Integrand size = 253, antiderivative size = 34 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=\frac {e^x}{\log \left (\left (3-e+\frac {4 \left (e^4+x+\frac {e^{-x} x}{3}\right )}{x}\right )^2\right )} \]
Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=\frac {e^x}{\log \left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right )^2}{9 x^2}\right )} \]
Integrate[(24*E^(4 + 2*x) + 8*E^x*x^2 + (4*E^x*x^2 + E^(2*x)*(12*E^4*x + 2 1*x^2 - 3*E*x^2))*Log[(16*x^2 + E^x*(96*E^4*x + 168*x^2 - 24*E*x^2) + E^(2 *x)*(144*E^8 + 441*x^2 - 126*E*x^2 + 9*E^2*x^2 + E^4*(504*x - 72*E*x)))/(9 *E^(2*x)*x^2)])/((4*x^2 + E^x*(12*E^4*x + 21*x^2 - 3*E*x^2))*Log[(16*x^2 + E^x*(96*E^4*x + 168*x^2 - 24*E*x^2) + E^(2*x)*(144*E^8 + 441*x^2 - 126*E* x^2 + 9*E^2*x^2 + E^4*(504*x - 72*E*x)))/(9*E^(2*x)*x^2)]^2),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.
| \(\displaystyle \int \frac {8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (-3 e x^2+21 x^2+12 e^4 x\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (-24 e x^2+168 x^2+96 e^4 x\right )+e^{2 x} \left (9 e^2 x^2-126 e x^2+441 x^2+e^4 (504 x-72 e x)+144 e^8\right )\right )}{9 x^2}\right )+24 e^{2 x+4}}{\left (4 x^2+e^x \left (-3 e x^2+21 x^2+12 e^4 x\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (-24 e x^2+168 x^2+96 e^4 x\right )+e^{2 x} \left (9 e^2 x^2-126 e x^2+441 x^2+e^4 (504 x-72 e x)+144 e^8\right )\right )}{9 x^2}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (-3 e x^2+21 x^2+12 e^4 x\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (-24 e x^2+168 x^2+96 e^4 x\right )+e^{2 x} \left (9 e^2 x^2-126 e x^2+441 x^2+e^4 (504 x-72 e x)+144 e^8\right )\right )}{9 x^2}\right )+24 e^{2 x+4}}{\left (4 x^2+e^x \left (-3 e x^2+21 x^2+12 e^4 x\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (21 \left (1-\frac {e}{7}\right ) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 x \left (-\left ((7-e) x^2\right )-4 e^4 x+4 e^4\right )}{3 \left ((7-e) x+4 e^4\right )^2 \left (21 \left (1-\frac {e}{7}\right ) e^x x+4 x+12 e^{x+4}\right ) \log ^2\left (\frac {e^{-2 x} \left (-3 (e-7) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}+\frac {e^x \left (7 \left (1-\frac {e}{7}\right ) x^2 \log \left (\frac {e^{-2 x} \left (21 e^x x-3 e^{x+1} x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )+4 e^4 x \log \left (\frac {e^{-2 x} \left (21 e^x x-3 e^{x+1} x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )+8 e^4\right )}{x \left ((7-e) x+4 e^4\right ) \log ^2\left (\frac {e^{-2 x} \left (-3 (e-7) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}+\frac {8 \left ((7-e) x^2+4 e^4 x-4 e^4\right )}{3 \left ((7-e) x+4 e^4\right )^2 \log ^2\left (\frac {e^{-2 x} \left (-3 (e-7) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 \int \frac {1}{\log ^2\left (\frac {e^{-2 x} \left (-3 (-7+e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}dx}{3 (7-e)}+2 \int \frac {e^x}{x \log ^2\left (\frac {e^{-2 x} \left (-3 (-7+e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}dx-\frac {32}{3} e^4 \int \frac {1}{\left ((7-e) x+4 e^4\right )^2 \log ^2\left (\frac {e^{-2 x} \left (-3 (-7+e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}dx-\frac {32 e^4 \int \frac {1}{\left ((7-e) x+4 e^4\right ) \log ^2\left (\frac {e^{-2 x} \left (-3 (-7+e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}dx}{3 (7-e)}-2 (7-e) \int \frac {e^x}{\left ((7-e) x+4 e^4\right ) \log ^2\left (\frac {e^{-2 x} \left (-3 (-7+e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}dx+\frac {128 e^4 \int \frac {1}{\left (21 \left (1-\frac {e}{7}\right ) e^x x+4 x+12 e^{x+4}\right ) \log ^2\left (\frac {e^{-2 x} \left (-3 (-7+e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}dx}{3 (7-e)^2}-\frac {32 \int \frac {x}{\left (21 \left (1-\frac {e}{7}\right ) e^x x+4 x+12 e^{x+4}\right ) \log ^2\left (\frac {e^{-2 x} \left (-3 (-7+e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}dx}{3 (7-e)}-\frac {512 e^8 \int \frac {1}{\left ((7-e) x+4 e^4\right )^2 \left (21 \left (1-\frac {e}{7}\right ) e^x x+4 x+12 e^{x+4}\right ) \log ^2\left (\frac {e^{-2 x} \left (-3 (-7+e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}dx}{3 (7-e)}+\frac {128 e^4 \left (7-e-4 e^4\right ) \int \frac {1}{\left ((7-e) x+4 e^4\right ) \left (21 \left (1-\frac {e}{7}\right ) e^x x+4 x+12 e^{x+4}\right ) \log ^2\left (\frac {e^{-2 x} \left (-3 (-7+e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}dx}{3 (7-e)^2}+\int \frac {e^x}{\log \left (\frac {e^{-2 x} \left (-3 (-7+e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}dx\) |
Int[(24*E^(4 + 2*x) + 8*E^x*x^2 + (4*E^x*x^2 + E^(2*x)*(12*E^4*x + 21*x^2 - 3*E*x^2))*Log[(16*x^2 + E^x*(96*E^4*x + 168*x^2 - 24*E*x^2) + E^(2*x)*(1 44*E^8 + 441*x^2 - 126*E*x^2 + 9*E^2*x^2 + E^4*(504*x - 72*E*x)))/(9*E^(2* x)*x^2)])/((4*x^2 + E^x*(12*E^4*x + 21*x^2 - 3*E*x^2))*Log[(16*x^2 + E^x*( 96*E^4*x + 168*x^2 - 24*E*x^2) + E^(2*x)*(144*E^8 + 441*x^2 - 126*E*x^2 + 9*E^2*x^2 + E^4*(504*x - 72*E*x)))/(9*E^(2*x)*x^2)]^2),x]
3.6.77.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(87\) vs. \(2(30)=60\).
Time = 1.66 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.59
| method | result | size |
| parallelrisch | \(\frac {{\mathrm e}^{x}}{\ln \left (\frac {\left (\left (144 \,{\mathrm e}^{8}+\left (-72 x \,{\mathrm e}+504 x \right ) {\mathrm e}^{4}+9 x^{2} {\mathrm e}^{2}-126 x^{2} {\mathrm e}+441 x^{2}\right ) {\mathrm e}^{2 x}+\left (96 x \,{\mathrm e}^{4}-24 x^{2} {\mathrm e}+168 x^{2}\right ) {\mathrm e}^{x}+16 x^{2}\right ) {\mathrm e}^{-2 x}}{9 x^{2}}\right )}\) | \(88\) |
| risch | \(\text {Expression too large to display}\) | \(709\) |
int((((12*x*exp(4)-3*x^2*exp(1)+21*x^2)*exp(x)^2+4*exp(x)*x^2)*ln(1/9*((14 4*exp(4)^2+(-72*x*exp(1)+504*x)*exp(4)+9*x^2*exp(1)^2-126*x^2*exp(1)+441*x ^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168*x^2)*exp(x)+16*x^2)/exp(x)^2/x ^2)+24*exp(4)*exp(x)^2+8*exp(x)*x^2)/((12*x*exp(4)-3*x^2*exp(1)+21*x^2)*ex p(x)+4*x^2)/ln(1/9*((144*exp(4)^2+(-72*x*exp(1)+504*x)*exp(4)+9*x^2*exp(1) ^2-126*x^2*exp(1)+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168*x^2)*ex p(x)+16*x^2)/exp(x)^2/x^2)^2,x,method=_RETURNVERBOSE)
exp(x)/ln(1/9*((144*exp(4)^2+(-72*x*exp(1)+504*x)*exp(4)+9*x^2*exp(1)^2-12 6*x^2*exp(1)+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168*x^2)*exp(x)+ 16*x^2)/exp(x)^2/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (35) = 70\).
Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.38 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=\frac {e^{x}}{\log \left (\frac {{\left (16 \, x^{2} + 9 \, {\left (x^{2} e^{2} - 14 \, x^{2} e + 49 \, x^{2} - 8 \, x e^{5} + 56 \, x e^{4} + 16 \, e^{8}\right )} e^{\left (2 \, x\right )} - 24 \, {\left (x^{2} e - 7 \, x^{2} - 4 \, x e^{4}\right )} e^{x}\right )} e^{\left (-2 \, x\right )}}{9 \, x^{2}}\right )} \]
integrate((((12*x*exp(4)-3*x^2*exp(1)+21*x^2)*exp(x)^2+4*exp(x)*x^2)*log(1 /9*((144*exp(4)^2+(-72*x*exp(1)+504*x)*exp(4)+9*x^2*exp(1)^2-126*x^2*exp(1 )+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168*x^2)*exp(x)+16*x^2)/exp (x)^2/x^2)+24*exp(4)*exp(x)^2+8*exp(x)*x^2)/((12*x*exp(4)-3*x^2*exp(1)+21* x^2)*exp(x)+4*x^2)/log(1/9*((144*exp(4)^2+(-72*x*exp(1)+504*x)*exp(4)+9*x^ 2*exp(1)^2-126*x^2*exp(1)+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168 *x^2)*exp(x)+16*x^2)/exp(x)^2/x^2)^2,x, algorithm=\
e^x/log(1/9*(16*x^2 + 9*(x^2*e^2 - 14*x^2*e + 49*x^2 - 8*x*e^5 + 56*x*e^4 + 16*e^8)*e^(2*x) - 24*(x^2*e - 7*x^2 - 4*x*e^4)*e^x)*e^(-2*x)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (29) = 58\).
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.71 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=\frac {e^{x}}{\log {\left (\frac {\left (\frac {16 x^{2}}{9} + \frac {\left (- 24 e x^{2} + 168 x^{2} + 96 x e^{4}\right ) e^{x}}{9} + \frac {\left (- 126 e x^{2} + 9 x^{2} e^{2} + 441 x^{2} + \left (- 72 e x + 504 x\right ) e^{4} + 144 e^{8}\right ) e^{2 x}}{9}\right ) e^{- 2 x}}{x^{2}} \right )}} \]
integrate((((12*x*exp(4)-3*x**2*exp(1)+21*x**2)*exp(x)**2+4*exp(x)*x**2)*l n(1/9*((144*exp(4)**2+(-72*x*exp(1)+504*x)*exp(4)+9*x**2*exp(1)**2-126*x** 2*exp(1)+441*x**2)*exp(x)**2+(96*x*exp(4)-24*x**2*exp(1)+168*x**2)*exp(x)+ 16*x**2)/exp(x)**2/x**2)+24*exp(4)*exp(x)**2+8*exp(x)*x**2)/((12*x*exp(4)- 3*x**2*exp(1)+21*x**2)*exp(x)+4*x**2)/ln(1/9*((144*exp(4)**2+(-72*x*exp(1) +504*x)*exp(4)+9*x**2*exp(1)**2-126*x**2*exp(1)+441*x**2)*exp(x)**2+(96*x* exp(4)-24*x**2*exp(1)+168*x**2)*exp(x)+16*x**2)/exp(x)**2/x**2)**2,x)
exp(x)/log((16*x**2/9 + (-24*E*x**2 + 168*x**2 + 96*x*exp(4))*exp(x)/9 + ( -126*E*x**2 + 9*x**2*exp(2) + 441*x**2 + (-72*E*x + 504*x)*exp(4) + 144*ex p(8))*exp(2*x)/9)*exp(-2*x)/x**2)
Time = 0.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=-\frac {e^{x}}{2 \, {\left (x + \log \left (3\right ) - \log \left (3 \, {\left (x {\left (e - 7\right )} - 4 \, e^{4}\right )} e^{x} - 4 \, x\right ) + \log \left (x\right )\right )}} \]
integrate((((12*x*exp(4)-3*x^2*exp(1)+21*x^2)*exp(x)^2+4*exp(x)*x^2)*log(1 /9*((144*exp(4)^2+(-72*x*exp(1)+504*x)*exp(4)+9*x^2*exp(1)^2-126*x^2*exp(1 )+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168*x^2)*exp(x)+16*x^2)/exp (x)^2/x^2)+24*exp(4)*exp(x)^2+8*exp(x)*x^2)/((12*x*exp(4)-3*x^2*exp(1)+21* x^2)*exp(x)+4*x^2)/log(1/9*((144*exp(4)^2+(-72*x*exp(1)+504*x)*exp(4)+9*x^ 2*exp(1)^2-126*x^2*exp(1)+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168 *x^2)*exp(x)+16*x^2)/exp(x)^2/x^2)^2,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (35) = 70\).
Time = 4.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.15 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=-\frac {e^{x}}{\log \left (9 \, x^{2}\right ) - \log \left ({\left (441 \, x^{2} e^{\left (2 \, x\right )} + 9 \, x^{2} e^{\left (2 \, x + 2\right )} - 126 \, x^{2} e^{\left (2 \, x + 1\right )} - 24 \, x^{2} e^{\left (x + 1\right )} + 168 \, x^{2} e^{x} + 16 \, x^{2} - 72 \, x e^{\left (2 \, x + 5\right )} + 504 \, x e^{\left (2 \, x + 4\right )} + 96 \, x e^{\left (x + 4\right )} + 144 \, e^{\left (2 \, x + 8\right )}\right )} e^{\left (-2 \, x\right )}\right )} \]
integrate((((12*x*exp(4)-3*x^2*exp(1)+21*x^2)*exp(x)^2+4*exp(x)*x^2)*log(1 /9*((144*exp(4)^2+(-72*x*exp(1)+504*x)*exp(4)+9*x^2*exp(1)^2-126*x^2*exp(1 )+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168*x^2)*exp(x)+16*x^2)/exp (x)^2/x^2)+24*exp(4)*exp(x)^2+8*exp(x)*x^2)/((12*x*exp(4)-3*x^2*exp(1)+21* x^2)*exp(x)+4*x^2)/log(1/9*((144*exp(4)^2+(-72*x*exp(1)+504*x)*exp(4)+9*x^ 2*exp(1)^2-126*x^2*exp(1)+441*x^2)*exp(x)^2+(96*x*exp(4)-24*x^2*exp(1)+168 *x^2)*exp(x)+16*x^2)/exp(x)^2/x^2)^2,x, algorithm=\
-e^x/(log(9*x^2) - log((441*x^2*e^(2*x) + 9*x^2*e^(2*x + 2) - 126*x^2*e^(2 *x + 1) - 24*x^2*e^(x + 1) + 168*x^2*e^x + 16*x^2 - 72*x*e^(2*x + 5) + 504 *x*e^(2*x + 4) + 96*x*e^(x + 4) + 144*e^(2*x + 8))*e^(-2*x)))
Time = 8.87 (sec) , antiderivative size = 326, normalized size of antiderivative = 9.59 \[ \int \frac {24 e^{4+2 x}+8 e^x x^2+\left (4 e^x x^2+e^{2 x} \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log \left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )}{\left (4 x^2+e^x \left (12 e^4 x+21 x^2-3 e x^2\right )\right ) \log ^2\left (\frac {e^{-2 x} \left (16 x^2+e^x \left (96 e^4 x+168 x^2-24 e x^2\right )+e^{2 x} \left (144 e^8+441 x^2-126 e x^2+9 e^2 x^2+e^4 (504 x-72 e x)\right )\right )}{9 x^2}\right )} \, dx=\frac {{\mathrm {e}}^x+\frac {x\,{\mathrm {e}}^x\,\ln \left (\frac {{\mathrm {e}}^{-2\,x}\,\left (\frac {{\mathrm {e}}^{2\,x}\,\left (144\,{\mathrm {e}}^8+{\mathrm {e}}^4\,\left (504\,x-72\,x\,\mathrm {e}\right )-126\,x^2\,\mathrm {e}+9\,x^2\,{\mathrm {e}}^2+441\,x^2\right )}{9}+\frac {{\mathrm {e}}^x\,\left (96\,x\,{\mathrm {e}}^4-24\,x^2\,\mathrm {e}+168\,x^2\right )}{9}+\frac {16\,x^2}{9}\right )}{x^2}\right )\,\left (4\,x+12\,{\mathrm {e}}^{x+4}-3\,x\,{\mathrm {e}}^{x+1}+21\,x\,{\mathrm {e}}^x\right )}{8\,\left (3\,{\mathrm {e}}^{x+4}+x^2\right )}}{\ln \left (\frac {{\mathrm {e}}^{-2\,x}\,\left (\frac {{\mathrm {e}}^{2\,x}\,\left (144\,{\mathrm {e}}^8+{\mathrm {e}}^4\,\left (504\,x-72\,x\,\mathrm {e}\right )-126\,x^2\,\mathrm {e}+9\,x^2\,{\mathrm {e}}^2+441\,x^2\right )}{9}+\frac {{\mathrm {e}}^x\,\left (96\,x\,{\mathrm {e}}^4-24\,x^2\,\mathrm {e}+168\,x^2\right )}{9}+\frac {16\,x^2}{9}\right )}{x^2}\right )}-\frac {x^2\,{\mathrm {e}}^{-4}}{6}+\frac {x^3\,{\mathrm {e}}^{-4}}{6}-{\mathrm {e}}^x\,\left (\frac {x}{2}-\frac {x^2\,{\mathrm {e}}^{-4}\,\left (\mathrm {e}-7\right )}{8}\right )-\frac {x^4\,{\mathrm {e}}^{-8}\,\left (\mathrm {e}-7\right )}{24}+\frac {{\mathrm {e}}^{-8}\,\left (2\,x^7\,\mathrm {e}+8\,x^5\,{\mathrm {e}}^4-x^8\,\mathrm {e}-12\,x^6\,{\mathrm {e}}^4+4\,x^7\,{\mathrm {e}}^4-14\,x^7+7\,x^8\right )}{72\,\left (2\,x\,{\mathrm {e}}^4-x^2\,{\mathrm {e}}^4\right )\,\left ({\mathrm {e}}^x+\frac {x^2\,{\mathrm {e}}^{-4}}{3}\right )} \]
int((24*exp(2*x)*exp(4) + 8*x^2*exp(x) + log((exp(-2*x)*((exp(2*x)*(144*ex p(8) + exp(4)*(504*x - 72*x*exp(1)) - 126*x^2*exp(1) + 9*x^2*exp(2) + 441* x^2))/9 + (exp(x)*(96*x*exp(4) - 24*x^2*exp(1) + 168*x^2))/9 + (16*x^2)/9) )/x^2)*(exp(2*x)*(12*x*exp(4) - 3*x^2*exp(1) + 21*x^2) + 4*x^2*exp(x)))/(l og((exp(-2*x)*((exp(2*x)*(144*exp(8) + exp(4)*(504*x - 72*x*exp(1)) - 126* x^2*exp(1) + 9*x^2*exp(2) + 441*x^2))/9 + (exp(x)*(96*x*exp(4) - 24*x^2*ex p(1) + 168*x^2))/9 + (16*x^2)/9))/x^2)^2*(exp(x)*(12*x*exp(4) - 3*x^2*exp( 1) + 21*x^2) + 4*x^2)),x)
(exp(x) + (x*exp(x)*log((exp(-2*x)*((exp(2*x)*(144*exp(8) + exp(4)*(504*x - 72*x*exp(1)) - 126*x^2*exp(1) + 9*x^2*exp(2) + 441*x^2))/9 + (exp(x)*(96 *x*exp(4) - 24*x^2*exp(1) + 168*x^2))/9 + (16*x^2)/9))/x^2)*(4*x + 12*exp( x + 4) - 3*x*exp(x + 1) + 21*x*exp(x)))/(8*(3*exp(x + 4) + x^2)))/log((exp (-2*x)*((exp(2*x)*(144*exp(8) + exp(4)*(504*x - 72*x*exp(1)) - 126*x^2*exp (1) + 9*x^2*exp(2) + 441*x^2))/9 + (exp(x)*(96*x*exp(4) - 24*x^2*exp(1) + 168*x^2))/9 + (16*x^2)/9))/x^2) - (x^2*exp(-4))/6 + (x^3*exp(-4))/6 - exp( x)*(x/2 - (x^2*exp(-4)*(exp(1) - 7))/8) - (x^4*exp(-8)*(exp(1) - 7))/24 + (exp(-8)*(2*x^7*exp(1) + 8*x^5*exp(4) - x^8*exp(1) - 12*x^6*exp(4) + 4*x^7 *exp(4) - 14*x^7 + 7*x^8))/(72*(2*x*exp(4) - x^2*exp(4))*(exp(x) + (x^2*ex p(-4))/3))