Integrand size = 328, antiderivative size = 33 \[ \int \frac {e^{2 e^4} \left (36-24 e^{2 x}+4 e^{4 x}\right )+36 x^2+3 x^3+\left (72 x+6 x^2\right ) \log (3)+36 \log ^2(3)+e^{2 x} \left (-24 x^2-x^3+2 x^4+\left (-48 x-2 x^2+2 x^3\right ) \log (3)-24 \log ^2(3)\right )+e^{4 x} \left (4 x^2+8 x \log (3)+4 \log ^2(3)\right )+e^{e^4} \left (-72 x-6 x^2+e^{4 x} (-8 x-8 \log (3))-72 \log (3)+e^{2 x} \left (48 x+2 x^2-2 x^3+48 \log (3)\right )\right )}{9 x^3+e^{2 e^4} \left (9 x-6 e^{2 x} x+e^{4 x} x\right )+18 x^2 \log (3)+9 x \log ^2(3)+e^{2 x} \left (-6 x^3-12 x^2 \log (3)-6 x \log ^2(3)\right )+e^{4 x} \left (x^3+2 x^2 \log (3)+x \log ^2(3)\right )+e^{e^4} \left (-18 x^2-18 x \log (3)+e^{4 x} \left (-2 x^2-2 x \log (3)\right )+e^{2 x} \left (12 x^2+12 x \log (3)\right )\right )} \, dx=\frac {x^2}{\left (3-e^{2 x}\right ) \left (-e^{e^4}+x+\log (3)\right )}+4 \log (x) \] Output:
x^2/(ln(3)+x-exp(exp(4)))/(3-exp(2*x))+4*ln(x)
Time = 0.14 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42 \[ \int \frac {e^{2 e^4} \left (36-24 e^{2 x}+4 e^{4 x}\right )+36 x^2+3 x^3+\left (72 x+6 x^2\right ) \log (3)+36 \log ^2(3)+e^{2 x} \left (-24 x^2-x^3+2 x^4+\left (-48 x-2 x^2+2 x^3\right ) \log (3)-24 \log ^2(3)\right )+e^{4 x} \left (4 x^2+8 x \log (3)+4 \log ^2(3)\right )+e^{e^4} \left (-72 x-6 x^2+e^{4 x} (-8 x-8 \log (3))-72 \log (3)+e^{2 x} \left (48 x+2 x^2-2 x^3+48 \log (3)\right )\right )}{9 x^3+e^{2 e^4} \left (9 x-6 e^{2 x} x+e^{4 x} x\right )+18 x^2 \log (3)+9 x \log ^2(3)+e^{2 x} \left (-6 x^3-12 x^2 \log (3)-6 x \log ^2(3)\right )+e^{4 x} \left (x^3+2 x^2 \log (3)+x \log ^2(3)\right )+e^{e^4} \left (-18 x^2-18 x \log (3)+e^{4 x} \left (-2 x^2-2 x \log (3)\right )+e^{2 x} \left (12 x^2+12 x \log (3)\right )\right )} \, dx=-\frac {x^2 \left (-2 e^{e^4}+2 x+\log (9)\right )}{2 \left (-3+e^{2 x}\right ) \left (-e^{e^4}+x+\log (3)\right )^2}+4 \log (x) \] Input:
Integrate[(E^(2*E^4)*(36 - 24*E^(2*x) + 4*E^(4*x)) + 36*x^2 + 3*x^3 + (72* x + 6*x^2)*Log[3] + 36*Log[3]^2 + E^(2*x)*(-24*x^2 - x^3 + 2*x^4 + (-48*x - 2*x^2 + 2*x^3)*Log[3] - 24*Log[3]^2) + E^(4*x)*(4*x^2 + 8*x*Log[3] + 4*L og[3]^2) + E^E^4*(-72*x - 6*x^2 + E^(4*x)*(-8*x - 8*Log[3]) - 72*Log[3] + E^(2*x)*(48*x + 2*x^2 - 2*x^3 + 48*Log[3])))/(9*x^3 + E^(2*E^4)*(9*x - 6*E ^(2*x)*x + E^(4*x)*x) + 18*x^2*Log[3] + 9*x*Log[3]^2 + E^(2*x)*(-6*x^3 - 1 2*x^2*Log[3] - 6*x*Log[3]^2) + E^(4*x)*(x^3 + 2*x^2*Log[3] + x*Log[3]^2) + E^E^4*(-18*x^2 - 18*x*Log[3] + E^(4*x)*(-2*x^2 - 2*x*Log[3]) + E^(2*x)*(1 2*x^2 + 12*x*Log[3]))),x]
Output:
-1/2*(x^2*(-2*E^E^4 + 2*x + Log[9]))/((-3 + E^(2*x))*(-E^E^4 + x + Log[3]) ^2) + 4*Log[x]
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.
| \(\displaystyle \int \frac {3 x^3+36 x^2+e^{4 x} \left (4 x^2+8 x \log (3)+4 \log ^2(3)\right )+\left (6 x^2+72 x\right ) \log (3)+e^{e^4} \left (-6 x^2+e^{2 x} \left (-2 x^3+2 x^2+48 x+48 \log (3)\right )-72 x+e^{4 x} (-8 x-8 \log (3))-72 \log (3)\right )+e^{2 x} \left (2 x^4-x^3-24 x^2+\left (2 x^3-2 x^2-48 x\right ) \log (3)-24 \log ^2(3)\right )+e^{2 e^4} \left (-24 e^{2 x}+4 e^{4 x}+36\right )+36 \log ^2(3)}{9 x^3+18 x^2 \log (3)+e^{e^4} \left (-18 x^2+e^{4 x} \left (-2 x^2-2 x \log (3)\right )+e^{2 x} \left (12 x^2+12 x \log (3)\right )-18 x \log (3)\right )+e^{2 x} \left (-6 x^3-12 x^2 \log (3)-6 x \log ^2(3)\right )+e^{4 x} \left (x^3+2 x^2 \log (3)+x \log ^2(3)\right )+e^{2 e^4} \left (-6 e^{2 x} x+e^{4 x} x+9 x\right )+9 x \log ^2(3)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {3 x^3+36 x^2+e^{4 x} \left (4 x^2+8 x \log (3)+4 \log ^2(3)\right )+\left (6 x^2+72 x\right ) \log (3)+e^{e^4} \left (-6 x^2+e^{2 x} \left (-2 x^3+2 x^2+48 x+48 \log (3)\right )-72 x+e^{4 x} (-8 x-8 \log (3))-72 \log (3)\right )+e^{2 x} \left (2 x^4-x^3-24 x^2+\left (2 x^3-2 x^2-48 x\right ) \log (3)-24 \log ^2(3)\right )+e^{2 e^4} \left (-24 e^{2 x}+4 e^{4 x}+36\right )+36 \log ^2(3)}{\left (3-e^{2 x}\right )^2 x \left (-x+e^{e^4}-\log (3)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {6 x^2}{\left (e^{2 x}-3\right )^2 \left (x-e^{e^4}+\log (3)\right )}+\frac {x \left (-2 x^2+x \left (1+2 e^{e^4}-\log (9)\right )-2 e^{e^4}+\log (9)\right )}{\left (3-e^{2 x}\right ) \left (-x+e^{e^4}-\log (3)\right )^2}+\frac {4}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\left (2 e^{2 e^4}+6 \log ^2(3)-\log ^2(9)-e^{e^4} \log (81)\right ) \int \frac {1}{\left (-3+e^{2 x}\right ) \left (-x-\log (3)+e^{e^4}\right )}dx+\left (e^{e^4}-\log (3)\right )^2 \int \frac {1}{\left (-3+e^{2 x}\right ) \left (-x-\log (3)+e^{e^4}\right )^2}dx-6 \left (e^{e^4}-\log (3)\right )^2 \int \frac {1}{\left (-3+e^{2 x}\right )^2 \left (-x-\log (3)+e^{e^4}\right )}dx+\frac {x}{3-e^{2 x}}-\frac {x}{3}+\frac {1}{3} x \left (1-2 e^{e^4}+\frac {2 \log ^2(9)}{\log (81)}\right )-\frac {1}{6} \left (1-2 e^{e^4}+\frac {2 \log ^2(9)}{\log (81)}\right ) \log \left (3-e^{2 x}\right )+\frac {2}{3} x \left (e^{e^4}-\log (3)\right )-\frac {1}{3} \left (e^{e^4}-\log (3)\right ) \log \left (3-e^{2 x}\right )+\frac {1}{6} \log \left (3-e^{2 x}\right )+4 \log (x)+\frac {e^{e^4}-\log (3)}{3-e^{2 x}}\) |
Input:
Int[(E^(2*E^4)*(36 - 24*E^(2*x) + 4*E^(4*x)) + 36*x^2 + 3*x^3 + (72*x + 6* x^2)*Log[3] + 36*Log[3]^2 + E^(2*x)*(-24*x^2 - x^3 + 2*x^4 + (-48*x - 2*x^ 2 + 2*x^3)*Log[3] - 24*Log[3]^2) + E^(4*x)*(4*x^2 + 8*x*Log[3] + 4*Log[3]^ 2) + E^E^4*(-72*x - 6*x^2 + E^(4*x)*(-8*x - 8*Log[3]) - 72*Log[3] + E^(2*x )*(48*x + 2*x^2 - 2*x^3 + 48*Log[3])))/(9*x^3 + E^(2*E^4)*(9*x - 6*E^(2*x) *x + E^(4*x)*x) + 18*x^2*Log[3] + 9*x*Log[3]^2 + E^(2*x)*(-6*x^3 - 12*x^2* Log[3] - 6*x*Log[3]^2) + E^(4*x)*(x^3 + 2*x^2*Log[3] + x*Log[3]^2) + E^E^4 *(-18*x^2 - 18*x*Log[3] + E^(4*x)*(-2*x^2 - 2*x*Log[3]) + E^(2*x)*(12*x^2 + 12*x*Log[3]))),x]
Output:
$Aborted
Time = 7.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91
| method | result | size |
| norman | \(-\frac {x^{2}}{\left ({\mathrm e}^{2 x}-3\right ) \left (\ln \left (3\right )+x -{\mathrm e}^{{\mathrm e}^{4}}\right )}+4 \ln \left (x \right )\) | \(30\) |
| risch | \(-\frac {x^{2}}{\left ({\mathrm e}^{2 x}-3\right ) \left (\ln \left (3\right )+x -{\mathrm e}^{{\mathrm e}^{4}}\right )}+4 \ln \left (x \right )\) | \(30\) |
| parallelrisch | \(\frac {4 \ln \left (3\right ) \ln \left (x \right ) {\mathrm e}^{2 x}-4 \ln \left (x \right ) {\mathrm e}^{2 x} {\mathrm e}^{{\mathrm e}^{4}}+4 \ln \left (x \right ) {\mathrm e}^{2 x} x -12 \ln \left (3\right ) \ln \left (x \right )+12 \ln \left (x \right ) {\mathrm e}^{{\mathrm e}^{4}}-12 x \ln \left (x \right )-x^{2}}{\left ({\mathrm e}^{2 x}-3\right ) \left (\ln \left (3\right )+x -{\mathrm e}^{{\mathrm e}^{4}}\right )}\) | \(75\) |
Input:
int(((4*exp(2*x)^2-24*exp(2*x)+36)*exp(exp(4))^2+((-8*ln(3)-8*x)*exp(2*x)^ 2+(48*ln(3)-2*x^3+2*x^2+48*x)*exp(2*x)-72*ln(3)-6*x^2-72*x)*exp(exp(4))+(4 *ln(3)^2+8*x*ln(3)+4*x^2)*exp(2*x)^2+(-24*ln(3)^2+(2*x^3-2*x^2-48*x)*ln(3) +2*x^4-x^3-24*x^2)*exp(2*x)+36*ln(3)^2+(6*x^2+72*x)*ln(3)+3*x^3+36*x^2)/(( x*exp(2*x)^2-6*x*exp(2*x)+9*x)*exp(exp(4))^2+((-2*x*ln(3)-2*x^2)*exp(2*x)^ 2+(12*x*ln(3)+12*x^2)*exp(2*x)-18*x*ln(3)-18*x^2)*exp(exp(4))+(x*ln(3)^2+2 *x^2*ln(3)+x^3)*exp(2*x)^2+(-6*x*ln(3)^2-12*x^2*ln(3)-6*x^3)*exp(2*x)+9*x* ln(3)^2+18*x^2*ln(3)+9*x^3),x,method=_RETURNVERBOSE)
Output:
-x^2/(exp(2*x)-3)/(ln(3)+x-exp(exp(4)))+4*ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (29) = 58\).
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.06 \[ \int \frac {e^{2 e^4} \left (36-24 e^{2 x}+4 e^{4 x}\right )+36 x^2+3 x^3+\left (72 x+6 x^2\right ) \log (3)+36 \log ^2(3)+e^{2 x} \left (-24 x^2-x^3+2 x^4+\left (-48 x-2 x^2+2 x^3\right ) \log (3)-24 \log ^2(3)\right )+e^{4 x} \left (4 x^2+8 x \log (3)+4 \log ^2(3)\right )+e^{e^4} \left (-72 x-6 x^2+e^{4 x} (-8 x-8 \log (3))-72 \log (3)+e^{2 x} \left (48 x+2 x^2-2 x^3+48 \log (3)\right )\right )}{9 x^3+e^{2 e^4} \left (9 x-6 e^{2 x} x+e^{4 x} x\right )+18 x^2 \log (3)+9 x \log ^2(3)+e^{2 x} \left (-6 x^3-12 x^2 \log (3)-6 x \log ^2(3)\right )+e^{4 x} \left (x^3+2 x^2 \log (3)+x \log ^2(3)\right )+e^{e^4} \left (-18 x^2-18 x \log (3)+e^{4 x} \left (-2 x^2-2 x \log (3)\right )+e^{2 x} \left (12 x^2+12 x \log (3)\right )\right )} \, dx=-\frac {x^{2} - 4 \, {\left ({\left (x + \log \left (3\right )\right )} e^{\left (2 \, x\right )} - {\left (e^{\left (2 \, x\right )} - 3\right )} e^{\left (e^{4}\right )} - 3 \, x - 3 \, \log \left (3\right )\right )} \log \left (x\right )}{{\left (x + \log \left (3\right )\right )} e^{\left (2 \, x\right )} - {\left (e^{\left (2 \, x\right )} - 3\right )} e^{\left (e^{4}\right )} - 3 \, x - 3 \, \log \left (3\right )} \] Input:
integrate(((4*exp(2*x)^2-24*exp(2*x)+36)*exp(exp(4))^2+((-8*log(3)-8*x)*ex p(2*x)^2+(48*log(3)-2*x^3+2*x^2+48*x)*exp(2*x)-72*log(3)-6*x^2-72*x)*exp(e xp(4))+(4*log(3)^2+8*x*log(3)+4*x^2)*exp(2*x)^2+(-24*log(3)^2+(2*x^3-2*x^2 -48*x)*log(3)+2*x^4-x^3-24*x^2)*exp(2*x)+36*log(3)^2+(6*x^2+72*x)*log(3)+3 *x^3+36*x^2)/((x*exp(2*x)^2-6*x*exp(2*x)+9*x)*exp(exp(4))^2+((-2*x*log(3)- 2*x^2)*exp(2*x)^2+(12*x*log(3)+12*x^2)*exp(2*x)-18*x*log(3)-18*x^2)*exp(ex p(4))+(x*log(3)^2+2*x^2*log(3)+x^3)*exp(2*x)^2+(-6*x*log(3)^2-12*x^2*log(3 )-6*x^3)*exp(2*x)+9*x*log(3)^2+18*x^2*log(3)+9*x^3),x, algorithm="fricas")
Output:
-(x^2 - 4*((x + log(3))*e^(2*x) - (e^(2*x) - 3)*e^(e^4) - 3*x - 3*log(3))* log(x))/((x + log(3))*e^(2*x) - (e^(2*x) - 3)*e^(e^4) - 3*x - 3*log(3))
Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {e^{2 e^4} \left (36-24 e^{2 x}+4 e^{4 x}\right )+36 x^2+3 x^3+\left (72 x+6 x^2\right ) \log (3)+36 \log ^2(3)+e^{2 x} \left (-24 x^2-x^3+2 x^4+\left (-48 x-2 x^2+2 x^3\right ) \log (3)-24 \log ^2(3)\right )+e^{4 x} \left (4 x^2+8 x \log (3)+4 \log ^2(3)\right )+e^{e^4} \left (-72 x-6 x^2+e^{4 x} (-8 x-8 \log (3))-72 \log (3)+e^{2 x} \left (48 x+2 x^2-2 x^3+48 \log (3)\right )\right )}{9 x^3+e^{2 e^4} \left (9 x-6 e^{2 x} x+e^{4 x} x\right )+18 x^2 \log (3)+9 x \log ^2(3)+e^{2 x} \left (-6 x^3-12 x^2 \log (3)-6 x \log ^2(3)\right )+e^{4 x} \left (x^3+2 x^2 \log (3)+x \log ^2(3)\right )+e^{e^4} \left (-18 x^2-18 x \log (3)+e^{4 x} \left (-2 x^2-2 x \log (3)\right )+e^{2 x} \left (12 x^2+12 x \log (3)\right )\right )} \, dx=- \frac {x^{2}}{- 3 x + \left (x - e^{e^{4}} + \log {\left (3 \right )}\right ) e^{2 x} - 3 \log {\left (3 \right )} + 3 e^{e^{4}}} + 4 \log {\left (x \right )} \] Input:
integrate(((4*exp(2*x)**2-24*exp(2*x)+36)*exp(exp(4))**2+((-8*ln(3)-8*x)*e xp(2*x)**2+(48*ln(3)-2*x**3+2*x**2+48*x)*exp(2*x)-72*ln(3)-6*x**2-72*x)*ex p(exp(4))+(4*ln(3)**2+8*x*ln(3)+4*x**2)*exp(2*x)**2+(-24*ln(3)**2+(2*x**3- 2*x**2-48*x)*ln(3)+2*x**4-x**3-24*x**2)*exp(2*x)+36*ln(3)**2+(6*x**2+72*x) *ln(3)+3*x**3+36*x**2)/((x*exp(2*x)**2-6*x*exp(2*x)+9*x)*exp(exp(4))**2+(( -2*x*ln(3)-2*x**2)*exp(2*x)**2+(12*x*ln(3)+12*x**2)*exp(2*x)-18*x*ln(3)-18 *x**2)*exp(exp(4))+(x*ln(3)**2+2*x**2*ln(3)+x**3)*exp(2*x)**2+(-6*x*ln(3)* *2-12*x**2*ln(3)-6*x**3)*exp(2*x)+9*x*ln(3)**2+18*x**2*ln(3)+9*x**3),x)
Output:
-x**2/(-3*x + (x - exp(exp(4)) + log(3))*exp(2*x) - 3*log(3) + 3*exp(exp(4 ))) + 4*log(x)
Time = 0.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {e^{2 e^4} \left (36-24 e^{2 x}+4 e^{4 x}\right )+36 x^2+3 x^3+\left (72 x+6 x^2\right ) \log (3)+36 \log ^2(3)+e^{2 x} \left (-24 x^2-x^3+2 x^4+\left (-48 x-2 x^2+2 x^3\right ) \log (3)-24 \log ^2(3)\right )+e^{4 x} \left (4 x^2+8 x \log (3)+4 \log ^2(3)\right )+e^{e^4} \left (-72 x-6 x^2+e^{4 x} (-8 x-8 \log (3))-72 \log (3)+e^{2 x} \left (48 x+2 x^2-2 x^3+48 \log (3)\right )\right )}{9 x^3+e^{2 e^4} \left (9 x-6 e^{2 x} x+e^{4 x} x\right )+18 x^2 \log (3)+9 x \log ^2(3)+e^{2 x} \left (-6 x^3-12 x^2 \log (3)-6 x \log ^2(3)\right )+e^{4 x} \left (x^3+2 x^2 \log (3)+x \log ^2(3)\right )+e^{e^4} \left (-18 x^2-18 x \log (3)+e^{4 x} \left (-2 x^2-2 x \log (3)\right )+e^{2 x} \left (12 x^2+12 x \log (3)\right )\right )} \, dx=-\frac {x^{2}}{{\left (x - e^{\left (e^{4}\right )} + \log \left (3\right )\right )} e^{\left (2 \, x\right )} - 3 \, x + 3 \, e^{\left (e^{4}\right )} - 3 \, \log \left (3\right )} + 4 \, \log \left (x\right ) \] Input:
integrate(((4*exp(2*x)^2-24*exp(2*x)+36)*exp(exp(4))^2+((-8*log(3)-8*x)*ex p(2*x)^2+(48*log(3)-2*x^3+2*x^2+48*x)*exp(2*x)-72*log(3)-6*x^2-72*x)*exp(e xp(4))+(4*log(3)^2+8*x*log(3)+4*x^2)*exp(2*x)^2+(-24*log(3)^2+(2*x^3-2*x^2 -48*x)*log(3)+2*x^4-x^3-24*x^2)*exp(2*x)+36*log(3)^2+(6*x^2+72*x)*log(3)+3 *x^3+36*x^2)/((x*exp(2*x)^2-6*x*exp(2*x)+9*x)*exp(exp(4))^2+((-2*x*log(3)- 2*x^2)*exp(2*x)^2+(12*x*log(3)+12*x^2)*exp(2*x)-18*x*log(3)-18*x^2)*exp(ex p(4))+(x*log(3)^2+2*x^2*log(3)+x^3)*exp(2*x)^2+(-6*x*log(3)^2-12*x^2*log(3 )-6*x^3)*exp(2*x)+9*x*log(3)^2+18*x^2*log(3)+9*x^3),x, algorithm="maxima")
Output:
-x^2/((x - e^(e^4) + log(3))*e^(2*x) - 3*x + 3*e^(e^4) - 3*log(3)) + 4*log (x)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (29) = 58\).
Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.15 \[ \int \frac {e^{2 e^4} \left (36-24 e^{2 x}+4 e^{4 x}\right )+36 x^2+3 x^3+\left (72 x+6 x^2\right ) \log (3)+36 \log ^2(3)+e^{2 x} \left (-24 x^2-x^3+2 x^4+\left (-48 x-2 x^2+2 x^3\right ) \log (3)-24 \log ^2(3)\right )+e^{4 x} \left (4 x^2+8 x \log (3)+4 \log ^2(3)\right )+e^{e^4} \left (-72 x-6 x^2+e^{4 x} (-8 x-8 \log (3))-72 \log (3)+e^{2 x} \left (48 x+2 x^2-2 x^3+48 \log (3)\right )\right )}{9 x^3+e^{2 e^4} \left (9 x-6 e^{2 x} x+e^{4 x} x\right )+18 x^2 \log (3)+9 x \log ^2(3)+e^{2 x} \left (-6 x^3-12 x^2 \log (3)-6 x \log ^2(3)\right )+e^{4 x} \left (x^3+2 x^2 \log (3)+x \log ^2(3)\right )+e^{e^4} \left (-18 x^2-18 x \log (3)+e^{4 x} \left (-2 x^2-2 x \log (3)\right )+e^{2 x} \left (12 x^2+12 x \log (3)\right )\right )} \, dx=\frac {4 \, x e^{\left (2 \, x\right )} \log \left (2 \, x\right ) + 4 \, e^{\left (2 \, x\right )} \log \left (3\right ) \log \left (2 \, x\right ) - x^{2} - 12 \, x \log \left (2 \, x\right ) - 4 \, e^{\left (2 \, x + e^{4}\right )} \log \left (2 \, x\right ) + 12 \, e^{\left (e^{4}\right )} \log \left (2 \, x\right ) - 12 \, \log \left (3\right ) \log \left (2 \, x\right )}{x e^{\left (2 \, x\right )} + e^{\left (2 \, x\right )} \log \left (3\right ) - 3 \, x - e^{\left (2 \, x + e^{4}\right )} + 3 \, e^{\left (e^{4}\right )} - 3 \, \log \left (3\right )} \] Input:
integrate(((4*exp(2*x)^2-24*exp(2*x)+36)*exp(exp(4))^2+((-8*log(3)-8*x)*ex p(2*x)^2+(48*log(3)-2*x^3+2*x^2+48*x)*exp(2*x)-72*log(3)-6*x^2-72*x)*exp(e xp(4))+(4*log(3)^2+8*x*log(3)+4*x^2)*exp(2*x)^2+(-24*log(3)^2+(2*x^3-2*x^2 -48*x)*log(3)+2*x^4-x^3-24*x^2)*exp(2*x)+36*log(3)^2+(6*x^2+72*x)*log(3)+3 *x^3+36*x^2)/((x*exp(2*x)^2-6*x*exp(2*x)+9*x)*exp(exp(4))^2+((-2*x*log(3)- 2*x^2)*exp(2*x)^2+(12*x*log(3)+12*x^2)*exp(2*x)-18*x*log(3)-18*x^2)*exp(ex p(4))+(x*log(3)^2+2*x^2*log(3)+x^3)*exp(2*x)^2+(-6*x*log(3)^2-12*x^2*log(3 )-6*x^3)*exp(2*x)+9*x*log(3)^2+18*x^2*log(3)+9*x^3),x, algorithm="giac")
Output:
(4*x*e^(2*x)*log(2*x) + 4*e^(2*x)*log(3)*log(2*x) - x^2 - 12*x*log(2*x) - 4*e^(2*x + e^4)*log(2*x) + 12*e^(e^4)*log(2*x) - 12*log(3)*log(2*x))/(x*e^ (2*x) + e^(2*x)*log(3) - 3*x - e^(2*x + e^4) + 3*e^(e^4) - 3*log(3))
Time = 4.00 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {e^{2 e^4} \left (36-24 e^{2 x}+4 e^{4 x}\right )+36 x^2+3 x^3+\left (72 x+6 x^2\right ) \log (3)+36 \log ^2(3)+e^{2 x} \left (-24 x^2-x^3+2 x^4+\left (-48 x-2 x^2+2 x^3\right ) \log (3)-24 \log ^2(3)\right )+e^{4 x} \left (4 x^2+8 x \log (3)+4 \log ^2(3)\right )+e^{e^4} \left (-72 x-6 x^2+e^{4 x} (-8 x-8 \log (3))-72 \log (3)+e^{2 x} \left (48 x+2 x^2-2 x^3+48 \log (3)\right )\right )}{9 x^3+e^{2 e^4} \left (9 x-6 e^{2 x} x+e^{4 x} x\right )+18 x^2 \log (3)+9 x \log ^2(3)+e^{2 x} \left (-6 x^3-12 x^2 \log (3)-6 x \log ^2(3)\right )+e^{4 x} \left (x^3+2 x^2 \log (3)+x \log ^2(3)\right )+e^{e^4} \left (-18 x^2-18 x \log (3)+e^{4 x} \left (-2 x^2-2 x \log (3)\right )+e^{2 x} \left (12 x^2+12 x \log (3)\right )\right )} \, dx=4\,\ln \left (x\right )-\frac {x^2\,\ln \left (3\right )-x^2\,{\mathrm {e}}^{{\mathrm {e}}^4}+x^3}{\left ({\mathrm {e}}^{2\,x}-3\right )\,{\left (x+\ln \left (3\right )-{\mathrm {e}}^{{\mathrm {e}}^4}\right )}^2} \] Input:
int((exp(2*exp(4))*(4*exp(4*x) - 24*exp(2*x) + 36) - exp(exp(4))*(72*x + 7 2*log(3) - exp(2*x)*(48*x + 48*log(3) + 2*x^2 - 2*x^3) + exp(4*x)*(8*x + 8 *log(3)) + 6*x^2) + log(3)*(72*x + 6*x^2) + exp(4*x)*(8*x*log(3) + 4*log(3 )^2 + 4*x^2) + 36*log(3)^2 + 36*x^2 + 3*x^3 - exp(2*x)*(log(3)*(48*x + 2*x ^2 - 2*x^3) + 24*log(3)^2 + 24*x^2 + x^3 - 2*x^4))/(exp(4*x)*(x*log(3)^2 + 2*x^2*log(3) + x^3) - exp(2*x)*(6*x*log(3)^2 + 12*x^2*log(3) + 6*x^3) + 9 *x*log(3)^2 + 18*x^2*log(3) + 9*x^3 + exp(2*exp(4))*(9*x - 6*x*exp(2*x) + x*exp(4*x)) - exp(exp(4))*(18*x*log(3) + exp(4*x)*(2*x*log(3) + 2*x^2) - e xp(2*x)*(12*x*log(3) + 12*x^2) + 18*x^2)),x)
Output:
4*log(x) - (x^2*log(3) - x^2*exp(exp(4)) + x^3)/((exp(2*x) - 3)*(x + log(3 ) - exp(exp(4)))^2)
Time = 0.18 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.09 \[ \int \frac {e^{2 e^4} \left (36-24 e^{2 x}+4 e^{4 x}\right )+36 x^2+3 x^3+\left (72 x+6 x^2\right ) \log (3)+36 \log ^2(3)+e^{2 x} \left (-24 x^2-x^3+2 x^4+\left (-48 x-2 x^2+2 x^3\right ) \log (3)-24 \log ^2(3)\right )+e^{4 x} \left (4 x^2+8 x \log (3)+4 \log ^2(3)\right )+e^{e^4} \left (-72 x-6 x^2+e^{4 x} (-8 x-8 \log (3))-72 \log (3)+e^{2 x} \left (48 x+2 x^2-2 x^3+48 \log (3)\right )\right )}{9 x^3+e^{2 e^4} \left (9 x-6 e^{2 x} x+e^{4 x} x\right )+18 x^2 \log (3)+9 x \log ^2(3)+e^{2 x} \left (-6 x^3-12 x^2 \log (3)-6 x \log ^2(3)\right )+e^{4 x} \left (x^3+2 x^2 \log (3)+x \log ^2(3)\right )+e^{e^4} \left (-18 x^2-18 x \log (3)+e^{4 x} \left (-2 x^2-2 x \log (3)\right )+e^{2 x} \left (12 x^2+12 x \log (3)\right )\right )} \, dx=\frac {4 e^{e^{4}+2 x} \mathrm {log}\left (x \right )-12 e^{e^{4}} \mathrm {log}\left (x \right )-4 e^{2 x} \mathrm {log}\left (x \right ) \mathrm {log}\left (3\right )-4 e^{2 x} \mathrm {log}\left (x \right ) x +12 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right )+12 \,\mathrm {log}\left (x \right ) x +x^{2}}{e^{e^{4}+2 x}-3 e^{e^{4}}-e^{2 x} \mathrm {log}\left (3\right )-e^{2 x} x +3 \,\mathrm {log}\left (3\right )+3 x} \] Input:
int(((4*exp(2*x)^2-24*exp(2*x)+36)*exp(exp(4))^2+((-8*log(3)-8*x)*exp(2*x) ^2+(48*log(3)-2*x^3+2*x^2+48*x)*exp(2*x)-72*log(3)-6*x^2-72*x)*exp(exp(4)) +(4*log(3)^2+8*x*log(3)+4*x^2)*exp(2*x)^2+(-24*log(3)^2+(2*x^3-2*x^2-48*x) *log(3)+2*x^4-x^3-24*x^2)*exp(2*x)+36*log(3)^2+(6*x^2+72*x)*log(3)+3*x^3+3 6*x^2)/((x*exp(2*x)^2-6*x*exp(2*x)+9*x)*exp(exp(4))^2+((-2*x*log(3)-2*x^2) *exp(2*x)^2+(12*x*log(3)+12*x^2)*exp(2*x)-18*x*log(3)-18*x^2)*exp(exp(4))+ (x*log(3)^2+2*x^2*log(3)+x^3)*exp(2*x)^2+(-6*x*log(3)^2-12*x^2*log(3)-6*x^ 3)*exp(2*x)+9*x*log(3)^2+18*x^2*log(3)+9*x^3),x)
Output:
(4*e**(e**4 + 2*x)*log(x) - 12*e**(e**4)*log(x) - 4*e**(2*x)*log(x)*log(3) - 4*e**(2*x)*log(x)*x + 12*log(x)*log(3) + 12*log(x)*x + x**2)/(e**(e**4 + 2*x) - 3*e**(e**4) - e**(2*x)*log(3) - e**(2*x)*x + 3*log(3) + 3*x)