∫ −16x+(−4−4x) log(3)+ex(16+8 log(3))___________________ −4x2+(3x−e5∕3x−x2) log(3)+ex(4x+(−3+e5∕3+x) log(3))+(ex log(3)−x log(3)) log(−ex+x) dx

Optimal. Leaf size=32 \[ \log \left (\left (3-e^{5/3}-x-\frac {4 x}{\log (3)}-\log \left (-e^x+x\right )\right )^4\right ) \]

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Rubi [A] time = 0.41, antiderivative size = 34, normalized size of antiderivative = 1.06, number of steps used = 2, number of rules used = 2, integrand size = 91, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6688, 6684} \begin {gather*} 4 \log \left (x (4+\log (3))+\log (3) \log \left (x-e^x\right )-\left (3-e^{5/3}\right ) \log (3)\right ) \end {gather*}

Int[(-16*x + (-4 - 4*x)*Log[3] + E^x*(16 + 8*Log[3]))/(-4*x^2 + (3*x - E^(5/3)*x - x^2)*Log[3] + E^x*(4*x + (-

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /; !Fa

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 (\log (3)+x (4+\log (3)))+4 e^x (4+\log (9))}{\left (e^x-x\right ) \left (\left (-3+e^{5/3}\right ) \log (3)+x (4+\log (3))+\log (3) \log \left (-e^x+x\right )\right )} \, dx\\ &=4 \log \left (-\left (\left (3-e^{5/3}\right ) \log (3)\right )+x (4+\log (3))+\log (3) \log \left (-e^x+x\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [F] time = 0.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-16 x+(-4-4 x) \log (3)+e^x (16+8 \log (3))}{-4 x^2+\left (3 x-e^{5/3} x-x^2\right ) \log (3)+e^x \left (4 x+\left (-3+e^{5/3}+x\right ) \log (3)\right )+\left (e^x \log (3)-x \log (3)\right ) \log \left (-e^x+x\right )} \, dx \end {gather*}

Integrate[(-16*x + (-4 - 4*x)*Log[3] + E^x*(16 + 8*Log[3]))/(-4*x^2 + (3*x - E^(5/3)*x - x^2)*Log[3] + E^x*(4*

x + (-3 + E^(5/3) + x)*Log[3]) + (E^x*Log[3] - x*Log[3])*Log[-E^x + x]),x]

Integrate[(-16*x + (-4 - 4*x)*Log[3] + E^x*(16 + 8*Log[3]))/(-4*x^2 + (3*x - E^(5/3)*x - x^2)*Log[3] + E^x*(4*

x + (-3 + E^(5/3) + x)*Log[3]) + (E^x*Log[3] - x*Log[3])*Log[-E^x + x]), x]

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fricas [A] time = 0.56, size = 25, normalized size = 0.78 \begin {gather*} 4 \, \log \left ({\left (x + e^{\frac {5}{3}} - 3\right )} \log \relax (3) + \log \relax (3) \log \left (x - e^{x}\right ) + 4 \, x\right ) \end {gather*}

integrate(((8*log(3)+16)*exp(x)+(-4*x-4)*log(3)-16*x)/((log(3)*exp(x)-x*log(3))*log(x-exp(x))+((exp(5/3)+x-3)*

log(3)+4*x)*exp(x)+(-x*exp(5/3)-x^2+3*x)*log(3)-4*x^2),x, algorithm="fricas")

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giac [A] time = 0.20, size = 30, normalized size = 0.94 \begin {gather*} 4 \, \log \left (x \log \relax (3) + e^{\frac {5}{3}} \log \relax (3) + \log \relax (3) \log \left (x - e^{x}\right ) + 4 \, x - 3 \, \log \relax (3)\right ) \end {gather*}

integrate(((8*log(3)+16)*exp(x)+(-4*x-4)*log(3)-16*x)/((log(3)*exp(x)-x*log(3))*log(x-exp(x))+((exp(5/3)+x-3)*

log(3)+4*x)*exp(x)+(-x*exp(5/3)-x^2+3*x)*log(3)-4*x^2),x, algorithm="giac")

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method	result	size

norman	\(4 \ln \left ({\mathrm e}^{\frac {5}{3}} \ln \relax (3)+\ln \left (x -{\mathrm e}^{x}\right ) \ln \relax (3)+x \ln \relax (3)-3 \ln \relax (3)+4 x \right )\)	\(31\)
risch	\(4 \ln \left (\ln \left (x -{\mathrm e}^{x}\right )+\frac {{\mathrm e}^{\frac {5}{3}} \ln \relax (3)+x \ln \relax (3)-3 \ln \relax (3)+4 x}{\ln \relax (3)}\right )\)	\(34\)

int(((8*ln(3)+16)*exp(x)+(-4*x-4)*ln(3)-16*x)/((ln(3)*exp(x)-x*ln(3))*ln(x-exp(x))+((exp(5/3)+x-3)*ln(3)+4*x)*

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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

integrate(((8*log(3)+16)*exp(x)+(-4*x-4)*log(3)-16*x)/((log(3)*exp(x)-x*log(3))*log(x-exp(x))+((exp(5/3)+x-3)*

log(3)+4*x)*exp(x)+(-x*exp(5/3)-x^2+3*x)*log(3)-4*x^2),x, algorithm="maxima")

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B] time = 4.94, size = 28, normalized size = 0.88 \begin {gather*} 4\,\ln \left (4\,x+x\,\ln \relax (3)+\ln \relax (3)\,\ln \left (x-{\mathrm {e}}^x\right )+\ln \relax (3)\,\left ({\mathrm {e}}^{5/3}-3\right )\right ) \end {gather*}

int((16*x + log(3)*(4*x + 4) - exp(x)*(8*log(3) + 16))/(log(3)*(x*exp(5/3) - 3*x + x^2) + log(x - exp(x))*(x*l

og(3) - exp(x)*log(3)) - exp(x)*(4*x + log(3)*(x + exp(5/3) - 3)) + 4*x^2),x)

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sympy [A] time = 0.38, size = 34, normalized size = 1.06 \begin {gather*} 4 \log {\left (\frac {x \log {\relax (3 )} + 4 x - 3 \log {\relax (3 )} + e^{\frac {5}{3}} \log {\relax (3 )}}{\log {\relax (3 )}} + \log {\left (x - e^{x} \right )} \right )} \end {gather*}

integrate(((8*ln(3)+16)*exp(x)+(-4*x-4)*ln(3)-16*x)/((ln(3)*exp(x)-x*ln(3))*ln(x-exp(x))+((exp(5/3)+x-3)*ln(3)

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

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3.67.4 \(\int \frac {-16 x+(-4-4 x) \log (3)+e^x (16+8 \log (3))}{-4 x^2+(3 x-e^{5/3} x-x^2) \log (3)+e^x (4 x+(-3+e^{5/3}+x) \log (3))+(e^x \log (3)-x \log (3)) \log (-e^x+x)} \, dx\)