Integrand size = 93, antiderivative size = 27 \[ \int \frac {36 x^4-36 x^5+\left (12 x^2-48 x^3+126 x^4\right ) \log \left (\frac {e^x}{x}\right )+\left (36 x^2-144 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+\left (2-24 x+54 x^2\right ) \log ^3\left (\frac {e^x}{x}\right )}{81 \log ^3\left (\frac {e^x}{x}\right )} \, dx=\frac {2}{9} x \left (-\frac {1}{3}+x-\frac {x^2}{\log \left (\frac {e^x}{x}\right )}\right )^2 \]
Time = 61.60 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {36 x^4-36 x^5+\left (12 x^2-48 x^3+126 x^4\right ) \log \left (\frac {e^x}{x}\right )+\left (36 x^2-144 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+\left (2-24 x+54 x^2\right ) \log ^3\left (\frac {e^x}{x}\right )}{81 \log ^3\left (\frac {e^x}{x}\right )} \, dx=\frac {2 x \left (3 x^2+(1-3 x) \log \left (\frac {e^x}{x}\right )\right )^2}{81 \log ^2\left (\frac {e^x}{x}\right )} \]
Integrate[(36*x^4 - 36*x^5 + (12*x^2 - 48*x^3 + 126*x^4)*Log[E^x/x] + (36* x^2 - 144*x^3)*Log[E^x/x]^2 + (2 - 24*x + 54*x^2)*Log[E^x/x]^3)/(81*Log[E^ x/x]^3),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.
\(\displaystyle \int \frac {-36 x^5+36 x^4+\left (54 x^2-24 x+2\right ) \log ^3\left (\frac {e^x}{x}\right )+\left (36 x^2-144 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+\left (126 x^4-48 x^3+12 x^2\right ) \log \left (\frac {e^x}{x}\right )}{81 \log ^3\left (\frac {e^x}{x}\right )} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{81} \int \frac {2 \left (-18 x^5+18 x^4+\left (27 x^2-12 x+1\right ) \log ^3\left (\frac {e^x}{x}\right )+18 \left (x^2-4 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+3 \left (21 x^4-8 x^3+2 x^2\right ) \log \left (\frac {e^x}{x}\right )\right )}{\log ^3\left (\frac {e^x}{x}\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{81} \int \frac {-18 x^5+18 x^4+\left (27 x^2-12 x+1\right ) \log ^3\left (\frac {e^x}{x}\right )+18 \left (x^2-4 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+3 \left (21 x^4-8 x^3+2 x^2\right ) \log \left (\frac {e^x}{x}\right )}{\log ^3\left (\frac {e^x}{x}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {2}{81} \int \left (-\frac {18 (x-1) x^4}{\log ^3\left (\frac {e^x}{x}\right )}-\frac {18 (4 x-1) x^2}{\log \left (\frac {e^x}{x}\right )}+\frac {3 \left (21 x^2-8 x+2\right ) x^2}{\log ^2\left (\frac {e^x}{x}\right )}+27 x^2-12 x+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{81} \left (-18 \int \frac {x^5}{\log ^3\left (\frac {e^x}{x}\right )}dx+18 \int \frac {x^4}{\log ^3\left (\frac {e^x}{x}\right )}dx+63 \int \frac {x^4}{\log ^2\left (\frac {e^x}{x}\right )}dx-24 \int \frac {x^3}{\log ^2\left (\frac {e^x}{x}\right )}dx-72 \int \frac {x^3}{\log \left (\frac {e^x}{x}\right )}dx+6 \int \frac {x^2}{\log ^2\left (\frac {e^x}{x}\right )}dx+18 \int \frac {x^2}{\log \left (\frac {e^x}{x}\right )}dx+9 x^3-6 x^2+x\right )\) |
Int[(36*x^4 - 36*x^5 + (12*x^2 - 48*x^3 + 126*x^4)*Log[E^x/x] + (36*x^2 - 144*x^3)*Log[E^x/x]^2 + (2 - 24*x + 54*x^2)*Log[E^x/x]^3)/(81*Log[E^x/x]^3 ),x]
3.9.72.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs. \(2(22)=44\).
Time = 19.79 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.04
method | result | size |
parallelrisch | \(\frac {18 x^{5}-36 \ln \left (\frac {{\mathrm e}^{x}}{x}\right ) x^{4}+18 x^{3} \ln \left (\frac {{\mathrm e}^{x}}{x}\right )^{2}+12 \ln \left (\frac {{\mathrm e}^{x}}{x}\right ) x^{3}-12 x^{2} \ln \left (\frac {{\mathrm e}^{x}}{x}\right )^{2}+2 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )^{2} x}{81 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )^{2}}\) | \(82\) |
risch | \(\frac {2 x^{3}}{9}-\frac {4 x^{2}}{27}+\frac {2 x}{81}+\frac {8 x^{3} \left (3 i \pi x \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}-3 i \pi x \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-3 i \pi x \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )+3 i \pi x \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )-i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}+i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )+i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )-i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )+3 x^{2}+6 x \ln \left (x \right )-6 x \ln \left ({\mathrm e}^{x}\right )-2 \ln \left (x \right )+2 \ln \left ({\mathrm e}^{x}\right )\right )}{27 {\left (2 \ln \left (x \right )-2 \ln \left ({\mathrm e}^{x}\right )+i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}-i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )+i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )\right )}^{2}}\) | \(314\) |
int(1/81*((54*x^2-24*x+2)*ln(exp(x)/x)^3+(-144*x^3+36*x^2)*ln(exp(x)/x)^2+ (126*x^4-48*x^3+12*x^2)*ln(exp(x)/x)-36*x^5+36*x^4)/ln(exp(x)/x)^3,x,metho d=_RETURNVERBOSE)
1/81*(18*x^5-36*ln(exp(x)/x)*x^4+18*x^3*ln(exp(x)/x)^2+12*ln(exp(x)/x)*x^3 -12*x^2*ln(exp(x)/x)^2+2*ln(exp(x)/x)^2*x)/ln(exp(x)/x)^2
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.19 \[ \int \frac {36 x^4-36 x^5+\left (12 x^2-48 x^3+126 x^4\right ) \log \left (\frac {e^x}{x}\right )+\left (36 x^2-144 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+\left (2-24 x+54 x^2\right ) \log ^3\left (\frac {e^x}{x}\right )}{81 \log ^3\left (\frac {e^x}{x}\right )} \, dx=\frac {2 \, {\left (9 \, x^{5} + {\left (9 \, x^{3} - 6 \, x^{2} + x\right )} \log \left (\frac {e^{x}}{x}\right )^{2} - 6 \, {\left (3 \, x^{4} - x^{3}\right )} \log \left (\frac {e^{x}}{x}\right )\right )}}{81 \, \log \left (\frac {e^{x}}{x}\right )^{2}} \]
integrate(1/81*((54*x^2-24*x+2)*log(exp(x)/x)^3+(-144*x^3+36*x^2)*log(exp( x)/x)^2+(126*x^4-48*x^3+12*x^2)*log(exp(x)/x)-36*x^5+36*x^4)/log(exp(x)/x) ^3,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {36 x^4-36 x^5+\left (12 x^2-48 x^3+126 x^4\right ) \log \left (\frac {e^x}{x}\right )+\left (36 x^2-144 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+\left (2-24 x+54 x^2\right ) \log ^3\left (\frac {e^x}{x}\right )}{81 \log ^3\left (\frac {e^x}{x}\right )} \, dx=\frac {2 x^{3}}{9} - \frac {4 x^{2}}{27} + \frac {2 x}{81} + \frac {6 x^{5} + \left (- 12 x^{4} + 4 x^{3}\right ) \log {\left (\frac {e^{x}}{x} \right )}}{27 \log {\left (\frac {e^{x}}{x} \right )}^{2}} \]
integrate(1/81*((54*x**2-24*x+2)*ln(exp(x)/x)**3+(-144*x**3+36*x**2)*ln(ex p(x)/x)**2+(126*x**4-48*x**3+12*x**2)*ln(exp(x)/x)-36*x**5+36*x**4)/ln(exp (x)/x)**3,x)
2*x**3/9 - 4*x**2/27 + 2*x/81 + (6*x**5 + (-12*x**4 + 4*x**3)*log(exp(x)/x ))/(27*log(exp(x)/x)**2)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.11 \[ \int \frac {36 x^4-36 x^5+\left (12 x^2-48 x^3+126 x^4\right ) \log \left (\frac {e^x}{x}\right )+\left (36 x^2-144 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+\left (2-24 x+54 x^2\right ) \log ^3\left (\frac {e^x}{x}\right )}{81 \log ^3\left (\frac {e^x}{x}\right )} \, dx=\frac {2}{9} \, x^{3} - \frac {4}{27} \, x^{2} + \frac {2}{81} \, x - \frac {2 \, {\left (3 \, x^{5} - 2 \, x^{4} - 2 \, {\left (3 \, x^{4} - x^{3}\right )} \log \left (x\right )\right )}}{27 \, {\left (x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \]
integrate(1/81*((54*x^2-24*x+2)*log(exp(x)/x)^3+(-144*x^3+36*x^2)*log(exp( x)/x)^2+(126*x^4-48*x^3+12*x^2)*log(exp(x)/x)-36*x^5+36*x^4)/log(exp(x)/x) ^3,x, algorithm=\
2/9*x^3 - 4/27*x^2 + 2/81*x - 2/27*(3*x^5 - 2*x^4 - 2*(3*x^4 - x^3)*log(x) )/(x^2 - 2*x*log(x) + log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.07 \[ \int \frac {36 x^4-36 x^5+\left (12 x^2-48 x^3+126 x^4\right ) \log \left (\frac {e^x}{x}\right )+\left (36 x^2-144 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+\left (2-24 x+54 x^2\right ) \log ^3\left (\frac {e^x}{x}\right )}{81 \log ^3\left (\frac {e^x}{x}\right )} \, dx=\frac {2}{9} \, x^{3} - \frac {4}{27} \, x^{2} + \frac {2}{81} \, x - \frac {2 \, {\left (3 \, x^{5} - 6 \, x^{4} \log \left (x\right ) - 2 \, x^{4} + 2 \, x^{3} \log \left (x\right )\right )}}{27 \, {\left (x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \]
integrate(1/81*((54*x^2-24*x+2)*log(exp(x)/x)^3+(-144*x^3+36*x^2)*log(exp( x)/x)^2+(126*x^4-48*x^3+12*x^2)*log(exp(x)/x)-36*x^5+36*x^4)/log(exp(x)/x) ^3,x, algorithm=\
2/9*x^3 - 4/27*x^2 + 2/81*x - 2/27*(3*x^5 - 6*x^4*log(x) - 2*x^4 + 2*x^3*l og(x))/(x^2 - 2*x*log(x) + log(x)^2)
Time = 8.40 (sec) , antiderivative size = 303, normalized size of antiderivative = 11.22 \[ \int \frac {36 x^4-36 x^5+\left (12 x^2-48 x^3+126 x^4\right ) \log \left (\frac {e^x}{x}\right )+\left (36 x^2-144 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+\left (2-24 x+54 x^2\right ) \log ^3\left (\frac {e^x}{x}\right )}{81 \log ^3\left (\frac {e^x}{x}\right )} \, dx=\frac {245\,x}{81}+\frac {2\,\ln \left (x\right )}{3}+\frac {\frac {x\,\left (9\,x^4+8\,x^3-2\,x^2\right )}{27\,\left (x-1\right )}+\frac {x^3\,\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,\left (-12\,x^3+7\,x^2+12\,x-2\right )}{9\,{\left (x-1\right )}^3}+\frac {2\,x^3\,{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}^2\,\left (4\,x^2-6\,x+1\right )}{3\,{\left (x-1\right )}^3}}{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}+\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,\left (\frac {24\,x^5-92\,x^4+128\,x^3-\frac {230\,x^2}{3}+18\,x-\frac {2}{3}}{x^3-3\,x^2+3\,x-1}-\frac {\frac {80\,x^5}{3}-96\,x^4+128\,x^3-\frac {224\,x^2}{3}+16\,x}{x^3-3\,x^2+3\,x-1}\right )+\frac {\frac {38\,x^2}{9}-9\,x+\frac {38}{9}}{x^3-3\,x^2+3\,x-1}+\frac {\frac {2\,x^5}{9}-\frac {x^3\,\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,\left (21\,x^2-8\,x+2\right )}{27\,\left (x-1\right )}+\frac {2\,x^3\,{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}^2\,\left (4\,x-1\right )}{9\,\left (x-1\right )}}{{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}^2}+\frac {65\,x^2}{27}+\frac {2\,x^3}{3} \]
int(((log(exp(x)/x)^2*(36*x^2 - 144*x^3))/81 + (log(exp(x)/x)^3*(54*x^2 - 24*x + 2))/81 + (4*x^4)/9 - (4*x^5)/9 + (log(exp(x)/x)*(12*x^2 - 48*x^3 + 126*x^4))/81)/log(exp(x)/x)^3,x)
(245*x)/81 + (2*log(x))/3 + ((x*(8*x^3 - 2*x^2 + 9*x^4))/(27*(x - 1)) + (x ^3*log(exp(x)/x)*(12*x + 7*x^2 - 12*x^3 - 2))/(9*(x - 1)^3) + (2*x^3*log(e xp(x)/x)^2*(4*x^2 - 6*x + 1))/(3*(x - 1)^3))/log(exp(x)/x) + log(exp(x)/x) *((18*x - (230*x^2)/3 + 128*x^3 - 92*x^4 + 24*x^5 - 2/3)/(3*x - 3*x^2 + x^ 3 - 1) - (16*x - (224*x^2)/3 + 128*x^3 - 96*x^4 + (80*x^5)/3)/(3*x - 3*x^2 + x^3 - 1)) + ((38*x^2)/9 - 9*x + 38/9)/(3*x - 3*x^2 + x^3 - 1) + ((2*x^5 )/9 - (x^3*log(exp(x)/x)*(21*x^2 - 8*x + 2))/(27*(x - 1)) + (2*x^3*log(exp (x)/x)^2*(4*x - 1))/(9*(x - 1)))/log(exp(x)/x)^2 + (65*x^2)/27 + (2*x^3)/3