Integrand size = 186, antiderivative size = 32 \[ \int \frac {392+112 x+6 x^2+e^{e^{e^3}} \left (16 x+24 e^x x+12 e^{2 x} x+2 e^{3 x} x\right )+(112+16 x) \log (5)+8 \log ^2(5)+e^{3 x} \left (49+28 x+3 x^2+(14+4 x) \log (5)+\log ^2(5)\right )+e^{2 x} \left (294+140 x+26 x^2+2 x^3+\left (84+20 x+2 x^2\right ) \log (5)+6 \log ^2(5)\right )+e^x \left (588+224 x+43 x^2+2 x^3+\left (168+32 x+4 x^2\right ) \log (5)+12 \log ^2(5)\right )}{8+12 e^x+6 e^{2 x}+e^{3 x}} \, dx=x \left (e^{e^{e^3}} x+\left (-7-x+\frac {x}{2+e^x}-\log (5)\right )^2\right ) \]
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59 \[ \int \frac {392+112 x+6 x^2+e^{e^{e^3}} \left (16 x+24 e^x x+12 e^{2 x} x+2 e^{3 x} x\right )+(112+16 x) \log (5)+8 \log ^2(5)+e^{3 x} \left (49+28 x+3 x^2+(14+4 x) \log (5)+\log ^2(5)\right )+e^{2 x} \left (294+140 x+26 x^2+2 x^3+\left (84+20 x+2 x^2\right ) \log (5)+6 \log ^2(5)\right )+e^x \left (588+224 x+43 x^2+2 x^3+\left (168+32 x+4 x^2\right ) \log (5)+12 \log ^2(5)\right )}{8+12 e^x+6 e^{2 x}+e^{3 x}} \, dx=x \left (x^2+\frac {x^2}{\left (2+e^x\right )^2}+(7+\log (5))^2-\frac {2 x (7+x+\log (5))}{2+e^x}+x \left (14+e^{e^{e^3}}+\log (25)\right )\right ) \]
Integrate[(392 + 112*x + 6*x^2 + E^E^E^3*(16*x + 24*E^x*x + 12*E^(2*x)*x + 2*E^(3*x)*x) + (112 + 16*x)*Log[5] + 8*Log[5]^2 + E^(3*x)*(49 + 28*x + 3* x^2 + (14 + 4*x)*Log[5] + Log[5]^2) + E^(2*x)*(294 + 140*x + 26*x^2 + 2*x^ 3 + (84 + 20*x + 2*x^2)*Log[5] + 6*Log[5]^2) + E^x*(588 + 224*x + 43*x^2 + 2*x^3 + (168 + 32*x + 4*x^2)*Log[5] + 12*Log[5]^2))/(8 + 12*E^x + 6*E^(2* x) + E^(3*x)),x]
x*(x^2 + x^2/(2 + E^x)^2 + (7 + Log[5])^2 - (2*x*(7 + x + Log[5]))/(2 + E^ x) + x*(14 + E^E^E^3 + Log[25]))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.79 (sec) , antiderivative size = 381, normalized size of antiderivative = 11.91, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {7292, 7293, 2009}
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.
| \(\displaystyle \int \frac {6 x^2+e^{3 x} \left (3 x^2+28 x+(4 x+14) \log (5)+49+\log ^2(5)\right )+e^{2 x} \left (2 x^3+26 x^2+\left (2 x^2+20 x+84\right ) \log (5)+140 x+294+6 \log ^2(5)\right )+e^x \left (2 x^3+43 x^2+\left (4 x^2+32 x+168\right ) \log (5)+224 x+588+12 \log ^2(5)\right )+112 x+e^{e^{e^3}} \left (24 e^x x+12 e^{2 x} x+2 e^{3 x} x+16 x\right )+(16 x+112) \log (5)+392+8 \log ^2(5)}{12 e^x+6 e^{2 x}+e^{3 x}+8} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {6 x^2+e^{3 x} \left (3 x^2+28 x+(4 x+14) \log (5)+49+\log ^2(5)\right )+e^{2 x} \left (2 x^3+26 x^2+\left (2 x^2+20 x+84\right ) \log (5)+140 x+294+6 \log ^2(5)\right )+e^x \left (2 x^3+43 x^2+\left (4 x^2+32 x+168\right ) \log (5)+224 x+588+12 \log ^2(5)\right )+112 x+e^{e^{e^3}} \left (24 e^x x+12 e^{2 x} x+2 e^{3 x} x+16 x\right )+(16 x+112) \log (5)+392 \left (1+\frac {\log ^2(5)}{49}\right )}{\left (e^x+2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 x^3}{\left (e^x+2\right )^3}+3 x^2-\frac {x^2 (6 x+25+\log (625))}{\left (e^x+2\right )^2}+\frac {2 x \left (x^2+x (4+\log (5))-2 (7+\log (5))\right )}{e^x+2}+28 x \left (1+\frac {1}{14} \left (e^{e^{e^3}}+\log (25)\right )\right )+49 \left (1+\frac {1}{49} \log (5) (14+\log (5))\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {9}{2} x \operatorname {PolyLog}\left (2,-\frac {e^x}{2}\right )-\frac {3}{2} \operatorname {PolyLog}\left (2,-\frac {e^x}{2}\right )+\frac {9}{2} \operatorname {PolyLog}\left (3,-\frac {e^x}{2}\right )+\frac {1}{2} x (25+\log (625)) \operatorname {PolyLog}\left (2,-\frac {e^x}{2}\right )-2 x (4+\log (5)) \operatorname {PolyLog}\left (2,-\frac {e^x}{2}\right )-\frac {1}{2} (25+\log (625)) \operatorname {PolyLog}\left (2,-\frac {e^x}{2}\right )+2 (7+\log (5)) \operatorname {PolyLog}\left (2,-\frac {e^x}{2}\right )-\frac {1}{2} (25+\log (625)) \operatorname {PolyLog}\left (3,-\frac {e^x}{2}\right )+2 (4+\log (5)) \operatorname {PolyLog}\left (3,-\frac {e^x}{2}\right )-\frac {2 x^3}{e^x+2}+\frac {x^3}{\left (e^x+2\right )^2}+\frac {7 x^3}{4}-\frac {1}{12} x^3 (25+\log (625))+\frac {1}{3} x^3 (4+\log (5))-\frac {3 x^2}{2 \left (e^x+2\right )}+\frac {3 x^2}{4}+\frac {1}{4} x^2 (25+\log (625)) \log \left (\frac {e^x}{2}+1\right )-x^2 (4+\log (5)) \log \left (\frac {e^x}{2}+1\right )-\frac {9}{4} x^2 \log \left (\frac {e^x}{2}+1\right )-\frac {x^2 (25+\log (625))}{2 \left (e^x+2\right )}+\frac {1}{4} x^2 (25+\log (625))+x^2 \left (14+e^{e^{e^3}}+\log (25)\right )-x^2 (7+\log (5))-\frac {1}{2} x (25+\log (625)) \log \left (\frac {e^x}{2}+1\right )+2 x (7+\log (5)) \log \left (\frac {e^x}{2}+1\right )-\frac {3}{2} x \log \left (\frac {e^x}{2}+1\right )+x (7+\log (5))^2\) |
Int[(392 + 112*x + 6*x^2 + E^E^E^3*(16*x + 24*E^x*x + 12*E^(2*x)*x + 2*E^( 3*x)*x) + (112 + 16*x)*Log[5] + 8*Log[5]^2 + E^(3*x)*(49 + 28*x + 3*x^2 + (14 + 4*x)*Log[5] + Log[5]^2) + E^(2*x)*(294 + 140*x + 26*x^2 + 2*x^3 + (8 4 + 20*x + 2*x^2)*Log[5] + 6*Log[5]^2) + E^x*(588 + 224*x + 43*x^2 + 2*x^3 + (168 + 32*x + 4*x^2)*Log[5] + 12*Log[5]^2))/(8 + 12*E^x + 6*E^(2*x) + E ^(3*x)),x]
(3*x^2)/4 - (3*x^2)/(2*(2 + E^x)) + (7*x^3)/4 + x^3/(2 + E^x)^2 - (2*x^3)/ (2 + E^x) + (x^3*(4 + Log[5]))/3 - x^2*(7 + Log[5]) + x*(7 + Log[5])^2 + x ^2*(14 + E^E^E^3 + Log[25]) + (x^2*(25 + Log[625]))/4 - (x^2*(25 + Log[625 ]))/(2*(2 + E^x)) - (x^3*(25 + Log[625]))/12 - (3*x*Log[1 + E^x/2])/2 - (9 *x^2*Log[1 + E^x/2])/4 - x^2*(4 + Log[5])*Log[1 + E^x/2] + 2*x*(7 + Log[5] )*Log[1 + E^x/2] - (x*(25 + Log[625])*Log[1 + E^x/2])/2 + (x^2*(25 + Log[6 25])*Log[1 + E^x/2])/4 - (3*PolyLog[2, -1/2*E^x])/2 - (9*x*PolyLog[2, -1/2 *E^x])/2 - 2*x*(4 + Log[5])*PolyLog[2, -1/2*E^x] + 2*(7 + Log[5])*PolyLog[ 2, -1/2*E^x] - ((25 + Log[625])*PolyLog[2, -1/2*E^x])/2 + (x*(25 + Log[625 ])*PolyLog[2, -1/2*E^x])/2 + (9*PolyLog[3, -1/2*E^x])/2 + 2*(4 + Log[5])*P olyLog[3, -1/2*E^x] - ((25 + Log[625])*PolyLog[3, -1/2*E^x])/2
3.5.20.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(28)=56\).
Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31
| method | result | size |
| risch | \(x \ln \left (5\right )^{2}+2 x^{2} \ln \left (5\right )+{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3}}} x^{2}+x^{3}+14 x \ln \left (5\right )+14 x^{2}+49 x -\frac {x^{2} \left (2 \,{\mathrm e}^{x} \ln \left (5\right )+2 \,{\mathrm e}^{x} x +4 \ln \left (5\right )+3 x +14 \,{\mathrm e}^{x}+28\right )}{\left ({\mathrm e}^{x}+2\right )^{2}}\) | \(74\) |
| norman | \(\frac {x^{3}+\left (28+4 \ln \left (5\right )+4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3}}}\right ) x^{2}+\left (4 \ln \left (5\right )^{2}+56 \ln \left (5\right )+196\right ) x +{\mathrm e}^{2 x} x^{3}+\left (42+6 \ln \left (5\right )+4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3}}}\right ) x^{2} {\mathrm e}^{x}+\left (49+\ln \left (5\right )^{2}+14 \ln \left (5\right )\right ) x \,{\mathrm e}^{2 x}+\left (4 \ln \left (5\right )^{2}+56 \ln \left (5\right )+196\right ) x \,{\mathrm e}^{x}+\left (2 \ln \left (5\right )+{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3}}}+14\right ) x^{2} {\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x^{3}}{\left ({\mathrm e}^{x}+2\right )^{2}}\) | \(125\) |
| parallelrisch | \(\frac {\ln \left (5\right )^{2} x \,{\mathrm e}^{2 x}+2 \ln \left (5\right ) {\mathrm e}^{2 x} x^{2}+x^{2} {\mathrm e}^{2 x} {\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3}}}+{\mathrm e}^{2 x} x^{3}+4 x \ln \left (5\right )^{2} {\mathrm e}^{x}+6 x^{2} \ln \left (5\right ) {\mathrm e}^{x}+14 \ln \left (5\right ) {\mathrm e}^{2 x} x +4 \,{\mathrm e}^{x} x^{2} {\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3}}}+2 \,{\mathrm e}^{x} x^{3}+14 \,{\mathrm e}^{2 x} x^{2}+4 x \ln \left (5\right )^{2}+4 x^{2} \ln \left (5\right )+56 x \,{\mathrm e}^{x} \ln \left (5\right )+4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3}}} x^{2}+x^{3}+42 \,{\mathrm e}^{x} x^{2}+49 x \,{\mathrm e}^{2 x}+56 x \ln \left (5\right )+28 x^{2}+196 \,{\mathrm e}^{x} x +196 x}{{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+4}\) | \(175\) |
int(((2*x*exp(x)^3+12*x*exp(x)^2+24*exp(x)*x+16*x)*exp(exp(exp(3)))+(ln(5) ^2+(4*x+14)*ln(5)+3*x^2+28*x+49)*exp(x)^3+(6*ln(5)^2+(2*x^2+20*x+84)*ln(5) +2*x^3+26*x^2+140*x+294)*exp(x)^2+(12*ln(5)^2+(4*x^2+32*x+168)*ln(5)+2*x^3 +43*x^2+224*x+588)*exp(x)+8*ln(5)^2+(16*x+112)*ln(5)+6*x^2+112*x+392)/(exp (x)^3+6*exp(x)^2+12*exp(x)+8),x,method=_RETURNVERBOSE)
x*ln(5)^2+2*x^2*ln(5)+exp(exp(exp(3)))*x^2+x^3+14*x*ln(5)+14*x^2+49*x-x^2* (2*exp(x)*ln(5)+2*exp(x)*x+4*ln(5)+3*x+14*exp(x)+28)/(exp(x)+2)^2
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.31 \[ \int \frac {392+112 x+6 x^2+e^{e^{e^3}} \left (16 x+24 e^x x+12 e^{2 x} x+2 e^{3 x} x\right )+(112+16 x) \log (5)+8 \log ^2(5)+e^{3 x} \left (49+28 x+3 x^2+(14+4 x) \log (5)+\log ^2(5)\right )+e^{2 x} \left (294+140 x+26 x^2+2 x^3+\left (84+20 x+2 x^2\right ) \log (5)+6 \log ^2(5)\right )+e^x \left (588+224 x+43 x^2+2 x^3+\left (168+32 x+4 x^2\right ) \log (5)+12 \log ^2(5)\right )}{8+12 e^x+6 e^{2 x}+e^{3 x}} \, dx=\frac {x^{3} + 4 \, x \log \left (5\right )^{2} + 28 \, x^{2} + {\left (x^{3} + x \log \left (5\right )^{2} + 14 \, x^{2} + 2 \, {\left (x^{2} + 7 \, x\right )} \log \left (5\right ) + 49 \, x\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{3} + 2 \, x \log \left (5\right )^{2} + 21 \, x^{2} + {\left (3 \, x^{2} + 28 \, x\right )} \log \left (5\right ) + 98 \, x\right )} e^{x} + {\left (x^{2} e^{\left (2 \, x\right )} + 4 \, x^{2} e^{x} + 4 \, x^{2}\right )} e^{\left (e^{\left (e^{3}\right )}\right )} + 4 \, {\left (x^{2} + 14 \, x\right )} \log \left (5\right ) + 196 \, x}{e^{\left (2 \, x\right )} + 4 \, e^{x} + 4} \]
integrate(((2*x*exp(x)^3+12*x*exp(x)^2+24*exp(x)*x+16*x)*exp(exp(exp(3)))+ (log(5)^2+(4*x+14)*log(5)+3*x^2+28*x+49)*exp(x)^3+(6*log(5)^2+(2*x^2+20*x+ 84)*log(5)+2*x^3+26*x^2+140*x+294)*exp(x)^2+(12*log(5)^2+(4*x^2+32*x+168)* log(5)+2*x^3+43*x^2+224*x+588)*exp(x)+8*log(5)^2+(16*x+112)*log(5)+6*x^2+1 12*x+392)/(exp(x)^3+6*exp(x)^2+12*exp(x)+8),x, algorithm=\
(x^3 + 4*x*log(5)^2 + 28*x^2 + (x^3 + x*log(5)^2 + 14*x^2 + 2*(x^2 + 7*x)* log(5) + 49*x)*e^(2*x) + 2*(x^3 + 2*x*log(5)^2 + 21*x^2 + (3*x^2 + 28*x)*l og(5) + 98*x)*e^x + (x^2*e^(2*x) + 4*x^2*e^x + 4*x^2)*e^(e^(e^3)) + 4*(x^2 + 14*x)*log(5) + 196*x)/(e^(2*x) + 4*e^x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (24) = 48\).
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.72 \[ \int \frac {392+112 x+6 x^2+e^{e^{e^3}} \left (16 x+24 e^x x+12 e^{2 x} x+2 e^{3 x} x\right )+(112+16 x) \log (5)+8 \log ^2(5)+e^{3 x} \left (49+28 x+3 x^2+(14+4 x) \log (5)+\log ^2(5)\right )+e^{2 x} \left (294+140 x+26 x^2+2 x^3+\left (84+20 x+2 x^2\right ) \log (5)+6 \log ^2(5)\right )+e^x \left (588+224 x+43 x^2+2 x^3+\left (168+32 x+4 x^2\right ) \log (5)+12 \log ^2(5)\right )}{8+12 e^x+6 e^{2 x}+e^{3 x}} \, dx=x^{3} + x^{2} \cdot \left (2 \log {\left (5 \right )} + 14 + e^{e^{e^{3}}}\right ) + x \left (\log {\left (5 \right )}^{2} + 14 \log {\left (5 \right )} + 49\right ) + \frac {- 3 x^{3} - 28 x^{2} - 4 x^{2} \log {\left (5 \right )} + \left (- 2 x^{3} - 14 x^{2} - 2 x^{2} \log {\left (5 \right )}\right ) e^{x}}{e^{2 x} + 4 e^{x} + 4} \]
integrate(((2*x*exp(x)**3+12*x*exp(x)**2+24*exp(x)*x+16*x)*exp(exp(exp(3)) )+(ln(5)**2+(4*x+14)*ln(5)+3*x**2+28*x+49)*exp(x)**3+(6*ln(5)**2+(2*x**2+2 0*x+84)*ln(5)+2*x**3+26*x**2+140*x+294)*exp(x)**2+(12*ln(5)**2+(4*x**2+32* x+168)*ln(5)+2*x**3+43*x**2+224*x+588)*exp(x)+8*ln(5)**2+(16*x+112)*ln(5)+ 6*x**2+112*x+392)/(exp(x)**3+6*exp(x)**2+12*exp(x)+8),x)
x**3 + x**2*(2*log(5) + 14 + exp(exp(exp(3)))) + x*(log(5)**2 + 14*log(5) + 49) + (-3*x**3 - 28*x**2 - 4*x**2*log(5) + (-2*x**3 - 14*x**2 - 2*x**2*l og(5))*exp(x))/(exp(2*x) + 4*exp(x) + 4)
Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (25) = 50\).
Time = 0.32 (sec) , antiderivative size = 207, normalized size of antiderivative = 6.47 \[ \int \frac {392+112 x+6 x^2+e^{e^{e^3}} \left (16 x+24 e^x x+12 e^{2 x} x+2 e^{3 x} x\right )+(112+16 x) \log (5)+8 \log ^2(5)+e^{3 x} \left (49+28 x+3 x^2+(14+4 x) \log (5)+\log ^2(5)\right )+e^{2 x} \left (294+140 x+26 x^2+2 x^3+\left (84+20 x+2 x^2\right ) \log (5)+6 \log ^2(5)\right )+e^x \left (588+224 x+43 x^2+2 x^3+\left (168+32 x+4 x^2\right ) \log (5)+12 \log ^2(5)\right )}{8+12 e^x+6 e^{2 x}+e^{3 x}} \, dx={\left (x + \frac {2 \, {\left (e^{x} + 3\right )}}{e^{\left (2 \, x\right )} + 4 \, e^{x} + 4} - \log \left (e^{x} + 2\right )\right )} \log \left (5\right )^{2} + 14 \, {\left (x + \frac {2 \, {\left (e^{x} + 3\right )}}{e^{\left (2 \, x\right )} + 4 \, e^{x} + 4} - \log \left (e^{x} + 2\right )\right )} \log \left (5\right ) + {\left (\log \left (5\right )^{2} + 14 \, \log \left (5\right ) + 49\right )} \log \left (e^{x} + 2\right ) + 49 \, x + \frac {x^{3} + 4 \, x^{2} {\left (e^{\left (e^{\left (e^{3}\right )}\right )} + \log \left (5\right ) + 7\right )} + {\left (x^{3} + x^{2} {\left (e^{\left (e^{\left (e^{3}\right )}\right )} + 2 \, \log \left (5\right ) + 14\right )}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{3} + x^{2} {\left (2 \, e^{\left (e^{\left (e^{3}\right )}\right )} + 3 \, \log \left (5\right ) + 21\right )} - \log \left (5\right )^{2} - 14 \, \log \left (5\right ) - 49\right )} e^{x} - 6 \, \log \left (5\right )^{2} - 84 \, \log \left (5\right ) - 294}{e^{\left (2 \, x\right )} + 4 \, e^{x} + 4} + \frac {98 \, {\left (e^{x} + 3\right )}}{e^{\left (2 \, x\right )} + 4 \, e^{x} + 4} - 49 \, \log \left (e^{x} + 2\right ) \]
integrate(((2*x*exp(x)^3+12*x*exp(x)^2+24*exp(x)*x+16*x)*exp(exp(exp(3)))+ (log(5)^2+(4*x+14)*log(5)+3*x^2+28*x+49)*exp(x)^3+(6*log(5)^2+(2*x^2+20*x+ 84)*log(5)+2*x^3+26*x^2+140*x+294)*exp(x)^2+(12*log(5)^2+(4*x^2+32*x+168)* log(5)+2*x^3+43*x^2+224*x+588)*exp(x)+8*log(5)^2+(16*x+112)*log(5)+6*x^2+1 12*x+392)/(exp(x)^3+6*exp(x)^2+12*exp(x)+8),x, algorithm=\
(x + 2*(e^x + 3)/(e^(2*x) + 4*e^x + 4) - log(e^x + 2))*log(5)^2 + 14*(x + 2*(e^x + 3)/(e^(2*x) + 4*e^x + 4) - log(e^x + 2))*log(5) + (log(5)^2 + 14* log(5) + 49)*log(e^x + 2) + 49*x + (x^3 + 4*x^2*(e^(e^(e^3)) + log(5) + 7) + (x^3 + x^2*(e^(e^(e^3)) + 2*log(5) + 14))*e^(2*x) + 2*(x^3 + x^2*(2*e^( e^(e^3)) + 3*log(5) + 21) - log(5)^2 - 14*log(5) - 49)*e^x - 6*log(5)^2 - 84*log(5) - 294)/(e^(2*x) + 4*e^x + 4) + 98*(e^x + 3)/(e^(2*x) + 4*e^x + 4 ) - 49*log(e^x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (25) = 50\).
Time = 0.31 (sec) , antiderivative size = 174, normalized size of antiderivative = 5.44 \[ \int \frac {392+112 x+6 x^2+e^{e^{e^3}} \left (16 x+24 e^x x+12 e^{2 x} x+2 e^{3 x} x\right )+(112+16 x) \log (5)+8 \log ^2(5)+e^{3 x} \left (49+28 x+3 x^2+(14+4 x) \log (5)+\log ^2(5)\right )+e^{2 x} \left (294+140 x+26 x^2+2 x^3+\left (84+20 x+2 x^2\right ) \log (5)+6 \log ^2(5)\right )+e^x \left (588+224 x+43 x^2+2 x^3+\left (168+32 x+4 x^2\right ) \log (5)+12 \log ^2(5)\right )}{8+12 e^x+6 e^{2 x}+e^{3 x}} \, dx=\frac {x^{3} e^{\left (2 \, x\right )} + 2 \, x^{3} e^{x} + 2 \, x^{2} e^{\left (2 \, x\right )} \log \left (5\right ) + 6 \, x^{2} e^{x} \log \left (5\right ) + x e^{\left (2 \, x\right )} \log \left (5\right )^{2} + 4 \, x e^{x} \log \left (5\right )^{2} + x^{3} + 14 \, x^{2} e^{\left (2 \, x\right )} + x^{2} e^{\left (2 \, x + e^{\left (e^{3}\right )}\right )} + 4 \, x^{2} e^{\left (x + e^{\left (e^{3}\right )}\right )} + 42 \, x^{2} e^{x} + 4 \, x^{2} e^{\left (e^{\left (e^{3}\right )}\right )} + 4 \, x^{2} \log \left (5\right ) + 14 \, x e^{\left (2 \, x\right )} \log \left (5\right ) + 56 \, x e^{x} \log \left (5\right ) + 4 \, x \log \left (5\right )^{2} + 28 \, x^{2} + 49 \, x e^{\left (2 \, x\right )} + 196 \, x e^{x} + 56 \, x \log \left (5\right ) + 196 \, x}{e^{\left (2 \, x\right )} + 4 \, e^{x} + 4} \]
integrate(((2*x*exp(x)^3+12*x*exp(x)^2+24*exp(x)*x+16*x)*exp(exp(exp(3)))+ (log(5)^2+(4*x+14)*log(5)+3*x^2+28*x+49)*exp(x)^3+(6*log(5)^2+(2*x^2+20*x+ 84)*log(5)+2*x^3+26*x^2+140*x+294)*exp(x)^2+(12*log(5)^2+(4*x^2+32*x+168)* log(5)+2*x^3+43*x^2+224*x+588)*exp(x)+8*log(5)^2+(16*x+112)*log(5)+6*x^2+1 12*x+392)/(exp(x)^3+6*exp(x)^2+12*exp(x)+8),x, algorithm=\
(x^3*e^(2*x) + 2*x^3*e^x + 2*x^2*e^(2*x)*log(5) + 6*x^2*e^x*log(5) + x*e^( 2*x)*log(5)^2 + 4*x*e^x*log(5)^2 + x^3 + 14*x^2*e^(2*x) + x^2*e^(2*x + e^( e^3)) + 4*x^2*e^(x + e^(e^3)) + 42*x^2*e^x + 4*x^2*e^(e^(e^3)) + 4*x^2*log (5) + 14*x*e^(2*x)*log(5) + 56*x*e^x*log(5) + 4*x*log(5)^2 + 28*x^2 + 49*x *e^(2*x) + 196*x*e^x + 56*x*log(5) + 196*x)/(e^(2*x) + 4*e^x + 4)
Time = 0.23 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.94 \[ \int \frac {392+112 x+6 x^2+e^{e^{e^3}} \left (16 x+24 e^x x+12 e^{2 x} x+2 e^{3 x} x\right )+(112+16 x) \log (5)+8 \log ^2(5)+e^{3 x} \left (49+28 x+3 x^2+(14+4 x) \log (5)+\log ^2(5)\right )+e^{2 x} \left (294+140 x+26 x^2+2 x^3+\left (84+20 x+2 x^2\right ) \log (5)+6 \log ^2(5)\right )+e^x \left (588+224 x+43 x^2+2 x^3+\left (168+32 x+4 x^2\right ) \log (5)+12 \log ^2(5)\right )}{8+12 e^x+6 e^{2 x}+e^{3 x}} \, dx=\frac {2\,x^3\,{\mathrm {e}}^x+x\,\left (56\,\ln \left (5\right )+4\,{\ln \left (5\right )}^2+196\right )+x^3\,{\mathrm {e}}^{2\,x}+x^2\,\left (4\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^3}}+\ln \left (625\right )+28\right )+x^3+x\,{\mathrm {e}}^{2\,x}\,\left (14\,\ln \left (5\right )+{\ln \left (5\right )}^2+49\right )+x\,{\mathrm {e}}^x\,\left (56\,\ln \left (5\right )+4\,{\ln \left (5\right )}^2+196\right )+x^2\,{\mathrm {e}}^{2\,x}\,\left ({\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^3}}+\ln \left (25\right )+14\right )+x^2\,{\mathrm {e}}^x\,\left (4\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^3}}+6\,\ln \left (5\right )+42\right )}{{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^x+4} \]
int((112*x + log(5)*(16*x + 112) + exp(x)*(224*x + log(5)*(32*x + 4*x^2 + 168) + 12*log(5)^2 + 43*x^2 + 2*x^3 + 588) + exp(exp(exp(3)))*(16*x + 12*x *exp(2*x) + 2*x*exp(3*x) + 24*x*exp(x)) + exp(2*x)*(140*x + log(5)*(20*x + 2*x^2 + 84) + 6*log(5)^2 + 26*x^2 + 2*x^3 + 294) + 8*log(5)^2 + 6*x^2 + e xp(3*x)*(28*x + log(5)*(4*x + 14) + log(5)^2 + 3*x^2 + 49) + 392)/(6*exp(2 *x) + exp(3*x) + 12*exp(x) + 8),x)
(2*x^3*exp(x) + x*(56*log(5) + 4*log(5)^2 + 196) + x^3*exp(2*x) + x^2*(4*e xp(exp(exp(3))) + log(625) + 28) + x^3 + x*exp(2*x)*(14*log(5) + log(5)^2 + 49) + x*exp(x)*(56*log(5) + 4*log(5)^2 + 196) + x^2*exp(2*x)*(exp(exp(ex p(3))) + log(25) + 14) + x^2*exp(x)*(4*exp(exp(exp(3))) + 6*log(5) + 42))/ (exp(2*x) + 4*exp(x) + 4)