Integrand size = 71, antiderivative size = 25 \[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=4 \left (-4+e^{3-x \log \left (\frac {e^5 x}{3}-\log (x)\right )}\right ) \] Output:
4*exp(-x*ln(-ln(x)+1/3*x*exp(5))+3)-16
\[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=\int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx \] Input:
Integrate[(E^(3 - x*Log[(E^5*x - 3*Log[x])/3])*(-12 + 4*E^5*x + (4*E^5*x - 12*Log[x])*Log[(E^5*x - 3*Log[x])/3]))/(-(E^5*x) + 3*Log[x]),x]
Output:
Integrate[(E^(3 - x*Log[(E^5*x - 3*Log[x])/3])*(-12 + 4*E^5*x + (4*E^5*x - 12*Log[x])*Log[(E^5*x - 3*Log[x])/3]))/(-(E^5*x) + 3*Log[x]), x]
Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {7257}
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.
\(\displaystyle \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )-12\right )}{3 \log (x)-e^5 x} \, dx\) |
\(\Big \downarrow \) 7257 |
\(\displaystyle 4 e^3 3^x \left (e^5 x-3 \log (x)\right )^{-x}\) |
Input:
Int[(E^(3 - x*Log[(E^5*x - 3*Log[x])/3])*(-12 + 4*E^5*x + (4*E^5*x - 12*Lo g[x])*Log[(E^5*x - 3*Log[x])/3]))/(-(E^5*x) + 3*Log[x]),x]
Output:
(4*3^x*E^3)/(E^5*x - 3*Log[x])^x
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Time = 1.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76
method | result | size |
risch | \(4 \left (-\ln \left (x \right )+\frac {x \,{\mathrm e}^{5}}{3}\right )^{-x} {\mathrm e}^{3}\) | \(19\) |
parallelrisch | \(4 \,{\mathrm e}^{-x \ln \left (-\ln \left (x \right )+\frac {x \,{\mathrm e}^{5}}{3}\right )+3}\) | \(20\) |
Input:
int(((-12*ln(x)+4*x*exp(5))*ln(-ln(x)+1/3*x*exp(5))+4*x*exp(5)-12)*exp(-x* ln(-ln(x)+1/3*x*exp(5))+3)/(3*ln(x)-x*exp(5)),x,method=_RETURNVERBOSE)
Output:
4*(-ln(x)+1/3*x*exp(5))^(-x)*exp(3)
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=4 \, e^{\left (-x \log \left (\frac {1}{3} \, x e^{5} - \log \left (x\right )\right ) + 3\right )} \] Input:
integrate(((-12*log(x)+4*x*exp(5))*log(-log(x)+1/3*x*exp(5))+4*x*exp(5)-12 )*exp(-x*log(-log(x)+1/3*x*exp(5))+3)/(3*log(x)-x*exp(5)),x, algorithm="fr icas")
Output:
4*e^(-x*log(1/3*x*e^5 - log(x)) + 3)
Time = 0.49 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=4 e^{- x \log {\left (\frac {x e^{5}}{3} - \log {\left (x \right )} \right )} + 3} \] Input:
integrate(((-12*ln(x)+4*x*exp(5))*ln(-ln(x)+1/3*x*exp(5))+4*x*exp(5)-12)*e xp(-x*ln(-ln(x)+1/3*x*exp(5))+3)/(3*ln(x)-x*exp(5)),x)
Output:
4*exp(-x*log(x*exp(5)/3 - log(x)) + 3)
Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=4 \, e^{\left (x \log \left (3\right ) - x \log \left (x e^{5} - 3 \, \log \left (x\right )\right ) + 3\right )} \] Input:
integrate(((-12*log(x)+4*x*exp(5))*log(-log(x)+1/3*x*exp(5))+4*x*exp(5)-12 )*exp(-x*log(-log(x)+1/3*x*exp(5))+3)/(3*log(x)-x*exp(5)),x, algorithm="ma xima")
Output:
4*e^(x*log(3) - x*log(x*e^5 - 3*log(x)) + 3)
\[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=\int { -\frac {4 \, {\left (x e^{5} + {\left (x e^{5} - 3 \, \log \left (x\right )\right )} \log \left (\frac {1}{3} \, x e^{5} - \log \left (x\right )\right ) - 3\right )} e^{\left (-x \log \left (\frac {1}{3} \, x e^{5} - \log \left (x\right )\right ) + 3\right )}}{x e^{5} - 3 \, \log \left (x\right )} \,d x } \] Input:
integrate(((-12*log(x)+4*x*exp(5))*log(-log(x)+1/3*x*exp(5))+4*x*exp(5)-12 )*exp(-x*log(-log(x)+1/3*x*exp(5))+3)/(3*log(x)-x*exp(5)),x, algorithm="gi ac")
Output:
integrate(-4*(x*e^5 + (x*e^5 - 3*log(x))*log(1/3*x*e^5 - log(x)) - 3)*e^(- x*log(1/3*x*e^5 - log(x)) + 3)/(x*e^5 - 3*log(x)), x)
Time = 3.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=\frac {4\,{\mathrm {e}}^3}{{\left (\frac {x\,{\mathrm {e}}^5}{3}-\ln \left (x\right )\right )}^x} \] Input:
int(-(exp(3 - x*log((x*exp(5))/3 - log(x)))*(log((x*exp(5))/3 - log(x))*(1 2*log(x) - 4*x*exp(5)) - 4*x*exp(5) + 12))/(3*log(x) - x*exp(5)),x)
Output:
(4*exp(3))/((x*exp(5))/3 - log(x))^x
\[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=4 e^{3} \left (-3 \left (\int \frac {3^{x}}{3 \left (-3 \,\mathrm {log}\left (x \right )+e^{5} x \right )^{x} \mathrm {log}\left (x \right )-\left (-3 \,\mathrm {log}\left (x \right )+e^{5} x \right )^{x} e^{5} x}d x \right )-3 \left (\int \frac {3^{x} \mathrm {log}\left (-\mathrm {log}\left (x \right )+\frac {e^{5} x}{3}\right ) \mathrm {log}\left (x \right )}{3 \left (-3 \,\mathrm {log}\left (x \right )+e^{5} x \right )^{x} \mathrm {log}\left (x \right )-\left (-3 \,\mathrm {log}\left (x \right )+e^{5} x \right )^{x} e^{5} x}d x \right )+\left (\int \frac {3^{x} \mathrm {log}\left (-\mathrm {log}\left (x \right )+\frac {e^{5} x}{3}\right ) x}{3 \left (-3 \,\mathrm {log}\left (x \right )+e^{5} x \right )^{x} \mathrm {log}\left (x \right )-\left (-3 \,\mathrm {log}\left (x \right )+e^{5} x \right )^{x} e^{5} x}d x \right ) e^{5}+\left (\int \frac {3^{x} x}{3 \left (-3 \,\mathrm {log}\left (x \right )+e^{5} x \right )^{x} \mathrm {log}\left (x \right )-\left (-3 \,\mathrm {log}\left (x \right )+e^{5} x \right )^{x} e^{5} x}d x \right ) e^{5}\right ) \] Input:
int(((-12*log(x)+4*x*exp(5))*log(-log(x)+1/3*x*exp(5))+4*x*exp(5)-12)*exp( -x*log(-log(x)+1/3*x*exp(5))+3)/(3*log(x)-x*exp(5)),x)
Output:
4*e**3*( - 3*int(3**x/(3*( - 3*log(x) + e**5*x)**x*log(x) - ( - 3*log(x) + e**5*x)**x*e**5*x),x) - 3*int((3**x*log(( - 3*log(x) + e**5*x)/3)*log(x)) /(3*( - 3*log(x) + e**5*x)**x*log(x) - ( - 3*log(x) + e**5*x)**x*e**5*x),x ) + int((3**x*log(( - 3*log(x) + e**5*x)/3)*x)/(3*( - 3*log(x) + e**5*x)** x*log(x) - ( - 3*log(x) + e**5*x)**x*e**5*x),x)*e**5 + int((3**x*x)/(3*( - 3*log(x) + e**5*x)**x*log(x) - ( - 3*log(x) + e**5*x)**x*e**5*x),x)*e**5)