Integrand size = 161, antiderivative size = 30 \[ \int \frac {-\frac {22500 e^8}{x}-22500 e^{20} x^3+22500 e^{20} x^3 \log (x)-900 e^{10} x \log ^2(x)+900 e^{10} x \log ^3(x)+\log ^5(x)+\frac {e^6 \left (-90000 e^5 x^3+22500 e^5 x^3 \log (x)\right )}{x^3}+\frac {e^4 \left (-135000 e^{10} x^3+67500 e^{10} x^3 \log (x)-900 x \log ^2(x)\right )}{x^2}+\frac {e^2 \left (-90000 e^{15} x^3+67500 e^{15} x^3 \log (x)-1800 e^5 x \log ^2(x)+900 e^5 x \log ^3(x)\right )}{x}}{\log ^5(x)} \, dx=x+9 \left (1+\frac {25 \left (e^5+\frac {e^2}{x}\right )^2 x^2}{\log ^2(x)}\right )^2 \] Output:
3*(25*x^2*(exp(5)+exp(-ln(x)+2))^2/ln(x)^2+1)*(75*x^2*(exp(5)+exp(-ln(x)+2 ))^2/ln(x)^2+3)+x
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {-\frac {22500 e^8}{x}-22500 e^{20} x^3+22500 e^{20} x^3 \log (x)-900 e^{10} x \log ^2(x)+900 e^{10} x \log ^3(x)+\log ^5(x)+\frac {e^6 \left (-90000 e^5 x^3+22500 e^5 x^3 \log (x)\right )}{x^3}+\frac {e^4 \left (-135000 e^{10} x^3+67500 e^{10} x^3 \log (x)-900 x \log ^2(x)\right )}{x^2}+\frac {e^2 \left (-90000 e^{15} x^3+67500 e^{15} x^3 \log (x)-1800 e^5 x \log ^2(x)+900 e^5 x \log ^3(x)\right )}{x}}{\log ^5(x)} \, dx=x+\frac {5625 e^8 \left (1+e^3 x\right )^4}{\log ^4(x)}+\frac {450 e^4 \left (1+e^3 x\right )^2}{\log ^2(x)} \] Input:
Integrate[((-22500*E^8)/x - 22500*E^20*x^3 + 22500*E^20*x^3*Log[x] - 900*E ^10*x*Log[x]^2 + 900*E^10*x*Log[x]^3 + Log[x]^5 + (E^6*(-90000*E^5*x^3 + 2 2500*E^5*x^3*Log[x]))/x^3 + (E^4*(-135000*E^10*x^3 + 67500*E^10*x^3*Log[x] - 900*x*Log[x]^2))/x^2 + (E^2*(-90000*E^15*x^3 + 67500*E^15*x^3*Log[x] - 1800*E^5*x*Log[x]^2 + 900*E^5*x*Log[x]^3))/x)/Log[x]^5,x]
Output:
x + (5625*E^8*(1 + E^3*x)^4)/Log[x]^4 + (450*E^4*(1 + E^3*x)^2)/Log[x]^2
Leaf count is larger than twice the leaf count of optimal. \(377\) vs. \(2(30)=60\).
Time = 1.77 (sec) , antiderivative size = 377, normalized size of antiderivative = 12.57, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {7239, 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.
| \(\displaystyle \int \frac {-22500 e^{20} x^3+\frac {e^2 \left (-90000 e^{15} x^3+67500 e^{15} x^3 \log (x)+900 e^5 x \log ^3(x)-1800 e^5 x \log ^2(x)\right )}{x}+22500 e^{20} x^3 \log (x)+\frac {e^6 \left (22500 e^5 x^3 \log (x)-90000 e^5 x^3\right )}{x^3}+\frac {e^4 \left (-135000 e^{10} x^3+67500 e^{10} x^3 \log (x)-900 x \log ^2(x)\right )}{x^2}-\frac {22500 e^8}{x}+\log ^5(x)+900 e^{10} x \log ^3(x)-900 e^{10} x \log ^2(x)}{\log ^5(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-22500 e^8 \left (e^3 x+1\right )^4+x \log ^5(x)+900 e^7 x \left (e^3 x+1\right ) \log ^3(x)-900 e^4 \left (e^3 x+1\right )^2 \log ^2(x)+22500 e^{11} x \left (e^3 x+1\right )^3 \log (x)}{x \log ^5(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {22500 e^8 \left (e^3 x+1\right )^4}{x \log ^5(x)}+\frac {22500 e^{11} \left (e^3 x+1\right )^3}{\log ^4(x)}-\frac {900 e^4 \left (e^3 x+1\right )^2}{x \log ^3(x)}+\frac {900 e^7 \left (e^3 x+1\right )}{\log ^2(x)}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5625 e^{20} x^4}{\log ^4(x)}+\frac {7500 e^{20} x^4}{\log ^3(x)}+\frac {15000 e^{20} x^4}{\log ^2(x)}+\frac {60000 e^{20} x^4}{\log (x)}+\frac {22500 e^{17} x^3}{\log ^4(x)}+\frac {22500 e^{17} x^3}{\log ^3(x)}+\frac {33750 e^{17} x^3}{\log ^2(x)}+\frac {101250 e^{17} x^3}{\log (x)}+\frac {33750 e^{14} x^2}{\log ^4(x)}+\frac {22500 e^{14} x^2}{\log ^3(x)}+\frac {22500 e^{14} x^2}{\log ^2(x)}+\frac {450 e^{10} x^2}{\log ^2(x)}+\frac {45000 e^{14} x^2}{\log (x)}+\frac {900 e^{10} x^2}{\log (x)}+x+\frac {22500 e^{11} x}{\log ^4(x)}+\frac {5625 e^8}{\log ^4(x)}-\frac {7500 e^{11} \left (e^3 x+1\right )^3 x}{\log ^3(x)}+\frac {7500 e^{11} x}{\log ^3(x)}-\frac {15000 e^{11} \left (e^3 x+1\right )^3 x}{\log ^2(x)}+\frac {11250 e^{11} \left (e^3 x+1\right )^2 x}{\log ^2(x)}+\frac {3750 e^{11} x}{\log ^2(x)}+\frac {900 e^7 x}{\log ^2(x)}+\frac {450 e^4}{\log ^2(x)}-\frac {60000 e^{11} \left (e^3 x+1\right )^3 x}{\log (x)}+\frac {78750 e^{11} \left (e^3 x+1\right )^2 x}{\log (x)}-\frac {22500 e^{11} \left (e^3 x+1\right ) x}{\log (x)}-\frac {900 e^7 \left (e^3 x+1\right ) x}{\log (x)}+\frac {3750 e^{11} x}{\log (x)}+\frac {900 e^7 x}{\log (x)}\) |
Input:
Int[((-22500*E^8)/x - 22500*E^20*x^3 + 22500*E^20*x^3*Log[x] - 900*E^10*x* Log[x]^2 + 900*E^10*x*Log[x]^3 + Log[x]^5 + (E^6*(-90000*E^5*x^3 + 22500*E ^5*x^3*Log[x]))/x^3 + (E^4*(-135000*E^10*x^3 + 67500*E^10*x^3*Log[x] - 900 *x*Log[x]^2))/x^2 + (E^2*(-90000*E^15*x^3 + 67500*E^15*x^3*Log[x] - 1800*E ^5*x*Log[x]^2 + 900*E^5*x*Log[x]^3))/x)/Log[x]^5,x]
Output:
x + (5625*E^8)/Log[x]^4 + (22500*E^11*x)/Log[x]^4 + (33750*E^14*x^2)/Log[x ]^4 + (22500*E^17*x^3)/Log[x]^4 + (5625*E^20*x^4)/Log[x]^4 + (7500*E^11*x) /Log[x]^3 + (22500*E^14*x^2)/Log[x]^3 + (22500*E^17*x^3)/Log[x]^3 + (7500* E^20*x^4)/Log[x]^3 - (7500*E^11*x*(1 + E^3*x)^3)/Log[x]^3 + (450*E^4)/Log[ x]^2 + (900*E^7*x)/Log[x]^2 + (3750*E^11*x)/Log[x]^2 + (450*E^10*x^2)/Log[ x]^2 + (22500*E^14*x^2)/Log[x]^2 + (33750*E^17*x^3)/Log[x]^2 + (15000*E^20 *x^4)/Log[x]^2 + (11250*E^11*x*(1 + E^3*x)^2)/Log[x]^2 - (15000*E^11*x*(1 + E^3*x)^3)/Log[x]^2 + (900*E^7*x)/Log[x] + (3750*E^11*x)/Log[x] + (900*E^ 10*x^2)/Log[x] + (45000*E^14*x^2)/Log[x] + (101250*E^17*x^3)/Log[x] + (600 00*E^20*x^4)/Log[x] - (900*E^7*x*(1 + E^3*x))/Log[x] - (22500*E^11*x*(1 + E^3*x))/Log[x] + (78750*E^11*x*(1 + E^3*x)^2)/Log[x] - (60000*E^11*x*(1 + E^3*x)^3)/Log[x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 2.53 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27
| method | result | size |
| risch | \(x +\frac {225 \,{\mathrm e}^{4} \left (25 x^{4} {\mathrm e}^{16}+100 \,{\mathrm e}^{13} x^{3}+150 x^{2} {\mathrm e}^{10}+2 x^{2} {\mathrm e}^{6} \ln \left (x \right )^{2}+100 \,{\mathrm e}^{7} x +4 \,{\mathrm e}^{3} \ln \left (x \right )^{2} x +25 \,{\mathrm e}^{4}+2 \ln \left (x \right )^{2}\right )}{\ln \left (x \right )^{4}}\) | \(68\) |
| parallelrisch | \(\frac {5625 \,{\mathrm e}^{20} x^{4}+22500 \,{\mathrm e}^{15} x^{4} {\mathrm e}^{-\ln \left (x \right )+2}+33750 \,{\mathrm e}^{4} {\mathrm e}^{10} x^{2}+22500 x \,{\mathrm e}^{5} {\mathrm e}^{6}+5625 \,{\mathrm e}^{8}+450 x^{2} \ln \left (x \right )^{2} {\mathrm e}^{10}+900 \,{\mathrm e}^{5} x^{2} \ln \left (x \right )^{2} {\mathrm e}^{-\ln \left (x \right )+2}+450 \,{\mathrm e}^{4} \ln \left (x \right )^{2}+x \ln \left (x \right )^{4}}{\ln \left (x \right )^{4}}\) | \(135\) |
| default | \(x +\frac {22500 \,{\mathrm e}^{6} {\mathrm e}^{5} x}{\ln \left (x \right )^{4}}+\frac {33750 \,{\mathrm e}^{4} {\mathrm e}^{10} x^{2}}{\ln \left (x \right )^{4}}+\frac {450 \,{\mathrm e}^{4}}{\ln \left (x \right )^{2}}+\frac {900 \,{\mathrm e}^{2} {\mathrm e}^{5} x}{\ln \left (x \right )^{2}}+\frac {22500 \,{\mathrm e}^{2} x^{3} {\mathrm e}^{15}}{\ln \left (x \right )^{4}}-900 \,{\mathrm e}^{10} \left (-\frac {x^{2}}{2 \ln \left (x \right )^{2}}-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )\right )+900 \,{\mathrm e}^{10} \left (-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )\right )-22500 \,{\mathrm e}^{20} \left (-\frac {x^{4}}{4 \ln \left (x \right )^{4}}-\frac {x^{4}}{3 \ln \left (x \right )^{3}}-\frac {2 x^{4}}{3 \ln \left (x \right )^{2}}-\frac {8 x^{4}}{3 \ln \left (x \right )}-\frac {32 \,\operatorname {expIntegral}_{1}\left (-4 \ln \left (x \right )\right )}{3}\right )+22500 \,{\mathrm e}^{20} \left (-\frac {x^{4}}{3 \ln \left (x \right )^{3}}-\frac {2 x^{4}}{3 \ln \left (x \right )^{2}}-\frac {8 x^{4}}{3 \ln \left (x \right )}-\frac {32 \,\operatorname {expIntegral}_{1}\left (-4 \ln \left (x \right )\right )}{3}\right )+\frac {5625 \,{\mathrm e}^{8}}{\ln \left (x \right )^{4}}\) | \(217\) |
| parts | \(x +\frac {22500 \,{\mathrm e}^{6} {\mathrm e}^{5} x}{\ln \left (x \right )^{4}}+\frac {33750 \,{\mathrm e}^{4} {\mathrm e}^{10} x^{2}}{\ln \left (x \right )^{4}}+\frac {450 \,{\mathrm e}^{4}}{\ln \left (x \right )^{2}}+\frac {900 \,{\mathrm e}^{2} {\mathrm e}^{5} x}{\ln \left (x \right )^{2}}+\frac {22500 \,{\mathrm e}^{2} x^{3} {\mathrm e}^{15}}{\ln \left (x \right )^{4}}-900 \,{\mathrm e}^{10} \left (-\frac {x^{2}}{2 \ln \left (x \right )^{2}}-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )\right )+900 \,{\mathrm e}^{10} \left (-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )\right )-22500 \,{\mathrm e}^{20} \left (-\frac {x^{4}}{4 \ln \left (x \right )^{4}}-\frac {x^{4}}{3 \ln \left (x \right )^{3}}-\frac {2 x^{4}}{3 \ln \left (x \right )^{2}}-\frac {8 x^{4}}{3 \ln \left (x \right )}-\frac {32 \,\operatorname {expIntegral}_{1}\left (-4 \ln \left (x \right )\right )}{3}\right )+22500 \,{\mathrm e}^{20} \left (-\frac {x^{4}}{3 \ln \left (x \right )^{3}}-\frac {2 x^{4}}{3 \ln \left (x \right )^{2}}-\frac {8 x^{4}}{3 \ln \left (x \right )}-\frac {32 \,\operatorname {expIntegral}_{1}\left (-4 \ln \left (x \right )\right )}{3}\right )+\frac {5625 \,{\mathrm e}^{8}}{\ln \left (x \right )^{4}}\) | \(217\) |
Input:
int((-22500*x^3*exp(-ln(x)+2)^4+(22500*x^3*exp(5)*ln(x)-90000*x^3*exp(5))* exp(-ln(x)+2)^3+(-900*x*ln(x)^2+67500*x^3*exp(5)^2*ln(x)-135000*x^3*exp(5) ^2)*exp(-ln(x)+2)^2+(900*x*exp(5)*ln(x)^3-1800*x*exp(5)*ln(x)^2+67500*x^3* exp(5)^3*ln(x)-90000*x^3*exp(5)^3)*exp(-ln(x)+2)+ln(x)^5+900*x*exp(5)^2*ln (x)^3-900*x*exp(5)^2*ln(x)^2+22500*x^3*exp(5)^4*ln(x)-22500*x^3*exp(5)^4)/ ln(x)^5,x,method=_RETURNVERBOSE)
Output:
x+225*exp(4)*(25*x^4*exp(16)+100*exp(13)*x^3+150*x^2*exp(10)+2*x^2*exp(6)* ln(x)^2+100*exp(7)*x+4*exp(3)*ln(x)^2*x+25*exp(4)+2*ln(x)^2)/ln(x)^4
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (29) = 58\).
Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07 \[ \int \frac {-\frac {22500 e^8}{x}-22500 e^{20} x^3+22500 e^{20} x^3 \log (x)-900 e^{10} x \log ^2(x)+900 e^{10} x \log ^3(x)+\log ^5(x)+\frac {e^6 \left (-90000 e^5 x^3+22500 e^5 x^3 \log (x)\right )}{x^3}+\frac {e^4 \left (-135000 e^{10} x^3+67500 e^{10} x^3 \log (x)-900 x \log ^2(x)\right )}{x^2}+\frac {e^2 \left (-90000 e^{15} x^3+67500 e^{15} x^3 \log (x)-1800 e^5 x \log ^2(x)+900 e^5 x \log ^3(x)\right )}{x}}{\log ^5(x)} \, dx=\frac {5625 \, x^{4} e^{20} + x \log \left (x\right )^{4} + 22500 \, x^{3} e^{17} + 33750 \, x^{2} e^{14} + 450 \, {\left (x^{2} e^{10} + 2 \, x e^{7} + e^{4}\right )} \log \left (x\right )^{2} + 22500 \, x e^{11} + 5625 \, e^{8}}{\log \left (x\right )^{4}} \] Input:
integrate((-22500*x^3*exp(-log(x)+2)^4+(22500*x^3*exp(5)*log(x)-90000*x^3* exp(5))*exp(-log(x)+2)^3+(-900*x*log(x)^2+67500*x^3*exp(5)^2*log(x)-135000 *x^3*exp(5)^2)*exp(-log(x)+2)^2+(900*x*exp(5)*log(x)^3-1800*x*exp(5)*log(x )^2+67500*x^3*exp(5)^3*log(x)-90000*x^3*exp(5)^3)*exp(-log(x)+2)+log(x)^5+ 900*x*exp(5)^2*log(x)^3-900*x*exp(5)^2*log(x)^2+22500*x^3*exp(5)^4*log(x)- 22500*x^3*exp(5)^4)/log(x)^5,x, algorithm="fricas")
Output:
(5625*x^4*e^20 + x*log(x)^4 + 22500*x^3*e^17 + 33750*x^2*e^14 + 450*(x^2*e ^10 + 2*x*e^7 + e^4)*log(x)^2 + 22500*x*e^11 + 5625*e^8)/log(x)^4
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27 \[ \int \frac {-\frac {22500 e^8}{x}-22500 e^{20} x^3+22500 e^{20} x^3 \log (x)-900 e^{10} x \log ^2(x)+900 e^{10} x \log ^3(x)+\log ^5(x)+\frac {e^6 \left (-90000 e^5 x^3+22500 e^5 x^3 \log (x)\right )}{x^3}+\frac {e^4 \left (-135000 e^{10} x^3+67500 e^{10} x^3 \log (x)-900 x \log ^2(x)\right )}{x^2}+\frac {e^2 \left (-90000 e^{15} x^3+67500 e^{15} x^3 \log (x)-1800 e^5 x \log ^2(x)+900 e^5 x \log ^3(x)\right )}{x}}{\log ^5(x)} \, dx=x + \frac {5625 x^{4} e^{20} + 22500 x^{3} e^{17} + 33750 x^{2} e^{14} + 22500 x e^{11} + \left (450 x^{2} e^{10} + 900 x e^{7} + 450 e^{4}\right ) \log {\left (x \right )}^{2} + 5625 e^{8}}{\log {\left (x \right )}^{4}} \] Input:
integrate((-22500*x**3*exp(-ln(x)+2)**4+(22500*x**3*exp(5)*ln(x)-90000*x** 3*exp(5))*exp(-ln(x)+2)**3+(-900*x*ln(x)**2+67500*x**3*exp(5)**2*ln(x)-135 000*x**3*exp(5)**2)*exp(-ln(x)+2)**2+(900*x*exp(5)*ln(x)**3-1800*x*exp(5)* ln(x)**2+67500*x**3*exp(5)**3*ln(x)-90000*x**3*exp(5)**3)*exp(-ln(x)+2)+ln (x)**5+900*x*exp(5)**2*ln(x)**3-900*x*exp(5)**2*ln(x)**2+22500*x**3*exp(5) **4*ln(x)-22500*x**3*exp(5)**4)/ln(x)**5,x)
Output:
x + (5625*x**4*exp(20) + 22500*x**3*exp(17) + 33750*x**2*exp(14) + 22500*x *exp(11) + (450*x**2*exp(10) + 900*x*exp(7) + 450*exp(4))*log(x)**2 + 5625 *exp(8))/log(x)**4
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.60 \[ \int \frac {-\frac {22500 e^8}{x}-22500 e^{20} x^3+22500 e^{20} x^3 \log (x)-900 e^{10} x \log ^2(x)+900 e^{10} x \log ^3(x)+\log ^5(x)+\frac {e^6 \left (-90000 e^5 x^3+22500 e^5 x^3 \log (x)\right )}{x^3}+\frac {e^4 \left (-135000 e^{10} x^3+67500 e^{10} x^3 \log (x)-900 x \log ^2(x)\right )}{x^2}+\frac {e^2 \left (-90000 e^{15} x^3+67500 e^{15} x^3 \log (x)-1800 e^5 x \log ^2(x)+900 e^5 x \log ^3(x)\right )}{x}}{\log ^5(x)} \, dx=900 \, e^{7} \Gamma \left (-1, -\log \left (x\right )\right ) + 1800 \, e^{10} \Gamma \left (-1, -2 \, \log \left (x\right )\right ) + 1800 \, e^{7} \Gamma \left (-2, -\log \left (x\right )\right ) + 3600 \, e^{10} \Gamma \left (-2, -2 \, \log \left (x\right )\right ) + 22500 \, e^{11} \Gamma \left (-3, -\log \left (x\right )\right ) + 540000 \, e^{14} \Gamma \left (-3, -2 \, \log \left (x\right )\right ) + 1822500 \, e^{17} \Gamma \left (-3, -3 \, \log \left (x\right )\right ) + 1440000 \, e^{20} \Gamma \left (-3, -4 \, \log \left (x\right )\right ) + 90000 \, e^{11} \Gamma \left (-4, -\log \left (x\right )\right ) + 2160000 \, e^{14} \Gamma \left (-4, -2 \, \log \left (x\right )\right ) + 7290000 \, e^{17} \Gamma \left (-4, -3 \, \log \left (x\right )\right ) + 5760000 \, e^{20} \Gamma \left (-4, -4 \, \log \left (x\right )\right ) + x + \frac {450 \, e^{4}}{\log \left (x\right )^{2}} + \frac {5625 \, e^{8}}{\log \left (x\right )^{4}} \] Input:
integrate((-22500*x^3*exp(-log(x)+2)^4+(22500*x^3*exp(5)*log(x)-90000*x^3* exp(5))*exp(-log(x)+2)^3+(-900*x*log(x)^2+67500*x^3*exp(5)^2*log(x)-135000 *x^3*exp(5)^2)*exp(-log(x)+2)^2+(900*x*exp(5)*log(x)^3-1800*x*exp(5)*log(x )^2+67500*x^3*exp(5)^3*log(x)-90000*x^3*exp(5)^3)*exp(-log(x)+2)+log(x)^5+ 900*x*exp(5)^2*log(x)^3-900*x*exp(5)^2*log(x)^2+22500*x^3*exp(5)^4*log(x)- 22500*x^3*exp(5)^4)/log(x)^5,x, algorithm="maxima")
Output:
900*e^7*gamma(-1, -log(x)) + 1800*e^10*gamma(-1, -2*log(x)) + 1800*e^7*gam ma(-2, -log(x)) + 3600*e^10*gamma(-2, -2*log(x)) + 22500*e^11*gamma(-3, -l og(x)) + 540000*e^14*gamma(-3, -2*log(x)) + 1822500*e^17*gamma(-3, -3*log( x)) + 1440000*e^20*gamma(-3, -4*log(x)) + 90000*e^11*gamma(-4, -log(x)) + 2160000*e^14*gamma(-4, -2*log(x)) + 7290000*e^17*gamma(-4, -3*log(x)) + 57 60000*e^20*gamma(-4, -4*log(x)) + x + 450*e^4/log(x)^2 + 5625*e^8/log(x)^4
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (29) = 58\).
Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \frac {-\frac {22500 e^8}{x}-22500 e^{20} x^3+22500 e^{20} x^3 \log (x)-900 e^{10} x \log ^2(x)+900 e^{10} x \log ^3(x)+\log ^5(x)+\frac {e^6 \left (-90000 e^5 x^3+22500 e^5 x^3 \log (x)\right )}{x^3}+\frac {e^4 \left (-135000 e^{10} x^3+67500 e^{10} x^3 \log (x)-900 x \log ^2(x)\right )}{x^2}+\frac {e^2 \left (-90000 e^{15} x^3+67500 e^{15} x^3 \log (x)-1800 e^5 x \log ^2(x)+900 e^5 x \log ^3(x)\right )}{x}}{\log ^5(x)} \, dx=\frac {5625 \, x^{4} e^{20} + 450 \, x^{2} e^{10} \log \left (x\right )^{2} + x \log \left (x\right )^{4} + 22500 \, x^{3} e^{17} + 900 \, x e^{7} \log \left (x\right )^{2} + 33750 \, x^{2} e^{14} + 450 \, e^{4} \log \left (x\right )^{2} + 22500 \, x e^{11} + 5625 \, e^{8}}{\log \left (x\right )^{4}} \] Input:
integrate((-22500*x^3*exp(-log(x)+2)^4+(22500*x^3*exp(5)*log(x)-90000*x^3* exp(5))*exp(-log(x)+2)^3+(-900*x*log(x)^2+67500*x^3*exp(5)^2*log(x)-135000 *x^3*exp(5)^2)*exp(-log(x)+2)^2+(900*x*exp(5)*log(x)^3-1800*x*exp(5)*log(x )^2+67500*x^3*exp(5)^3*log(x)-90000*x^3*exp(5)^3)*exp(-log(x)+2)+log(x)^5+ 900*x*exp(5)^2*log(x)^3-900*x*exp(5)^2*log(x)^2+22500*x^3*exp(5)^4*log(x)- 22500*x^3*exp(5)^4)/log(x)^5,x, algorithm="giac")
Output:
(5625*x^4*e^20 + 450*x^2*e^10*log(x)^2 + x*log(x)^4 + 22500*x^3*e^17 + 900 *x*e^7*log(x)^2 + 33750*x^2*e^14 + 450*e^4*log(x)^2 + 22500*x*e^11 + 5625* e^8)/log(x)^4
Time = 0.41 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.00 \[ \int \frac {-\frac {22500 e^8}{x}-22500 e^{20} x^3+22500 e^{20} x^3 \log (x)-900 e^{10} x \log ^2(x)+900 e^{10} x \log ^3(x)+\log ^5(x)+\frac {e^6 \left (-90000 e^5 x^3+22500 e^5 x^3 \log (x)\right )}{x^3}+\frac {e^4 \left (-135000 e^{10} x^3+67500 e^{10} x^3 \log (x)-900 x \log ^2(x)\right )}{x^2}+\frac {e^2 \left (-90000 e^{15} x^3+67500 e^{15} x^3 \log (x)-1800 e^5 x \log ^2(x)+900 e^5 x \log ^3(x)\right )}{x}}{\log ^5(x)} \, dx=x+\frac {5625\,{\mathrm {e}}^8+22500\,x\,{\mathrm {e}}^{11}+33750\,x^2\,{\mathrm {e}}^{14}+22500\,x^3\,{\mathrm {e}}^{17}+5625\,x^4\,{\mathrm {e}}^{20}+{\ln \left (x\right )}^2\,\left (450\,{\mathrm {e}}^{10}\,x^2+900\,{\mathrm {e}}^7\,x+450\,{\mathrm {e}}^4\right )}{{\ln \left (x\right )}^4} \] Input:
int(-(exp(4 - 2*log(x))*(900*x*log(x)^2 + 135000*x^3*exp(10) - 67500*x^3*e xp(10)*log(x)) + 22500*x^3*exp(8 - 4*log(x)) - log(x)^5 + 22500*x^3*exp(20 ) + exp(6 - 3*log(x))*(90000*x^3*exp(5) - 22500*x^3*exp(5)*log(x)) + exp(2 - log(x))*(90000*x^3*exp(15) + 1800*x*exp(5)*log(x)^2 - 900*x*exp(5)*log( x)^3 - 67500*x^3*exp(15)*log(x)) + 900*x*exp(10)*log(x)^2 - 900*x*exp(10)* log(x)^3 - 22500*x^3*exp(20)*log(x))/log(x)^5,x)
Output:
x + (5625*exp(8) + 22500*x*exp(11) + 33750*x^2*exp(14) + 22500*x^3*exp(17) + 5625*x^4*exp(20) + log(x)^2*(450*exp(4) + 900*x*exp(7) + 450*x^2*exp(10 )))/log(x)^4
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.60 \[ \int \frac {-\frac {22500 e^8}{x}-22500 e^{20} x^3+22500 e^{20} x^3 \log (x)-900 e^{10} x \log ^2(x)+900 e^{10} x \log ^3(x)+\log ^5(x)+\frac {e^6 \left (-90000 e^5 x^3+22500 e^5 x^3 \log (x)\right )}{x^3}+\frac {e^4 \left (-135000 e^{10} x^3+67500 e^{10} x^3 \log (x)-900 x \log ^2(x)\right )}{x^2}+\frac {e^2 \left (-90000 e^{15} x^3+67500 e^{15} x^3 \log (x)-1800 e^5 x \log ^2(x)+900 e^5 x \log ^3(x)\right )}{x}}{\log ^5(x)} \, dx=\frac {\mathrm {log}\left (x \right )^{4} x +450 \mathrm {log}\left (x \right )^{2} e^{10} x^{2}+900 \mathrm {log}\left (x \right )^{2} e^{7} x +450 \mathrm {log}\left (x \right )^{2} e^{4}+5625 e^{20} x^{4}+22500 e^{17} x^{3}+33750 e^{14} x^{2}+22500 e^{11} x +5625 e^{8}}{\mathrm {log}\left (x \right )^{4}} \] Input:
int((-22500*x^3*exp(-log(x)+2)^4+(22500*x^3*exp(5)*log(x)-90000*x^3*exp(5) )*exp(-log(x)+2)^3+(-900*x*log(x)^2+67500*x^3*exp(5)^2*log(x)-135000*x^3*e xp(5)^2)*exp(-log(x)+2)^2+(900*x*exp(5)*log(x)^3-1800*x*exp(5)*log(x)^2+67 500*x^3*exp(5)^3*log(x)-90000*x^3*exp(5)^3)*exp(-log(x)+2)+log(x)^5+900*x* exp(5)^2*log(x)^3-900*x*exp(5)^2*log(x)^2+22500*x^3*exp(5)^4*log(x)-22500* x^3*exp(5)^4)/log(x)^5,x)
Output:
(log(x)**4*x + 450*log(x)**2*e**10*x**2 + 900*log(x)**2*e**7*x + 450*log(x )**2*e**4 + 5625*e**20*x**4 + 22500*e**17*x**3 + 33750*e**14*x**2 + 22500* e**11*x + 5625*e**8)/log(x)**4