Integrand size = 113, antiderivative size = 29 \[ \int \frac {e^{-11+2 \log (x) \log (5 x)} \left (e^5 x-e^{5+\frac {x}{e^5}} x+\left (-2 e^5 x+e^{\frac {x}{e^5}} \left (2 e^5 x+x^2\right )\right ) \log (x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log ^2(x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log (x) \log (5 x)\right )}{\log ^2(x)} \, dx=\frac {e^{-6+2 \log (x) \log (5 x)} \left (-1+e^{\frac {x}{e^5}}\right ) x^2}{\log (x)} \] Output:
x^2*exp(ln(x)*ln(5*x)-3)^2*(exp(x/exp(5))-1)/ln(x)
Time = 1.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-11+2 \log (x) \log (5 x)} \left (e^5 x-e^{5+\frac {x}{e^5}} x+\left (-2 e^5 x+e^{\frac {x}{e^5}} \left (2 e^5 x+x^2\right )\right ) \log (x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log ^2(x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log (x) \log (5 x)\right )}{\log ^2(x)} \, dx=\frac {e^{-6+2 \log ^2(x)} \left (-1+e^{\frac {x}{e^5}}\right ) x^{2+\log (25)}}{\log (x)} \] Input:
Integrate[(E^(-11 + 2*Log[x]*Log[5*x])*(E^5*x - E^(5 + x/E^5)*x + (-2*E^5* x + E^(x/E^5)*(2*E^5*x + x^2))*Log[x] + (-2*E^5*x + 2*E^(5 + x/E^5)*x)*Log [x]^2 + (-2*E^5*x + 2*E^(5 + x/E^5)*x)*Log[x]*Log[5*x]))/Log[x]^2,x]
Output:
(E^(-6 + 2*Log[x]^2)*(-1 + E^(x/E^5))*x^(2 + Log[25]))/Log[x]
Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(29)=58\).
Time = 0.54 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.86, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {2726}
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.
| \(\displaystyle \int \frac {e^{2 \log (x) \log (5 x)-11} \left (\left (e^{\frac {x}{e^5}} \left (x^2+2 e^5 x\right )-2 e^5 x\right ) \log (x)-e^{\frac {x}{e^5}+5} x+e^5 x+\left (2 e^{\frac {x}{e^5}+5} x-2 e^5 x\right ) \log ^2(x)+\left (2 e^{\frac {x}{e^5}+5} x-2 e^5 x\right ) \log (5 x) \log (x)\right )}{\log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 2726 |
| \(\displaystyle -\frac {x^{2 \log (5 x)} \left (\left (e^5 x-e^{\frac {x}{e^5}+5} x\right ) \log ^2(x)+\left (e^5 x-e^{\frac {x}{e^5}+5} x\right ) \log (5 x) \log (x)\right )}{e^{11} \log ^2(x) \left (\frac {\log (x)}{x}+\frac {\log (5 x)}{x}\right )}\) |
Input:
Int[(E^(-11 + 2*Log[x]*Log[5*x])*(E^5*x - E^(5 + x/E^5)*x + (-2*E^5*x + E^ (x/E^5)*(2*E^5*x + x^2))*Log[x] + (-2*E^5*x + 2*E^(5 + x/E^5)*x)*Log[x]^2 + (-2*E^5*x + 2*E^(5 + x/E^5)*x)*Log[x]*Log[5*x]))/Log[x]^2,x]
Output:
-((x^(2*Log[5*x])*((E^5*x - E^(5 + x/E^5)*x)*Log[x]^2 + (E^5*x - E^(5 + x/ E^5)*x)*Log[x]*Log[5*x]))/(E^11*Log[x]^2*(Log[x]/x + Log[5*x]/x)))
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 157.93 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
| method | result | size |
| risch | \(\frac {x^{2} \left ({\mathrm e}^{x \,{\mathrm e}^{-5}}-1\right ) x^{2 \ln \left (5 x \right )} {\mathrm e}^{-6}}{\ln \left (x \right )}\) | \(27\) |
| parallelrisch | \(\frac {{\mathrm e}^{-5} \left (-x^{2} {\mathrm e}^{2 \ln \left (x \right ) \ln \left (5 x \right )-6} {\mathrm e}^{5}+{\mathrm e}^{5} {\mathrm e}^{2 \ln \left (x \right ) \ln \left (5 x \right )-6} x^{2} {\mathrm e}^{x \,{\mathrm e}^{-5}}\right )}{\ln \left (x \right )}\) | \(55\) |
Input:
int(((2*x*exp(5)*exp(x/exp(5))-2*x*exp(5))*ln(x)*ln(5*x)+(2*x*exp(5)*exp(x /exp(5))-2*x*exp(5))*ln(x)^2+((2*x*exp(5)+x^2)*exp(x/exp(5))-2*x*exp(5))*l n(x)-x*exp(5)*exp(x/exp(5))+x*exp(5))*exp(ln(x)*ln(5*x)-3)^2/exp(5)/ln(x)^ 2,x,method=_RETURNVERBOSE)
Output:
x^2*(exp(x*exp(-5))-1)/ln(x)*(x^(ln(5)+ln(x)))^2*exp(-6)
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {e^{-11+2 \log (x) \log (5 x)} \left (e^5 x-e^{5+\frac {x}{e^5}} x+\left (-2 e^5 x+e^{\frac {x}{e^5}} \left (2 e^5 x+x^2\right )\right ) \log (x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log ^2(x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log (x) \log (5 x)\right )}{\log ^2(x)} \, dx=-\frac {{\left (x^{2} e^{5} - x^{2} e^{\left ({\left (x + 5 \, e^{5}\right )} e^{\left (-5\right )}\right )}\right )} e^{\left (2 \, \log \left (5\right ) \log \left (x\right ) + 2 \, \log \left (x\right )^{2} - 11\right )}}{\log \left (x\right )} \] Input:
integrate(((2*x*exp(5)*exp(x/exp(5))-2*x*exp(5))*log(x)*log(5*x)+(2*x*exp( 5)*exp(x/exp(5))-2*x*exp(5))*log(x)^2+((2*x*exp(5)+x^2)*exp(x/exp(5))-2*x* exp(5))*log(x)-x*exp(5)*exp(x/exp(5))+x*exp(5))*exp(log(x)*log(5*x)-3)^2/e xp(5)/log(x)^2,x, algorithm="fricas")
Output:
-(x^2*e^5 - x^2*e^((x + 5*e^5)*e^(-5)))*e^(2*log(5)*log(x) + 2*log(x)^2 - 11)/log(x)
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-11+2 \log (x) \log (5 x)} \left (e^5 x-e^{5+\frac {x}{e^5}} x+\left (-2 e^5 x+e^{\frac {x}{e^5}} \left (2 e^5 x+x^2\right )\right ) \log (x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log ^2(x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log (x) \log (5 x)\right )}{\log ^2(x)} \, dx=\frac {\left (x^{2} e^{\frac {x}{e^{5}}} - x^{2}\right ) e^{2 \left (\log {\left (x \right )} + \log {\left (5 \right )}\right ) \log {\left (x \right )} - 6}}{\log {\left (x \right )}} \] Input:
integrate(((2*x*exp(5)*exp(x/exp(5))-2*x*exp(5))*ln(x)*ln(5*x)+(2*x*exp(5) *exp(x/exp(5))-2*x*exp(5))*ln(x)**2+((2*x*exp(5)+x**2)*exp(x/exp(5))-2*x*e xp(5))*ln(x)-x*exp(5)*exp(x/exp(5))+x*exp(5))*exp(ln(x)*ln(5*x)-3)**2/exp( 5)/ln(x)**2,x)
Output:
(x**2*exp(x*exp(-5)) - x**2)*exp(2*(log(x) + log(5))*log(x) - 6)/log(x)
Time = 0.18 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {e^{-11+2 \log (x) \log (5 x)} \left (e^5 x-e^{5+\frac {x}{e^5}} x+\left (-2 e^5 x+e^{\frac {x}{e^5}} \left (2 e^5 x+x^2\right )\right ) \log (x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log ^2(x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log (x) \log (5 x)\right )}{\log ^2(x)} \, dx=\frac {{\left (x^{2} e^{\left (x e^{\left (-5\right )} + 2 \, \log \left (5\right ) \log \left (x\right ) + 2 \, \log \left (x\right )^{2}\right )} - x^{2} e^{\left (2 \, \log \left (5\right ) \log \left (x\right ) + 2 \, \log \left (x\right )^{2}\right )}\right )} e^{\left (-6\right )}}{\log \left (x\right )} \] Input:
integrate(((2*x*exp(5)*exp(x/exp(5))-2*x*exp(5))*log(x)*log(5*x)+(2*x*exp( 5)*exp(x/exp(5))-2*x*exp(5))*log(x)^2+((2*x*exp(5)+x^2)*exp(x/exp(5))-2*x* exp(5))*log(x)-x*exp(5)*exp(x/exp(5))+x*exp(5))*exp(log(x)*log(5*x)-3)^2/e xp(5)/log(x)^2,x, algorithm="maxima")
Output:
(x^2*e^(x*e^(-5) + 2*log(5)*log(x) + 2*log(x)^2) - x^2*e^(2*log(5)*log(x) + 2*log(x)^2))*e^(-6)/log(x)
\[ \int \frac {e^{-11+2 \log (x) \log (5 x)} \left (e^5 x-e^{5+\frac {x}{e^5}} x+\left (-2 e^5 x+e^{\frac {x}{e^5}} \left (2 e^5 x+x^2\right )\right ) \log (x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log ^2(x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log (x) \log (5 x)\right )}{\log ^2(x)} \, dx=\int { -\frac {{\left (2 \, {\left (x e^{5} - x e^{\left (x e^{\left (-5\right )} + 5\right )}\right )} \log \left (5 \, x\right ) \log \left (x\right ) + 2 \, {\left (x e^{5} - x e^{\left (x e^{\left (-5\right )} + 5\right )}\right )} \log \left (x\right )^{2} - x e^{5} + x e^{\left (x e^{\left (-5\right )} + 5\right )} + {\left (2 \, x e^{5} - {\left (x^{2} + 2 \, x e^{5}\right )} e^{\left (x e^{\left (-5\right )}\right )}\right )} \log \left (x\right )\right )} e^{\left (2 \, \log \left (5 \, x\right ) \log \left (x\right ) - 11\right )}}{\log \left (x\right )^{2}} \,d x } \] Input:
integrate(((2*x*exp(5)*exp(x/exp(5))-2*x*exp(5))*log(x)*log(5*x)+(2*x*exp( 5)*exp(x/exp(5))-2*x*exp(5))*log(x)^2+((2*x*exp(5)+x^2)*exp(x/exp(5))-2*x* exp(5))*log(x)-x*exp(5)*exp(x/exp(5))+x*exp(5))*exp(log(x)*log(5*x)-3)^2/e xp(5)/log(x)^2,x, algorithm="giac")
Output:
integrate(-(2*(x*e^5 - x*e^(x*e^(-5) + 5))*log(5*x)*log(x) + 2*(x*e^5 - x* e^(x*e^(-5) + 5))*log(x)^2 - x*e^5 + x*e^(x*e^(-5) + 5) + (2*x*e^5 - (x^2 + 2*x*e^5)*e^(x*e^(-5)))*log(x))*e^(2*log(5*x)*log(x) - 11)/log(x)^2, x)
Time = 2.53 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-11+2 \log (x) \log (5 x)} \left (e^5 x-e^{5+\frac {x}{e^5}} x+\left (-2 e^5 x+e^{\frac {x}{e^5}} \left (2 e^5 x+x^2\right )\right ) \log (x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log ^2(x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log (x) \log (5 x)\right )}{\log ^2(x)} \, dx=\frac {x^{2\,\ln \left (5\right )+2}\,{\mathrm {e}}^{2\,{\ln \left (x\right )}^2-6}\,\left ({\mathrm {e}}^{x\,{\mathrm {e}}^{-5}}-1\right )}{\ln \left (x\right )} \] Input:
int(-(exp(2*log(5*x)*log(x) - 6)*exp(-5)*(log(x)^2*(2*x*exp(5) - 2*x*exp(5 )*exp(x*exp(-5))) - x*exp(5) + log(x)*(2*x*exp(5) - exp(x*exp(-5))*(2*x*ex p(5) + x^2)) + log(5*x)*log(x)*(2*x*exp(5) - 2*x*exp(5)*exp(x*exp(-5))) + x*exp(5)*exp(x*exp(-5))))/log(x)^2,x)
Output:
(x^(2*log(5) + 2)*exp(2*log(x)^2 - 6)*(exp(x*exp(-5)) - 1))/log(x)
\[ \int \frac {e^{-11+2 \log (x) \log (5 x)} \left (e^5 x-e^{5+\frac {x}{e^5}} x+\left (-2 e^5 x+e^{\frac {x}{e^5}} \left (2 e^5 x+x^2\right )\right ) \log (x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log ^2(x)+\left (-2 e^5 x+2 e^{5+\frac {x}{e^5}} x\right ) \log (x) \log (5 x)\right )}{\log ^2(x)} \, dx=\frac {\int \frac {x^{2 \,\mathrm {log}\left (x \right )} e^{\frac {x}{e^{5}}} 5^{2 \,\mathrm {log}\left (x \right )} x^{2}}{\mathrm {log}\left (x \right )}d x +2 \left (\int \frac {x^{2 \,\mathrm {log}\left (x \right )} e^{\frac {x}{e^{5}}} 5^{2 \,\mathrm {log}\left (x \right )} \mathrm {log}\left (5 x \right ) x}{\mathrm {log}\left (x \right )}d x \right ) e^{5}-\left (\int \frac {x^{2 \,\mathrm {log}\left (x \right )} e^{\frac {x}{e^{5}}} 5^{2 \,\mathrm {log}\left (x \right )} x}{\mathrm {log}\left (x \right )^{2}}d x \right ) e^{5}+2 \left (\int \frac {x^{2 \,\mathrm {log}\left (x \right )} e^{\frac {x}{e^{5}}} 5^{2 \,\mathrm {log}\left (x \right )} x}{\mathrm {log}\left (x \right )}d x \right ) e^{5}-2 \left (\int \frac {x^{2 \,\mathrm {log}\left (x \right )} 5^{2 \,\mathrm {log}\left (x \right )} \mathrm {log}\left (5 x \right ) x}{\mathrm {log}\left (x \right )}d x \right ) e^{5}+\left (\int \frac {x^{2 \,\mathrm {log}\left (x \right )} 5^{2 \,\mathrm {log}\left (x \right )} x}{\mathrm {log}\left (x \right )^{2}}d x \right ) e^{5}-2 \left (\int \frac {x^{2 \,\mathrm {log}\left (x \right )} 5^{2 \,\mathrm {log}\left (x \right )} x}{\mathrm {log}\left (x \right )}d x \right ) e^{5}+2 \left (\int x^{2 \,\mathrm {log}\left (x \right )} e^{\frac {x}{e^{5}}} 5^{2 \,\mathrm {log}\left (x \right )} x d x \right ) e^{5}-2 \left (\int x^{2 \,\mathrm {log}\left (x \right )} 5^{2 \,\mathrm {log}\left (x \right )} x d x \right ) e^{5}}{e^{11}} \] Input:
int(((2*x*exp(5)*exp(x/exp(5))-2*x*exp(5))*log(x)*log(5*x)+(2*x*exp(5)*exp (x/exp(5))-2*x*exp(5))*log(x)^2+((2*x*exp(5)+x^2)*exp(x/exp(5))-2*x*exp(5) )*log(x)-x*exp(5)*exp(x/exp(5))+x*exp(5))*exp(log(x)*log(5*x)-3)^2/exp(5)/ log(x)^2,x)
Output:
(int((x**(2*log(x))*e**(x/e**5)*5**(2*log(x))*x**2)/log(x),x) + 2*int((x** (2*log(x))*e**(x/e**5)*5**(2*log(x))*log(5*x)*x)/log(x),x)*e**5 - int((x** (2*log(x))*e**(x/e**5)*5**(2*log(x))*x)/log(x)**2,x)*e**5 + 2*int((x**(2*l og(x))*e**(x/e**5)*5**(2*log(x))*x)/log(x),x)*e**5 - 2*int((x**(2*log(x))* 5**(2*log(x))*log(5*x)*x)/log(x),x)*e**5 + int((x**(2*log(x))*5**(2*log(x) )*x)/log(x)**2,x)*e**5 - 2*int((x**(2*log(x))*5**(2*log(x))*x)/log(x),x)*e **5 + 2*int(x**(2*log(x))*e**(x/e**5)*5**(2*log(x))*x,x)*e**5 - 2*int(x**( 2*log(x))*5**(2*log(x))*x,x)*e**5)/e**11