Integrand size = 339, antiderivative size = 33 \[ \int \frac {\left (486 x-324 x^2-108 x^3+144 x^4-42 x^5+4 x^6+e^x \left (-162 x+216 x^2-108 x^3+24 x^4-2 x^5\right )+\left (324-108 x-216 x^2+168 x^3-44 x^4+4 x^5\right ) \log (x)\right ) \log \left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )+\left (-270 x^2+54 x^3+180 x^4-116 x^5+26 x^6-2 x^7+e^x \left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right )+\left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right ) \log (5)+\left (-540 x^2+648 x^3-288 x^4+56 x^5-4 x^6\right ) \log (x)+\left (-270 x+324 x^2-144 x^3+28 x^4-2 x^5\right ) \log ^2(x)\right ) \log ^2\left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )}{-e^{3 x} x+e^{2 x} \left (x^2+x^3-x \log (5)\right )+2 e^{2 x} x^2 \log (x)+e^{2 x} x \log ^2(x)} \, dx=e^{-2 x} (3-x)^4 \log ^2\left (e^x-x+\log (5)-(x+\log (x))^2\right ) \] Output:
(3-x)^4*ln(exp(x)+ln(5)-(x+ln(x))^2-x)^2/exp(x)^2
Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {\left (486 x-324 x^2-108 x^3+144 x^4-42 x^5+4 x^6+e^x \left (-162 x+216 x^2-108 x^3+24 x^4-2 x^5\right )+\left (324-108 x-216 x^2+168 x^3-44 x^4+4 x^5\right ) \log (x)\right ) \log \left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )+\left (-270 x^2+54 x^3+180 x^4-116 x^5+26 x^6-2 x^7+e^x \left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right )+\left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right ) \log (5)+\left (-540 x^2+648 x^3-288 x^4+56 x^5-4 x^6\right ) \log (x)+\left (-270 x+324 x^2-144 x^3+28 x^4-2 x^5\right ) \log ^2(x)\right ) \log ^2\left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )}{-e^{3 x} x+e^{2 x} \left (x^2+x^3-x \log (5)\right )+2 e^{2 x} x^2 \log (x)+e^{2 x} x \log ^2(x)} \, dx=e^{-2 x} (-3+x)^4 \log ^2\left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right ) \] Input:
Integrate[((486*x - 324*x^2 - 108*x^3 + 144*x^4 - 42*x^5 + 4*x^6 + E^x*(-1 62*x + 216*x^2 - 108*x^3 + 24*x^4 - 2*x^5) + (324 - 108*x - 216*x^2 + 168* x^3 - 44*x^4 + 4*x^5)*Log[x])*Log[E^x - x - x^2 + Log[5] - 2*x*Log[x] - Lo g[x]^2] + (-270*x^2 + 54*x^3 + 180*x^4 - 116*x^5 + 26*x^6 - 2*x^7 + E^x*(2 70*x - 324*x^2 + 144*x^3 - 28*x^4 + 2*x^5) + (270*x - 324*x^2 + 144*x^3 - 28*x^4 + 2*x^5)*Log[5] + (-540*x^2 + 648*x^3 - 288*x^4 + 56*x^5 - 4*x^6)*L og[x] + (-270*x + 324*x^2 - 144*x^3 + 28*x^4 - 2*x^5)*Log[x]^2)*Log[E^x - x - x^2 + Log[5] - 2*x*Log[x] - Log[x]^2]^2)/(-(E^(3*x)*x) + E^(2*x)*(x^2 + x^3 - x*Log[5]) + 2*E^(2*x)*x^2*Log[x] + E^(2*x)*x*Log[x]^2),x]
Output:
((-3 + x)^4*Log[E^x - x - x^2 + Log[5] - 2*x*Log[x] - Log[x]^2]^2)/E^(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.
| \(\displaystyle \int \frac {\left (4 x^6-42 x^5+144 x^4-108 x^3-324 x^2+e^x \left (-2 x^5+24 x^4-108 x^3+216 x^2-162 x\right )+\left (4 x^5-44 x^4+168 x^3-216 x^2-108 x+324\right ) \log (x)+486 x\right ) \log \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right )+\left (-2 x^7+26 x^6-116 x^5+180 x^4+54 x^3-270 x^2+e^x \left (2 x^5-28 x^4+144 x^3-324 x^2+270 x\right )+\left (-2 x^5+28 x^4-144 x^3+324 x^2-270 x\right ) \log ^2(x)+\left (2 x^5-28 x^4+144 x^3-324 x^2+270 x\right ) \log (5)+\left (-4 x^6+56 x^5-288 x^4+648 x^3-540 x^2\right ) \log (x)\right ) \log ^2\left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right )}{2 e^{2 x} x^2 \log (x)+e^{2 x} \left (x^3+x^2-x \log (5)\right )-e^{3 x} x+e^{2 x} x \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 x} \log \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right ) \left (-4 x^6+42 x^5-144 x^4+108 x^3+324 x^2+2 (x-5) (x-3)^3 x \left (x^2+x-e^x+\log ^2(x)+2 x \log (x)-\log (5)\right ) \log \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right )+2 e^x (x-3)^4 x-486 x-4 (x-3)^4 (x+1) \log (x)\right )}{x \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 e^{-2 x} (3-x)^4 \left (x^3-x^2+2 x^2 \log (x)+x \log ^2(x)-2 x \log (x)-3 x \left (1+\frac {\log (5)}{3}\right )-2 \log (x)\right ) \log \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right )}{x \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right )}-2 e^{-2 x} (x-3)^3 \log \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right ) \left (x \log \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right )-5 \log \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right )-x+3\right )\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {2 e^{-2 x} (3-x)^4 \left (x^3-x^2+2 x^2 \log (x)+x \log ^2(x)-2 x \log (x)-3 x \left (1+\frac {\log (5)}{3}\right )-2 \log (x)\right ) \log \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right )}{x \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right )}-2 e^{-2 x} (x-3)^3 \log \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right ) \left (x \log \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right )-5 \log \left (-x^2-x+e^x-\log ^2(x)-2 x \log (x)+\log (5)\right )-x+3\right )\right )dx\) |
Input:
Int[((486*x - 324*x^2 - 108*x^3 + 144*x^4 - 42*x^5 + 4*x^6 + E^x*(-162*x + 216*x^2 - 108*x^3 + 24*x^4 - 2*x^5) + (324 - 108*x - 216*x^2 + 168*x^3 - 44*x^4 + 4*x^5)*Log[x])*Log[E^x - x - x^2 + Log[5] - 2*x*Log[x] - Log[x]^2 ] + (-270*x^2 + 54*x^3 + 180*x^4 - 116*x^5 + 26*x^6 - 2*x^7 + E^x*(270*x - 324*x^2 + 144*x^3 - 28*x^4 + 2*x^5) + (270*x - 324*x^2 + 144*x^3 - 28*x^4 + 2*x^5)*Log[5] + (-540*x^2 + 648*x^3 - 288*x^4 + 56*x^5 - 4*x^6)*Log[x] + (-270*x + 324*x^2 - 144*x^3 + 28*x^4 - 2*x^5)*Log[x]^2)*Log[E^x - x - x^ 2 + Log[5] - 2*x*Log[x] - Log[x]^2]^2)/(-(E^(3*x)*x) + E^(2*x)*(x^2 + x^3 - x*Log[5]) + 2*E^(2*x)*x^2*Log[x] + E^(2*x)*x*Log[x]^2),x]
Output:
$Aborted
Time = 115.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55
| method | result | size |
| risch | \(\left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right ) {\mathrm e}^{-2 x} \ln \left (-\ln \left (x \right )^{2}-2 x \ln \left (x \right )+{\mathrm e}^{x}+\ln \left (5\right )-x^{2}-x \right )^{2}\) | \(51\) |
| parallelrisch | \(\left (\ln \left (-\ln \left (x \right )^{2}-2 x \ln \left (x \right )+{\mathrm e}^{x}+\ln \left (5\right )-x^{2}-x \right )^{2} x^{4}-12 \ln \left (-\ln \left (x \right )^{2}-2 x \ln \left (x \right )+{\mathrm e}^{x}+\ln \left (5\right )-x^{2}-x \right )^{2} x^{3}+54 \ln \left (-\ln \left (x \right )^{2}-2 x \ln \left (x \right )+{\mathrm e}^{x}+\ln \left (5\right )-x^{2}-x \right )^{2} x^{2}-108 x \ln \left (-\ln \left (x \right )^{2}-2 x \ln \left (x \right )+{\mathrm e}^{x}+\ln \left (5\right )-x^{2}-x \right )^{2}+81 \ln \left (-\ln \left (x \right )^{2}-2 x \ln \left (x \right )+{\mathrm e}^{x}+\ln \left (5\right )-x^{2}-x \right )^{2}\right ) {\mathrm e}^{-2 x}\) | \(161\) |
Input:
int((((-2*x^5+28*x^4-144*x^3+324*x^2-270*x)*ln(x)^2+(-4*x^6+56*x^5-288*x^4 +648*x^3-540*x^2)*ln(x)+(2*x^5-28*x^4+144*x^3-324*x^2+270*x)*exp(x)+(2*x^5 -28*x^4+144*x^3-324*x^2+270*x)*ln(5)-2*x^7+26*x^6-116*x^5+180*x^4+54*x^3-2 70*x^2)*ln(-ln(x)^2-2*x*ln(x)+exp(x)+ln(5)-x^2-x)^2+((4*x^5-44*x^4+168*x^3 -216*x^2-108*x+324)*ln(x)+(-2*x^5+24*x^4-108*x^3+216*x^2-162*x)*exp(x)+4*x ^6-42*x^5+144*x^4-108*x^3-324*x^2+486*x)*ln(-ln(x)^2-2*x*ln(x)+exp(x)+ln(5 )-x^2-x))/(x*exp(x)^2*ln(x)^2+2*x^2*exp(x)^2*ln(x)-x*exp(x)^3+(-x*ln(5)+x^ 3+x^2)*exp(x)^2),x,method=_RETURNVERBOSE)
Output:
(x^4-12*x^3+54*x^2-108*x+81)/exp(x)^2*ln(-ln(x)^2-2*x*ln(x)+exp(x)+ln(5)-x ^2-x)^2
Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {\left (486 x-324 x^2-108 x^3+144 x^4-42 x^5+4 x^6+e^x \left (-162 x+216 x^2-108 x^3+24 x^4-2 x^5\right )+\left (324-108 x-216 x^2+168 x^3-44 x^4+4 x^5\right ) \log (x)\right ) \log \left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )+\left (-270 x^2+54 x^3+180 x^4-116 x^5+26 x^6-2 x^7+e^x \left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right )+\left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right ) \log (5)+\left (-540 x^2+648 x^3-288 x^4+56 x^5-4 x^6\right ) \log (x)+\left (-270 x+324 x^2-144 x^3+28 x^4-2 x^5\right ) \log ^2(x)\right ) \log ^2\left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )}{-e^{3 x} x+e^{2 x} \left (x^2+x^3-x \log (5)\right )+2 e^{2 x} x^2 \log (x)+e^{2 x} x \log ^2(x)} \, dx={\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )} e^{\left (-2 \, x\right )} \log \left (-x^{2} - 2 \, x \log \left (x\right ) - \log \left (x\right )^{2} - x + e^{x} + \log \left (5\right )\right )^{2} \] Input:
integrate((((-2*x^5+28*x^4-144*x^3+324*x^2-270*x)*log(x)^2+(-4*x^6+56*x^5- 288*x^4+648*x^3-540*x^2)*log(x)+(2*x^5-28*x^4+144*x^3-324*x^2+270*x)*exp(x )+(2*x^5-28*x^4+144*x^3-324*x^2+270*x)*log(5)-2*x^7+26*x^6-116*x^5+180*x^4 +54*x^3-270*x^2)*log(-log(x)^2-2*x*log(x)+exp(x)+log(5)-x^2-x)^2+((4*x^5-4 4*x^4+168*x^3-216*x^2-108*x+324)*log(x)+(-2*x^5+24*x^4-108*x^3+216*x^2-162 *x)*exp(x)+4*x^6-42*x^5+144*x^4-108*x^3-324*x^2+486*x)*log(-log(x)^2-2*x*l og(x)+exp(x)+log(5)-x^2-x))/(x*exp(x)^2*log(x)^2+2*x^2*exp(x)^2*log(x)-x*e xp(x)^3+(-x*log(5)+x^3+x^2)*exp(x)^2),x, algorithm="fricas")
Output:
(x^4 - 12*x^3 + 54*x^2 - 108*x + 81)*e^(-2*x)*log(-x^2 - 2*x*log(x) - log( x)^2 - x + e^x + log(5))^2
Time = 1.55 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {\left (486 x-324 x^2-108 x^3+144 x^4-42 x^5+4 x^6+e^x \left (-162 x+216 x^2-108 x^3+24 x^4-2 x^5\right )+\left (324-108 x-216 x^2+168 x^3-44 x^4+4 x^5\right ) \log (x)\right ) \log \left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )+\left (-270 x^2+54 x^3+180 x^4-116 x^5+26 x^6-2 x^7+e^x \left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right )+\left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right ) \log (5)+\left (-540 x^2+648 x^3-288 x^4+56 x^5-4 x^6\right ) \log (x)+\left (-270 x+324 x^2-144 x^3+28 x^4-2 x^5\right ) \log ^2(x)\right ) \log ^2\left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )}{-e^{3 x} x+e^{2 x} \left (x^2+x^3-x \log (5)\right )+2 e^{2 x} x^2 \log (x)+e^{2 x} x \log ^2(x)} \, dx=\left (x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81\right ) e^{- 2 x} \log {\left (- x^{2} - 2 x \log {\left (x \right )} - x + e^{x} - \log {\left (x \right )}^{2} + \log {\left (5 \right )} \right )}^{2} \] Input:
integrate((((-2*x**5+28*x**4-144*x**3+324*x**2-270*x)*ln(x)**2+(-4*x**6+56 *x**5-288*x**4+648*x**3-540*x**2)*ln(x)+(2*x**5-28*x**4+144*x**3-324*x**2+ 270*x)*exp(x)+(2*x**5-28*x**4+144*x**3-324*x**2+270*x)*ln(5)-2*x**7+26*x** 6-116*x**5+180*x**4+54*x**3-270*x**2)*ln(-ln(x)**2-2*x*ln(x)+exp(x)+ln(5)- x**2-x)**2+((4*x**5-44*x**4+168*x**3-216*x**2-108*x+324)*ln(x)+(-2*x**5+24 *x**4-108*x**3+216*x**2-162*x)*exp(x)+4*x**6-42*x**5+144*x**4-108*x**3-324 *x**2+486*x)*ln(-ln(x)**2-2*x*ln(x)+exp(x)+ln(5)-x**2-x))/(x*exp(x)**2*ln( x)**2+2*x**2*exp(x)**2*ln(x)-x*exp(x)**3+(-x*ln(5)+x**3+x**2)*exp(x)**2),x )
Output:
(x**4 - 12*x**3 + 54*x**2 - 108*x + 81)*exp(-2*x)*log(-x**2 - 2*x*log(x) - x + exp(x) - log(x)**2 + log(5))**2
Time = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {\left (486 x-324 x^2-108 x^3+144 x^4-42 x^5+4 x^6+e^x \left (-162 x+216 x^2-108 x^3+24 x^4-2 x^5\right )+\left (324-108 x-216 x^2+168 x^3-44 x^4+4 x^5\right ) \log (x)\right ) \log \left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )+\left (-270 x^2+54 x^3+180 x^4-116 x^5+26 x^6-2 x^7+e^x \left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right )+\left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right ) \log (5)+\left (-540 x^2+648 x^3-288 x^4+56 x^5-4 x^6\right ) \log (x)+\left (-270 x+324 x^2-144 x^3+28 x^4-2 x^5\right ) \log ^2(x)\right ) \log ^2\left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )}{-e^{3 x} x+e^{2 x} \left (x^2+x^3-x \log (5)\right )+2 e^{2 x} x^2 \log (x)+e^{2 x} x \log ^2(x)} \, dx={\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )} e^{\left (-2 \, x\right )} \log \left (-x^{2} - 2 \, x \log \left (x\right ) - \log \left (x\right )^{2} - x + e^{x} + \log \left (5\right )\right )^{2} \] Input:
integrate((((-2*x^5+28*x^4-144*x^3+324*x^2-270*x)*log(x)^2+(-4*x^6+56*x^5- 288*x^4+648*x^3-540*x^2)*log(x)+(2*x^5-28*x^4+144*x^3-324*x^2+270*x)*exp(x )+(2*x^5-28*x^4+144*x^3-324*x^2+270*x)*log(5)-2*x^7+26*x^6-116*x^5+180*x^4 +54*x^3-270*x^2)*log(-log(x)^2-2*x*log(x)+exp(x)+log(5)-x^2-x)^2+((4*x^5-4 4*x^4+168*x^3-216*x^2-108*x+324)*log(x)+(-2*x^5+24*x^4-108*x^3+216*x^2-162 *x)*exp(x)+4*x^6-42*x^5+144*x^4-108*x^3-324*x^2+486*x)*log(-log(x)^2-2*x*l og(x)+exp(x)+log(5)-x^2-x))/(x*exp(x)^2*log(x)^2+2*x^2*exp(x)^2*log(x)-x*e xp(x)^3+(-x*log(5)+x^3+x^2)*exp(x)^2),x, algorithm="maxima")
Output:
(x^4 - 12*x^3 + 54*x^2 - 108*x + 81)*e^(-2*x)*log(-x^2 - 2*x*log(x) - log( x)^2 - x + e^x + log(5))^2
Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (29) = 58\).
Time = 0.59 (sec) , antiderivative size = 160, normalized size of antiderivative = 4.85 \[ \int \frac {\left (486 x-324 x^2-108 x^3+144 x^4-42 x^5+4 x^6+e^x \left (-162 x+216 x^2-108 x^3+24 x^4-2 x^5\right )+\left (324-108 x-216 x^2+168 x^3-44 x^4+4 x^5\right ) \log (x)\right ) \log \left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )+\left (-270 x^2+54 x^3+180 x^4-116 x^5+26 x^6-2 x^7+e^x \left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right )+\left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right ) \log (5)+\left (-540 x^2+648 x^3-288 x^4+56 x^5-4 x^6\right ) \log (x)+\left (-270 x+324 x^2-144 x^3+28 x^4-2 x^5\right ) \log ^2(x)\right ) \log ^2\left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )}{-e^{3 x} x+e^{2 x} \left (x^2+x^3-x \log (5)\right )+2 e^{2 x} x^2 \log (x)+e^{2 x} x \log ^2(x)} \, dx={\left (x^{4} \log \left (-x^{2} - 2 \, x \log \left (x\right ) - \log \left (x\right )^{2} - x + e^{x} + \log \left (5\right )\right )^{2} - 12 \, x^{3} \log \left (-x^{2} - 2 \, x \log \left (x\right ) - \log \left (x\right )^{2} - x + e^{x} + \log \left (5\right )\right )^{2} + 54 \, x^{2} \log \left (-x^{2} - 2 \, x \log \left (x\right ) - \log \left (x\right )^{2} - x + e^{x} + \log \left (5\right )\right )^{2} - 108 \, x \log \left (-x^{2} - 2 \, x \log \left (x\right ) - \log \left (x\right )^{2} - x + e^{x} + \log \left (5\right )\right )^{2} + 81 \, \log \left (-x^{2} - 2 \, x \log \left (x\right ) - \log \left (x\right )^{2} - x + e^{x} + \log \left (5\right )\right )^{2}\right )} e^{\left (-2 \, x\right )} \] Input:
integrate((((-2*x^5+28*x^4-144*x^3+324*x^2-270*x)*log(x)^2+(-4*x^6+56*x^5- 288*x^4+648*x^3-540*x^2)*log(x)+(2*x^5-28*x^4+144*x^3-324*x^2+270*x)*exp(x )+(2*x^5-28*x^4+144*x^3-324*x^2+270*x)*log(5)-2*x^7+26*x^6-116*x^5+180*x^4 +54*x^3-270*x^2)*log(-log(x)^2-2*x*log(x)+exp(x)+log(5)-x^2-x)^2+((4*x^5-4 4*x^4+168*x^3-216*x^2-108*x+324)*log(x)+(-2*x^5+24*x^4-108*x^3+216*x^2-162 *x)*exp(x)+4*x^6-42*x^5+144*x^4-108*x^3-324*x^2+486*x)*log(-log(x)^2-2*x*l og(x)+exp(x)+log(5)-x^2-x))/(x*exp(x)^2*log(x)^2+2*x^2*exp(x)^2*log(x)-x*e xp(x)^3+(-x*log(5)+x^3+x^2)*exp(x)^2),x, algorithm="giac")
Output:
(x^4*log(-x^2 - 2*x*log(x) - log(x)^2 - x + e^x + log(5))^2 - 12*x^3*log(- x^2 - 2*x*log(x) - log(x)^2 - x + e^x + log(5))^2 + 54*x^2*log(-x^2 - 2*x* log(x) - log(x)^2 - x + e^x + log(5))^2 - 108*x*log(-x^2 - 2*x*log(x) - lo g(x)^2 - x + e^x + log(5))^2 + 81*log(-x^2 - 2*x*log(x) - log(x)^2 - x + e ^x + log(5))^2)*e^(-2*x)
Timed out. \[ \int \frac {\left (486 x-324 x^2-108 x^3+144 x^4-42 x^5+4 x^6+e^x \left (-162 x+216 x^2-108 x^3+24 x^4-2 x^5\right )+\left (324-108 x-216 x^2+168 x^3-44 x^4+4 x^5\right ) \log (x)\right ) \log \left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )+\left (-270 x^2+54 x^3+180 x^4-116 x^5+26 x^6-2 x^7+e^x \left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right )+\left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right ) \log (5)+\left (-540 x^2+648 x^3-288 x^4+56 x^5-4 x^6\right ) \log (x)+\left (-270 x+324 x^2-144 x^3+28 x^4-2 x^5\right ) \log ^2(x)\right ) \log ^2\left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )}{-e^{3 x} x+e^{2 x} \left (x^2+x^3-x \log (5)\right )+2 e^{2 x} x^2 \log (x)+e^{2 x} x \log ^2(x)} \, dx=\int \frac {{\ln \left (\ln \left (5\right )-x+{\mathrm {e}}^x-{\ln \left (x\right )}^2-2\,x\,\ln \left (x\right )-x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (2\,x^5-28\,x^4+144\,x^3-324\,x^2+270\,x\right )-{\ln \left (x\right )}^2\,\left (2\,x^5-28\,x^4+144\,x^3-324\,x^2+270\,x\right )-\ln \left (x\right )\,\left (4\,x^6-56\,x^5+288\,x^4-648\,x^3+540\,x^2\right )-270\,x^2+54\,x^3+180\,x^4-116\,x^5+26\,x^6-2\,x^7+\ln \left (5\right )\,\left (2\,x^5-28\,x^4+144\,x^3-324\,x^2+270\,x\right )\right )-\ln \left (\ln \left (5\right )-x+{\mathrm {e}}^x-{\ln \left (x\right )}^2-2\,x\,\ln \left (x\right )-x^2\right )\,\left ({\mathrm {e}}^x\,\left (2\,x^5-24\,x^4+108\,x^3-216\,x^2+162\,x\right )-486\,x+\ln \left (x\right )\,\left (-4\,x^5+44\,x^4-168\,x^3+216\,x^2+108\,x-324\right )+324\,x^2+108\,x^3-144\,x^4+42\,x^5-4\,x^6\right )}{{\mathrm {e}}^{2\,x}\,\left (x^3+x^2-\ln \left (5\right )\,x\right )-x\,{\mathrm {e}}^{3\,x}+x\,{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^2+2\,x^2\,{\mathrm {e}}^{2\,x}\,\ln \left (x\right )} \,d x \] Input:
int((log(log(5) - x + exp(x) - log(x)^2 - 2*x*log(x) - x^2)^2*(exp(x)*(270 *x - 324*x^2 + 144*x^3 - 28*x^4 + 2*x^5) - log(x)^2*(270*x - 324*x^2 + 144 *x^3 - 28*x^4 + 2*x^5) - log(x)*(540*x^2 - 648*x^3 + 288*x^4 - 56*x^5 + 4* x^6) - 270*x^2 + 54*x^3 + 180*x^4 - 116*x^5 + 26*x^6 - 2*x^7 + log(5)*(270 *x - 324*x^2 + 144*x^3 - 28*x^4 + 2*x^5)) - log(log(5) - x + exp(x) - log( x)^2 - 2*x*log(x) - x^2)*(exp(x)*(162*x - 216*x^2 + 108*x^3 - 24*x^4 + 2*x ^5) - 486*x + log(x)*(108*x + 216*x^2 - 168*x^3 + 44*x^4 - 4*x^5 - 324) + 324*x^2 + 108*x^3 - 144*x^4 + 42*x^5 - 4*x^6))/(exp(2*x)*(x^2 - x*log(5) + x^3) - x*exp(3*x) + x*exp(2*x)*log(x)^2 + 2*x^2*exp(2*x)*log(x)),x)
Output:
int((log(log(5) - x + exp(x) - log(x)^2 - 2*x*log(x) - x^2)^2*(exp(x)*(270 *x - 324*x^2 + 144*x^3 - 28*x^4 + 2*x^5) - log(x)^2*(270*x - 324*x^2 + 144 *x^3 - 28*x^4 + 2*x^5) - log(x)*(540*x^2 - 648*x^3 + 288*x^4 - 56*x^5 + 4* x^6) - 270*x^2 + 54*x^3 + 180*x^4 - 116*x^5 + 26*x^6 - 2*x^7 + log(5)*(270 *x - 324*x^2 + 144*x^3 - 28*x^4 + 2*x^5)) - log(log(5) - x + exp(x) - log( x)^2 - 2*x*log(x) - x^2)*(exp(x)*(162*x - 216*x^2 + 108*x^3 - 24*x^4 + 2*x ^5) - 486*x + log(x)*(108*x + 216*x^2 - 168*x^3 + 44*x^4 - 4*x^5 - 324) + 324*x^2 + 108*x^3 - 144*x^4 + 42*x^5 - 4*x^6))/(exp(2*x)*(x^2 - x*log(5) + x^3) - x*exp(3*x) + x*exp(2*x)*log(x)^2 + 2*x^2*exp(2*x)*log(x)), x)
Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.64 \[ \int \frac {\left (486 x-324 x^2-108 x^3+144 x^4-42 x^5+4 x^6+e^x \left (-162 x+216 x^2-108 x^3+24 x^4-2 x^5\right )+\left (324-108 x-216 x^2+168 x^3-44 x^4+4 x^5\right ) \log (x)\right ) \log \left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )+\left (-270 x^2+54 x^3+180 x^4-116 x^5+26 x^6-2 x^7+e^x \left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right )+\left (270 x-324 x^2+144 x^3-28 x^4+2 x^5\right ) \log (5)+\left (-540 x^2+648 x^3-288 x^4+56 x^5-4 x^6\right ) \log (x)+\left (-270 x+324 x^2-144 x^3+28 x^4-2 x^5\right ) \log ^2(x)\right ) \log ^2\left (e^x-x-x^2+\log (5)-2 x \log (x)-\log ^2(x)\right )}{-e^{3 x} x+e^{2 x} \left (x^2+x^3-x \log (5)\right )+2 e^{2 x} x^2 \log (x)+e^{2 x} x \log ^2(x)} \, dx=\frac {\mathrm {log}\left (e^{x}-\mathrm {log}\left (x \right )^{2}-2 \,\mathrm {log}\left (x \right ) x +\mathrm {log}\left (5\right )-x^{2}-x \right )^{2} \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )}{e^{2 x}} \] Input:
int((((-2*x^5+28*x^4-144*x^3+324*x^2-270*x)*log(x)^2+(-4*x^6+56*x^5-288*x^ 4+648*x^3-540*x^2)*log(x)+(2*x^5-28*x^4+144*x^3-324*x^2+270*x)*exp(x)+(2*x ^5-28*x^4+144*x^3-324*x^2+270*x)*log(5)-2*x^7+26*x^6-116*x^5+180*x^4+54*x^ 3-270*x^2)*log(-log(x)^2-2*x*log(x)+exp(x)+log(5)-x^2-x)^2+((4*x^5-44*x^4+ 168*x^3-216*x^2-108*x+324)*log(x)+(-2*x^5+24*x^4-108*x^3+216*x^2-162*x)*ex p(x)+4*x^6-42*x^5+144*x^4-108*x^3-324*x^2+486*x)*log(-log(x)^2-2*x*log(x)+ exp(x)+log(5)-x^2-x))/(x*exp(x)^2*log(x)^2+2*x^2*exp(x)^2*log(x)-x*exp(x)^ 3+(-x*log(5)+x^3+x^2)*exp(x)^2),x)
Output:
(log(e**x - log(x)**2 - 2*log(x)*x + log(5) - x**2 - x)**2*(x**4 - 12*x**3 + 54*x**2 - 108*x + 81))/e**(2*x)