\(\int \frac {e^{3-x \log (\frac {1}{3} (e^5 x-3 \log (x)))} (-12+4 e^5 x+(4 e^5 x-12 \log (x)) \log (\frac {1}{3} (e^5 x-3 \log (x))))}{-e^5 x+3 \log (x)} \, dx\) [5911]

Optimal result

Integrand size = 71, antiderivative size = 25 \[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=4 \left (-4+e^{3-x \log \left (\frac {e^5 x}{3}-\log (x)\right )}\right ) \]

[Out]

4*exp(-x*ln(-ln(x)+1/3*x*exp(5))+3)-16

Rubi [A] (verified)

Time = 0.19 (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 = {6838} \[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=4 e^3 3^x \left (e^5 x-3 \log (x)\right )^{-x} \]

[In]

Int[(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]

[Out]

(4*3^x*E^3)/(E^5*x - 3*Log[x])^x

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = 4\ 3^x e^3 \left (e^5 x-3 \log (x)\right )^{-x} \\ \end{align*}

Mathematica [F]

\[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=\int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx \]

[In]

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]

[Out]

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]

Maple [A] (verified)

Time = 1.13 (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	\(4 \,{\mathrm e}^{-x \ln \left (-\ln \left (x \right )+\frac {x \,{\mathrm e}^{5}}{3}\right )+3}\)	\(20\)

[In]

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)

[Out]

4*(-ln(x)+1/3*x*exp(5))^(-x)*exp(3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=4 \, e^{\left (-x \log \left (\frac {1}{3} \, x e^{5} - \log \left (x\right )\right ) + 3\right )} \]

[In]

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="fricas")

[Out]

4*e^(-x*log(1/3*x*e^5 - log(x)) + 3)

Sympy [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=4 e^{- x \log {\left (\frac {x e^{5}}{3} - \log {\left (x \right )} \right )} + 3} \]

[In]

integrate(((-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)

[Out]

4*exp(-x*log(x*exp(5)/3 - log(x)) + 3)

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=4 \, e^{\left (x \log \left (3\right ) - x \log \left (x e^{5} - 3 \, \log \left (x\right )\right ) + 3\right )} \]

[In]

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="maxima")

[Out]

4*e^(x*log(3) - x*log(x*e^5 - 3*log(x)) + 3)

Giac [F]

\[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=\int { -\frac {4 \, {\left (x e^{5} + {\left (x e^{5} - 3 \, \log \left (x\right )\right )} \log \left (\frac {1}{3} \, x e^{5} - \log \left (x\right )\right ) - 3\right )} e^{\left (-x \log \left (\frac {1}{3} \, x e^{5} - \log \left (x\right )\right ) + 3\right )}}{x e^{5} - 3 \, \log \left (x\right )} \,d x } \]

[In]

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="giac")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 11.98 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {e^{3-x \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )} \left (-12+4 e^5 x+\left (4 e^5 x-12 \log (x)\right ) \log \left (\frac {1}{3} \left (e^5 x-3 \log (x)\right )\right )\right )}{-e^5 x+3 \log (x)} \, dx=\frac {4\,{\mathrm {e}}^3}{{\left (\frac {x\,{\mathrm {e}}^5}{3}-\ln \left (x\right )\right )}^x} \]

[In]

int(-(exp(3 - x*log((x*exp(5))/3 - log(x)))*(log((x*exp(5))/3 - log(x))*(12*log(x) - 4*x*exp(5)) - 4*x*exp(5)
+ 12))/(3*log(x) - x*exp(5)),x)

[Out]

(4*exp(3))/((x*exp(5))/3 - log(x))^x