (-9 x^2-6 e^{2 x} x-6 e^x (1+\log (4)) x-6 \log (4) x-e^{4 x}-e^{2 x} \left (1+\log ^2(4)+\log (256)\right )-2 e^{3 x} (1+\log (4))-2 e^x \log (4) (1+\log (4))-\log ^2(4)\right ) \log (x)}+\frac {e^{3 x}}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log (x)}+\frac {\log (64)}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log (x)}+\frac {e^{2 x} \log (16)}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log (x)}+\frac {3 e^x \left (1+\frac {\log ^2(4)}{3}\right )}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log (x)}+\frac {3 e^x}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log ^2(x)}+\frac {\log (64)}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log ^2(x)}+\frac {e^{3 x}}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log ^2(x) x}+\frac {e^{2 x} (1+\log (16))}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log ^2(x) x}+\frac {e^x \log (4) (2+\log (4))}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log ^2(x) x}+\frac {\log ^2(4)}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log ^2(x) x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (-\log (64) \int \frac {1}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log ^2(x)}dx-3 \int \frac {e^x}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log ^2(x)}dx-\log ^2(4) \int \frac {1}{x \left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log ^2(x)}dx-\log (4) (2+\log (4)) \int \frac {e^x}{x \left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log ^2(x)}dx-(1+\log (16)) \int \frac {e^{2 x}}{x \left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log ^2(x)}dx-\int \frac {e^{3 x}}{x \left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log ^2(x)}dx-3 \int \frac {e^x x}{\left (-9 x^2-6 e^{2 x} x-6 e^x (1+\log (4)) x-6 \log (4) x-e^{4 x}-e^{2 x} \left (1+\log ^2(4)+\log (256)\right )-2 e^{3 x} (1+\log (4))-2 e^x \log (4) (1+\log (4))-\log ^2(4)\right ) \log (x)}dx-\log (64) \int \frac {1}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log (x)}dx-\left (3+\log ^2(4)\right ) \int \frac {e^x}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log (x)}dx-\log (16) \int \frac {e^{2 x}}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log (x)}dx-\int \frac {e^{3 x}}{\left (9 x^2+6 e^{2 x} x+6 e^x (1+\log (4)) x+6 \log (4) x+e^{4 x}+e^{2 x} \left (1+\log ^2(4)+\log (256)\right )+2 e^{3 x} (1+\log (4))+2 e^x \log (4) (1+\log (4))+\log ^2(4)\right ) \log (x)}dx\right )\) |
Input:
Int[(-E^(3*x) + E^(2*x)*(-1 - 2*Log[4]) - 3*x*Log[4] - Log[4]^2 + E^x*(-3* x - 2*Log[4] - Log[4]^2) + (-(E^(3*x)*x) - 3*x*Log[4] - 2*E^(2*x)*x*Log[4] + E^x*(-3*x + 3*x^2 - x*Log[4]^2))*Log[x])/(3*(E^(4*x)*x + 9*x^3 + 6*x^2* Log[4] + x*Log[4]^2 + E^(3*x)*(2*x + 2*x*Log[4]) + E^(2*x)*(x + 6*x^2 + 4* x*Log[4] + x*Log[4]^2) + E^x*(6*x^2 + (2*x + 6*x^2)*Log[4] + 2*x*Log[4]^2) )*Log[x]^2),x]
Output:
$Aborted
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38
\[\frac {{\mathrm e}^{x}+2 \ln \left (2\right )}{3 \left ({\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} \ln \left (2\right )+{\mathrm e}^{x}+2 \ln \left (2\right )+3 x \right ) \ln \left (x \right )}\]
Input:
int(1/3*((-x*exp(x)^3-4*x*ln(2)*exp(x)^2+(-4*x*ln(2)^2+3*x^2-3*x)*exp(x)-6 *x*ln(2))*ln(x)-exp(x)^3+(-4*ln(2)-1)*exp(x)^2+(-4*ln(2)^2-4*ln(2)-3*x)*ex p(x)-4*ln(2)^2-6*x*ln(2))/ln(x)^2/(x*exp(x)^4+(4*x*ln(2)+2*x)*exp(x)^3+(4* x*ln(2)^2+8*x*ln(2)+6*x^2+x)*exp(x)^2+(8*x*ln(2)^2+2*(6*x^2+2*x)*ln(2)+6*x ^2)*exp(x)+4*x*ln(2)^2+12*x^2*ln(2)+9*x^3),x)
Output:
1/3*(exp(x)+2*ln(2))/(exp(2*x)+2*exp(x)*ln(2)+exp(x)+2*ln(2)+3*x)/ln(x)
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {-e^{3 x}+e^{2 x} (-1-2 \log (4))-3 x \log (4)-\log ^2(4)+e^x \left (-3 x-2 \log (4)-\log ^2(4)\right )+\left (-e^{3 x} x-3 x \log (4)-2 e^{2 x} x \log (4)+e^x \left (-3 x+3 x^2-x \log ^2(4)\right )\right ) \log (x)}{3 \left (e^{4 x} x+9 x^3+6 x^2 \log (4)+x \log ^2(4)+e^{3 x} (2 x+2 x \log (4))+e^{2 x} \left (x+6 x^2+4 x \log (4)+x \log ^2(4)\right )+e^x \left (6 x^2+\left (2 x+6 x^2\right ) \log (4)+2 x \log ^2(4)\right )\right ) \log ^2(x)} \, dx=\frac {e^{x} + 2 \, \log \left (2\right )}{3 \, {\left ({\left (2 \, \log \left (2\right ) + 1\right )} e^{x} + 3 \, x + e^{\left (2 \, x\right )} + 2 \, \log \left (2\right )\right )} \log \left (x\right )} \] Input:
integrate(1/3*((-x*exp(x)^3-4*x*log(2)*exp(x)^2+(-4*x*log(2)^2+3*x^2-3*x)* exp(x)-6*x*log(2))*log(x)-exp(x)^3+(-4*log(2)-1)*exp(x)^2+(-4*log(2)^2-4*l og(2)-3*x)*exp(x)-4*log(2)^2-6*x*log(2))/log(x)^2/(x*exp(x)^4+(4*x*log(2)+ 2*x)*exp(x)^3+(4*x*log(2)^2+8*x*log(2)+6*x^2+x)*exp(x)^2+(8*x*log(2)^2+2*( 6*x^2+2*x)*log(2)+6*x^2)*exp(x)+4*x*log(2)^2+12*x^2*log(2)+9*x^3),x, algor ithm="fricas")
Output:
1/3*(e^x + 2*log(2))/(((2*log(2) + 1)*e^x + 3*x + e^(2*x) + 2*log(2))*log( x))
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {-e^{3 x}+e^{2 x} (-1-2 \log (4))-3 x \log (4)-\log ^2(4)+e^x \left (-3 x-2 \log (4)-\log ^2(4)\right )+\left (-e^{3 x} x-3 x \log (4)-2 e^{2 x} x \log (4)+e^x \left (-3 x+3 x^2-x \log ^2(4)\right )\right ) \log (x)}{3 \left (e^{4 x} x+9 x^3+6 x^2 \log (4)+x \log ^2(4)+e^{3 x} (2 x+2 x \log (4))+e^{2 x} \left (x+6 x^2+4 x \log (4)+x \log ^2(4)\right )+e^x \left (6 x^2+\left (2 x+6 x^2\right ) \log (4)+2 x \log ^2(4)\right )\right ) \log ^2(x)} \, dx=\frac {e^{x} + 2 \log {\left (2 \right )}}{9 x \log {\left (x \right )} + \left (3 \log {\left (x \right )} + 6 \log {\left (2 \right )} \log {\left (x \right )}\right ) e^{x} + 3 e^{2 x} \log {\left (x \right )} + 6 \log {\left (2 \right )} \log {\left (x \right )}} \] Input:
integrate(1/3*((-x*exp(x)**3-4*x*ln(2)*exp(x)**2+(-4*x*ln(2)**2+3*x**2-3*x )*exp(x)-6*x*ln(2))*ln(x)-exp(x)**3+(-4*ln(2)-1)*exp(x)**2+(-4*ln(2)**2-4* ln(2)-3*x)*exp(x)-4*ln(2)**2-6*x*ln(2))/ln(x)**2/(x*exp(x)**4+(4*x*ln(2)+2 *x)*exp(x)**3+(4*x*ln(2)**2+8*x*ln(2)+6*x**2+x)*exp(x)**2+(8*x*ln(2)**2+2* (6*x**2+2*x)*ln(2)+6*x**2)*exp(x)+4*x*ln(2)**2+12*x**2*ln(2)+9*x**3),x)
Output:
(exp(x) + 2*log(2))/(9*x*log(x) + (3*log(x) + 6*log(2)*log(x))*exp(x) + 3* exp(2*x)*log(x) + 6*log(2)*log(x))
Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {-e^{3 x}+e^{2 x} (-1-2 \log (4))-3 x \log (4)-\log ^2(4)+e^x \left (-3 x-2 \log (4)-\log ^2(4)\right )+\left (-e^{3 x} x-3 x \log (4)-2 e^{2 x} x \log (4)+e^x \left (-3 x+3 x^2-x \log ^2(4)\right )\right ) \log (x)}{3 \left (e^{4 x} x+9 x^3+6 x^2 \log (4)+x \log ^2(4)+e^{3 x} (2 x+2 x \log (4))+e^{2 x} \left (x+6 x^2+4 x \log (4)+x \log ^2(4)\right )+e^x \left (6 x^2+\left (2 x+6 x^2\right ) \log (4)+2 x \log ^2(4)\right )\right ) \log ^2(x)} \, dx=\frac {e^{x} + 2 \, \log \left (2\right )}{3 \, {\left ({\left (2 \, \log \left (2\right ) + 1\right )} e^{x} \log \left (x\right ) + {\left (3 \, x + 2 \, \log \left (2\right )\right )} \log \left (x\right ) + e^{\left (2 \, x\right )} \log \left (x\right )\right )}} \] Input:
integrate(1/3*((-x*exp(x)^3-4*x*log(2)*exp(x)^2+(-4*x*log(2)^2+3*x^2-3*x)* exp(x)-6*x*log(2))*log(x)-exp(x)^3+(-4*log(2)-1)*exp(x)^2+(-4*log(2)^2-4*l og(2)-3*x)*exp(x)-4*log(2)^2-6*x*log(2))/log(x)^2/(x*exp(x)^4+(4*x*log(2)+ 2*x)*exp(x)^3+(4*x*log(2)^2+8*x*log(2)+6*x^2+x)*exp(x)^2+(8*x*log(2)^2+2*( 6*x^2+2*x)*log(2)+6*x^2)*exp(x)+4*x*log(2)^2+12*x^2*log(2)+9*x^3),x, algor ithm="maxima")
Output:
1/3*(e^x + 2*log(2))/((2*log(2) + 1)*e^x*log(x) + (3*x + 2*log(2))*log(x) + e^(2*x)*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 609 vs. \(2 (24) = 48\).
Time = 0.43 (sec) , antiderivative size = 609, normalized size of antiderivative = 23.42 \[ \int \frac {-e^{3 x}+e^{2 x} (-1-2 \log (4))-3 x \log (4)-\log ^2(4)+e^x \left (-3 x-2 \log (4)-\log ^2(4)\right )+\left (-e^{3 x} x-3 x \log (4)-2 e^{2 x} x \log (4)+e^x \left (-3 x+3 x^2-x \log ^2(4)\right )\right ) \log (x)}{3 \left (e^{4 x} x+9 x^3+6 x^2 \log (4)+x \log ^2(4)+e^{3 x} (2 x+2 x \log (4))+e^{2 x} \left (x+6 x^2+4 x \log (4)+x \log ^2(4)\right )+e^x \left (6 x^2+\left (2 x+6 x^2\right ) \log (4)+2 x \log ^2(4)\right )\right ) \log ^2(x)} \, dx=\text {Too large to display} \] Input:
integrate(1/3*((-x*exp(x)^3-4*x*log(2)*exp(x)^2+(-4*x*log(2)^2+3*x^2-3*x)* exp(x)-6*x*log(2))*log(x)-exp(x)^3+(-4*log(2)-1)*exp(x)^2+(-4*log(2)^2-4*l og(2)-3*x)*exp(x)-4*log(2)^2-6*x*log(2))/log(x)^2/(x*exp(x)^4+(4*x*log(2)+ 2*x)*exp(x)^3+(4*x*log(2)^2+8*x*log(2)+6*x^2+x)*exp(x)^2+(8*x*log(2)^2+2*( 6*x^2+2*x)*log(2)+6*x^2)*exp(x)+4*x*log(2)^2+12*x^2*log(2)+9*x^3),x, algor ithm="giac")
Output:
1/3*(12*x^2*e^x*log(2)^2*log(x) + 24*x^2*log(2)^3*log(x) + 8*x*e^x*log(2)^ 3*log(x) + 16*x*log(2)^4*log(x) - 36*x^3*e^x*log(x) - 72*x^3*log(2)*log(x) - 36*x^2*e^x*log(2)*log(x) - 72*x^2*log(2)^2*log(x) - 32*x*e^x*log(2)^2*l og(x) - 64*x*log(2)^3*log(x) - 12*x*e^x*log(2)^2 - 24*x*log(2)^3 - 8*e^x*l og(2)^3 - 16*log(2)^4 + 39*x^2*e^x*log(x) + 114*x^2*log(2)*log(x) + 8*x*e^ x*log(2)*log(x) + 28*x*log(2)^2*log(x) + 36*x^2*log(2) - 6*x*e^x*log(2) + 24*x*log(2)^2 - 4*e^x*log(2)^2 - 12*x*e^x*log(x) - 48*x*log(2)*log(x) - 24 *x*log(2) - 16*log(2)^2)/(24*x^2*e^x*log(2)^3*log(x)^2 + 16*x*e^x*log(2)^4 *log(x)^2 - 72*x^3*e^x*log(2)*log(x)^2 + 36*x^3*log(2)^2*log(x)^2 + 12*x^2 *e^(2*x)*log(2)^2*log(x)^2 - 60*x^2*e^x*log(2)^2*log(x)^2 + 48*x^2*log(2)^ 3*log(x)^2 + 8*x*e^(2*x)*log(2)^3*log(x)^2 - 32*x*e^x*log(2)^3*log(x)^2 + 16*x*log(2)^4*log(x)^2 - 108*x^4*log(x)^2 - 36*x^3*e^(2*x)*log(x)^2 - 36*x ^3*e^x*log(x)^2 - 180*x^3*log(2)*log(x)^2 - 36*x^2*e^(2*x)*log(2)*log(x)^2 + 42*x^2*e^x*log(2)*log(x)^2 - 132*x^2*log(2)^2*log(x)^2 - 20*x*e^(2*x)*l og(2)^2*log(x)^2 + 8*x*e^x*log(2)^2*log(x)^2 - 40*x*log(2)^3*log(x)^2 + 11 7*x^3*log(x)^2 + 39*x^2*e^(2*x)*log(x)^2 + 39*x^2*e^x*log(x)^2 + 120*x^2*l og(2)*log(x)^2 + 14*x*e^(2*x)*log(2)*log(x)^2 - 10*x*e^x*log(2)*log(x)^2 + 28*x*log(2)^2*log(x)^2 - 36*x^2*log(x)^2 - 12*x*e^(2*x)*log(x)^2 - 12*x*e ^x*log(x)^2 - 24*x*log(2)*log(x)^2)
Timed out. \[ \int \frac {-e^{3 x}+e^{2 x} (-1-2 \log (4))-3 x \log (4)-\log ^2(4)+e^x \left (-3 x-2 \log (4)-\log ^2(4)\right )+\left (-e^{3 x} x-3 x \log (4)-2 e^{2 x} x \log (4)+e^x \left (-3 x+3 x^2-x \log ^2(4)\right )\right ) \log (x)}{3 \left (e^{4 x} x+9 x^3+6 x^2 \log (4)+x \log ^2(4)+e^{3 x} (2 x+2 x \log (4))+e^{2 x} \left (x+6 x^2+4 x \log (4)+x \log ^2(4)\right )+e^x \left (6 x^2+\left (2 x+6 x^2\right ) \log (4)+2 x \log ^2(4)\right )\right ) \log ^2(x)} \, dx=\int -\frac {\frac {{\mathrm {e}}^{3\,x}}{3}+2\,x\,\ln \left (2\right )+\frac {{\mathrm {e}}^x\,\left (3\,x+4\,\ln \left (2\right )+4\,{\ln \left (2\right )}^2\right )}{3}+\frac {{\mathrm {e}}^{2\,x}\,\left (4\,\ln \left (2\right )+1\right )}{3}+\frac {\ln \left (x\right )\,\left (x\,{\mathrm {e}}^{3\,x}+6\,x\,\ln \left (2\right )+{\mathrm {e}}^x\,\left (3\,x+4\,x\,{\ln \left (2\right )}^2-3\,x^2\right )+4\,x\,{\mathrm {e}}^{2\,x}\,\ln \left (2\right )\right )}{3}+\frac {4\,{\ln \left (2\right )}^2}{3}}{{\ln \left (x\right )}^2\,\left (x\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{3\,x}\,\left (2\,x+4\,x\,\ln \left (2\right )\right )+{\mathrm {e}}^{2\,x}\,\left (x+8\,x\,\ln \left (2\right )+4\,x\,{\ln \left (2\right )}^2+6\,x^2\right )+4\,x\,{\ln \left (2\right )}^2+12\,x^2\,\ln \left (2\right )+{\mathrm {e}}^x\,\left (2\,\ln \left (2\right )\,\left (6\,x^2+2\,x\right )+8\,x\,{\ln \left (2\right )}^2+6\,x^2\right )+9\,x^3\right )} \,d x \] Input:
int(-(exp(3*x)/3 + 2*x*log(2) + (exp(x)*(3*x + 4*log(2) + 4*log(2)^2))/3 + (exp(2*x)*(4*log(2) + 1))/3 + (log(x)*(x*exp(3*x) + 6*x*log(2) + exp(x)*( 3*x + 4*x*log(2)^2 - 3*x^2) + 4*x*exp(2*x)*log(2)))/3 + (4*log(2)^2)/3)/(l og(x)^2*(x*exp(4*x) + exp(3*x)*(2*x + 4*x*log(2)) + exp(2*x)*(x + 8*x*log( 2) + 4*x*log(2)^2 + 6*x^2) + 4*x*log(2)^2 + 12*x^2*log(2) + exp(x)*(2*log( 2)*(2*x + 6*x^2) + 8*x*log(2)^2 + 6*x^2) + 9*x^3)),x)
Output:
int(-(exp(3*x)/3 + 2*x*log(2) + (exp(x)*(3*x + 4*log(2) + 4*log(2)^2))/3 + (exp(2*x)*(4*log(2) + 1))/3 + (log(x)*(x*exp(3*x) + 6*x*log(2) + exp(x)*( 3*x + 4*x*log(2)^2 - 3*x^2) + 4*x*exp(2*x)*log(2)))/3 + (4*log(2)^2)/3)/(l og(x)^2*(x*exp(4*x) + exp(3*x)*(2*x + 4*x*log(2)) + exp(2*x)*(x + 8*x*log( 2) + 4*x*log(2)^2 + 6*x^2) + 4*x*log(2)^2 + 12*x^2*log(2) + exp(x)*(2*log( 2)*(2*x + 6*x^2) + 8*x*log(2)^2 + 6*x^2) + 9*x^3)), x)
Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {-e^{3 x}+e^{2 x} (-1-2 \log (4))-3 x \log (4)-\log ^2(4)+e^x \left (-3 x-2 \log (4)-\log ^2(4)\right )+\left (-e^{3 x} x-3 x \log (4)-2 e^{2 x} x \log (4)+e^x \left (-3 x+3 x^2-x \log ^2(4)\right )\right ) \log (x)}{3 \left (e^{4 x} x+9 x^3+6 x^2 \log (4)+x \log ^2(4)+e^{3 x} (2 x+2 x \log (4))+e^{2 x} \left (x+6 x^2+4 x \log (4)+x \log ^2(4)\right )+e^x \left (6 x^2+\left (2 x+6 x^2\right ) \log (4)+2 x \log ^2(4)\right )\right ) \log ^2(x)} \, dx=\frac {e^{x}+2 \,\mathrm {log}\left (2\right )}{3 \,\mathrm {log}\left (x \right ) \left (e^{2 x}+2 e^{x} \mathrm {log}\left (2\right )+e^{x}+2 \,\mathrm {log}\left (2\right )+3 x \right )} \] Input:
int(1/3*((-x*exp(x)^3-4*x*log(2)*exp(x)^2+(-4*x*log(2)^2+3*x^2-3*x)*exp(x) -6*x*log(2))*log(x)-exp(x)^3+(-4*log(2)-1)*exp(x)^2+(-4*log(2)^2-4*log(2)- 3*x)*exp(x)-4*log(2)^2-6*x*log(2))/log(x)^2/(x*exp(x)^4+(4*x*log(2)+2*x)*e xp(x)^3+(4*x*log(2)^2+8*x*log(2)+6*x^2+x)*exp(x)^2+(8*x*log(2)^2+2*(6*x^2+ 2*x)*log(2)+6*x^2)*exp(x)+4*x*log(2)^2+12*x^2*log(2)+9*x^3),x)
Output:
(e**x + 2*log(2))/(3*log(x)*(e**(2*x) + 2*e**x*log(2) + e**x + 2*log(2) + 3*x))