\(\int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x (-51 x-14 x^2+2 x^3)+(69 x+30 e^x x+3 e^{2 x} x+2 x^2) \log (x)+(-15 x-3 e^x x) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+(75 x+30 e^x x+3 e^{2 x} x) \log (x)+(-15 x-3 e^x x) \log ^2(x)+x \log ^3(x)} \, dx\) [3317]

Optimal result

Integrand size = 162, antiderivative size = 19 \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=x+\frac {(-3+x)^2}{\left (5+e^x-\log (x)\right )^2} \]

[Out]

(-3+x)^2/(5+exp(x)-ln(x))^2+x

Rubi [F]

\[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=\int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx \]

[In]

Int[(-18 - 83*x - 15*E^(2*x)*x - E^(3*x)*x - 12*x^2 + E^x*(-51*x - 14*x^2 + 2*x^3) + (69*x + 30*E^x*x + 3*E^(2
*x)*x + 2*x^2)*Log[x] + (-15*x - 3*E^x*x)*Log[x]^2 + x*Log[x]^3)/(-125*x - 75*E^x*x - 15*E^(2*x)*x - E^(3*x)*x
+ (75*x + 30*E^x*x + 3*E^(2*x)*x)*Log[x] + (-15*x - 3*E^x*x)*Log[x]^2 + x*Log[x]^3),x]

[Out]

x + 78*Defer[Int][(5 + E^x - Log[x])^(-3), x] + 18*Defer[Int][1/(x*(5 + E^x - Log[x])^3), x] - 58*Defer[Int][x
/(5 + E^x - Log[x])^3, x] + 10*Defer[Int][x^2/(5 + E^x - Log[x])^3, x] - 24*Defer[Int][(5 + E^x - Log[x])^(-2)
, x] + 14*Defer[Int][x/(5 + E^x - Log[x])^2, x] - 2*Defer[Int][x^2/(5 + E^x - Log[x])^2, x] - 18*Defer[Int][Lo
g[x]/(5 + E^x - Log[x])^3, x] + 12*Defer[Int][(x*Log[x])/(5 + E^x - Log[x])^3, x] - 2*Defer[Int][(x^2*Log[x])/
(5 + E^x - Log[x])^3, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {18+\left (83+51 e^x+15 e^{2 x}+e^{3 x}\right ) x+2 \left (6+7 e^x\right ) x^2-2 e^x x^3-x \left (69+30 e^x+3 e^{2 x}+2 x\right ) \log (x)+3 \left (5+e^x\right ) x \log ^2(x)-x \log ^3(x)}{x \left (5+e^x-\log (x)\right )^3} \, dx \\ & = \int \left (1-\frac {2 \left (12-7 x+x^2\right )}{\left (5+e^x-\log (x)\right )^2}-\frac {2 (-3+x)^2 (-1-5 x+x \log (x))}{x \left (5+e^x-\log (x)\right )^3}\right ) \, dx \\ & = x-2 \int \frac {12-7 x+x^2}{\left (5+e^x-\log (x)\right )^2} \, dx-2 \int \frac {(-3+x)^2 (-1-5 x+x \log (x))}{x \left (5+e^x-\log (x)\right )^3} \, dx \\ & = x-2 \int \left (\frac {12}{\left (5+e^x-\log (x)\right )^2}-\frac {7 x}{\left (5+e^x-\log (x)\right )^2}+\frac {x^2}{\left (5+e^x-\log (x)\right )^2}\right ) \, dx-2 \int \left (-\frac {6 (-1-5 x+x \log (x))}{\left (5+e^x-\log (x)\right )^3}+\frac {9 (-1-5 x+x \log (x))}{x \left (5+e^x-\log (x)\right )^3}+\frac {x (-1-5 x+x \log (x))}{\left (5+e^x-\log (x)\right )^3}\right ) \, dx \\ & = x-2 \int \frac {x^2}{\left (5+e^x-\log (x)\right )^2} \, dx-2 \int \frac {x (-1-5 x+x \log (x))}{\left (5+e^x-\log (x)\right )^3} \, dx+12 \int \frac {-1-5 x+x \log (x)}{\left (5+e^x-\log (x)\right )^3} \, dx+14 \int \frac {x}{\left (5+e^x-\log (x)\right )^2} \, dx-18 \int \frac {-1-5 x+x \log (x)}{x \left (5+e^x-\log (x)\right )^3} \, dx-24 \int \frac {1}{\left (5+e^x-\log (x)\right )^2} \, dx \\ & = x-2 \int \frac {x^2}{\left (5+e^x-\log (x)\right )^2} \, dx-2 \int \left (-\frac {x}{\left (5+e^x-\log (x)\right )^3}-\frac {5 x^2}{\left (5+e^x-\log (x)\right )^3}+\frac {x^2 \log (x)}{\left (5+e^x-\log (x)\right )^3}\right ) \, dx+12 \int \left (-\frac {1}{\left (5+e^x-\log (x)\right )^3}-\frac {5 x}{\left (5+e^x-\log (x)\right )^3}+\frac {x \log (x)}{\left (5+e^x-\log (x)\right )^3}\right ) \, dx+14 \int \frac {x}{\left (5+e^x-\log (x)\right )^2} \, dx-18 \int \left (-\frac {5}{\left (5+e^x-\log (x)\right )^3}-\frac {1}{x \left (5+e^x-\log (x)\right )^3}+\frac {\log (x)}{\left (5+e^x-\log (x)\right )^3}\right ) \, dx-24 \int \frac {1}{\left (5+e^x-\log (x)\right )^2} \, dx \\ & = x+2 \int \frac {x}{\left (5+e^x-\log (x)\right )^3} \, dx-2 \int \frac {x^2}{\left (5+e^x-\log (x)\right )^2} \, dx-2 \int \frac {x^2 \log (x)}{\left (5+e^x-\log (x)\right )^3} \, dx+10 \int \frac {x^2}{\left (5+e^x-\log (x)\right )^3} \, dx-12 \int \frac {1}{\left (5+e^x-\log (x)\right )^3} \, dx+12 \int \frac {x \log (x)}{\left (5+e^x-\log (x)\right )^3} \, dx+14 \int \frac {x}{\left (5+e^x-\log (x)\right )^2} \, dx+18 \int \frac {1}{x \left (5+e^x-\log (x)\right )^3} \, dx-18 \int \frac {\log (x)}{\left (5+e^x-\log (x)\right )^3} \, dx-24 \int \frac {1}{\left (5+e^x-\log (x)\right )^2} \, dx-60 \int \frac {x}{\left (5+e^x-\log (x)\right )^3} \, dx+90 \int \frac {1}{\left (5+e^x-\log (x)\right )^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=x+\frac {(-3+x)^2}{\left (-5-e^x+\log (x)\right )^2} \]

[In]

Integrate[(-18 - 83*x - 15*E^(2*x)*x - E^(3*x)*x - 12*x^2 + E^x*(-51*x - 14*x^2 + 2*x^3) + (69*x + 30*E^x*x +
3*E^(2*x)*x + 2*x^2)*Log[x] + (-15*x - 3*E^x*x)*Log[x]^2 + x*Log[x]^3)/(-125*x - 75*E^x*x - 15*E^(2*x)*x - E^(
3*x)*x + (75*x + 30*E^x*x + 3*E^(2*x)*x)*Log[x] + (-15*x - 3*E^x*x)*Log[x]^2 + x*Log[x]^3),x]

[Out]

x + (-3 + x)^2/(-5 - E^x + Log[x])^2

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16

method	result	size
risch	\(x +\frac {x^{2}-6 x +9}{\left (5+{\mathrm e}^{x}-\ln \left (x \right )\right )^{2}}\)	\(22\)
parallelrisch	\(\frac {x \ln \left (x \right )^{2}-2 x \,{\mathrm e}^{x} \ln \left (x \right )+x \,{\mathrm e}^{2 x}-10 x \ln \left (x \right )+10 \,{\mathrm e}^{x} x +x^{2}+19 x +9}{\ln \left (x \right )^{2}-2 \,{\mathrm e}^{x} \ln \left (x \right )+{\mathrm e}^{2 x}-10 \ln \left (x \right )+10 \,{\mathrm e}^{x}+25}\)	\(65\)

[In]

int((x*ln(x)^3+(-3*exp(x)*x-15*x)*ln(x)^2+(3*x*exp(x)^2+30*exp(x)*x+2*x^2+69*x)*ln(x)-x*exp(x)^3-15*x*exp(x)^2
+(2*x^3-14*x^2-51*x)*exp(x)-12*x^2-83*x-18)/(x*ln(x)^3+(-3*exp(x)*x-15*x)*ln(x)^2+(3*x*exp(x)^2+30*exp(x)*x+75
*x)*ln(x)-x*exp(x)^3-15*x*exp(x)^2-75*exp(x)*x-125*x),x,method=_RETURNVERBOSE)

[Out]

x+(x^2-6*x+9)/(5+exp(x)-ln(x))^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (18) = 36\).

Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.53 \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=-\frac {x \log \left (x\right )^{2} + x^{2} + x e^{\left (2 \, x\right )} + 10 \, x e^{x} - 2 \, {\left (x e^{x} + 5 \, x\right )} \log \left (x\right ) + 19 \, x + 9}{2 \, {\left (e^{x} + 5\right )} \log \left (x\right ) - \log \left (x\right )^{2} - e^{\left (2 \, x\right )} - 10 \, e^{x} - 25} \]

[In]

integrate((x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*exp(x)*x+2*x^2+69*x)*log(x)-x*exp(x)^3-15*x
*exp(x)^2+(2*x^3-14*x^2-51*x)*exp(x)-12*x^2-83*x-18)/(x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*
exp(x)*x+75*x)*log(x)-x*exp(x)^3-15*x*exp(x)^2-75*exp(x)*x-125*x),x, algorithm="fricas")

[Out]

-(x*log(x)^2 + x^2 + x*e^(2*x) + 10*x*e^x - 2*(x*e^x + 5*x)*log(x) + 19*x + 9)/(2*(e^x + 5)*log(x) - log(x)^2
- e^(2*x) - 10*e^x - 25)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (15) = 30\).

Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=x + \frac {x^{2} - 6 x + 9}{\left (10 - 2 \log {\left (x \right )}\right ) e^{x} + e^{2 x} + \log {\left (x \right )}^{2} - 10 \log {\left (x \right )} + 25} \]

[In]

integrate((x*ln(x)**3+(-3*exp(x)*x-15*x)*ln(x)**2+(3*x*exp(x)**2+30*exp(x)*x+2*x**2+69*x)*ln(x)-x*exp(x)**3-15
*x*exp(x)**2+(2*x**3-14*x**2-51*x)*exp(x)-12*x**2-83*x-18)/(x*ln(x)**3+(-3*exp(x)*x-15*x)*ln(x)**2+(3*x*exp(x)
**2+30*exp(x)*x+75*x)*ln(x)-x*exp(x)**3-15*x*exp(x)**2-75*exp(x)*x-125*x),x)

[Out]

x + (x**2 - 6*x + 9)/((10 - 2*log(x))*exp(x) + exp(2*x) + log(x)**2 - 10*log(x) + 25)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (18) = 36\).

Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.53 \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=-\frac {x \log \left (x\right )^{2} + x^{2} + x e^{\left (2 \, x\right )} - 2 \, {\left (x \log \left (x\right ) - 5 \, x\right )} e^{x} - 10 \, x \log \left (x\right ) + 19 \, x + 9}{2 \, {\left (\log \left (x\right ) - 5\right )} e^{x} - \log \left (x\right )^{2} - e^{\left (2 \, x\right )} + 10 \, \log \left (x\right ) - 25} \]

[In]

integrate((x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*exp(x)*x+2*x^2+69*x)*log(x)-x*exp(x)^3-15*x
*exp(x)^2+(2*x^3-14*x^2-51*x)*exp(x)-12*x^2-83*x-18)/(x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*
exp(x)*x+75*x)*log(x)-x*exp(x)^3-15*x*exp(x)^2-75*exp(x)*x-125*x),x, algorithm="maxima")

[Out]

-(x*log(x)^2 + x^2 + x*e^(2*x) - 2*(x*log(x) - 5*x)*e^x - 10*x*log(x) + 19*x + 9)/(2*(log(x) - 5)*e^x - log(x)
^2 - e^(2*x) + 10*log(x) - 25)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (18) = 36\).

Time = 0.33 (sec) , antiderivative size = 99, normalized size of antiderivative = 5.21 \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=\frac {4 \, x e^{x} \log \left (x\right ) - 2 \, x \log \left (x\right )^{2} - 2 \, x^{2} - 2 \, x e^{\left (2 \, x\right )} - 20 \, x e^{x} + 20 \, x \log \left (x\right ) + 6 \, e^{x} \log \left (x\right ) - 3 \, \log \left (x\right )^{2} - 38 \, x - 3 \, e^{\left (2 \, x\right )} - 30 \, e^{x} + 30 \, \log \left (x\right ) - 93}{2 \, {\left (2 \, e^{x} \log \left (x\right ) - \log \left (x\right )^{2} - e^{\left (2 \, x\right )} - 10 \, e^{x} + 10 \, \log \left (x\right ) - 25\right )}} \]

[In]

integrate((x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*exp(x)*x+2*x^2+69*x)*log(x)-x*exp(x)^3-15*x
*exp(x)^2+(2*x^3-14*x^2-51*x)*exp(x)-12*x^2-83*x-18)/(x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*
exp(x)*x+75*x)*log(x)-x*exp(x)^3-15*x*exp(x)^2-75*exp(x)*x-125*x),x, algorithm="giac")

[Out]

1/2*(4*x*e^x*log(x) - 2*x*log(x)^2 - 2*x^2 - 2*x*e^(2*x) - 20*x*e^x + 20*x*log(x) + 6*e^x*log(x) - 3*log(x)^2
- 38*x - 3*e^(2*x) - 30*e^x + 30*log(x) - 93)/(2*e^x*log(x) - log(x)^2 - e^(2*x) - 10*e^x + 10*log(x) - 25)

Mupad [F(-1)]

Timed out. \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=\int \frac {83\,x+15\,x\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{3\,x}-x\,{\ln \left (x\right )}^3-\ln \left (x\right )\,\left (69\,x+3\,x\,{\mathrm {e}}^{2\,x}+30\,x\,{\mathrm {e}}^x+2\,x^2\right )+{\ln \left (x\right )}^2\,\left (15\,x+3\,x\,{\mathrm {e}}^x\right )+12\,x^2+{\mathrm {e}}^x\,\left (-2\,x^3+14\,x^2+51\,x\right )+18}{-x\,{\ln \left (x\right )}^3+\left (15\,x+3\,x\,{\mathrm {e}}^x\right )\,{\ln \left (x\right )}^2+\left (-75\,x-3\,x\,{\mathrm {e}}^{2\,x}-30\,x\,{\mathrm {e}}^x\right )\,\ln \left (x\right )+125\,x+15\,x\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{3\,x}+75\,x\,{\mathrm {e}}^x} \,d x \]

[In]

int((83*x + 15*x*exp(2*x) + x*exp(3*x) - x*log(x)^3 - log(x)*(69*x + 3*x*exp(2*x) + 30*x*exp(x) + 2*x^2) + log
(x)^2*(15*x + 3*x*exp(x)) + 12*x^2 + exp(x)*(51*x + 14*x^2 - 2*x^3) + 18)/(125*x + 15*x*exp(2*x) + x*exp(3*x)
- x*log(x)^3 - log(x)*(75*x + 3*x*exp(2*x) + 30*x*exp(x)) + log(x)^2*(15*x + 3*x*exp(x)) + 75*x*exp(x)),x)

[Out]

int((83*x + 15*x*exp(2*x) + x*exp(3*x) - x*log(x)^3 - log(x)*(69*x + 3*x*exp(2*x) + 30*x*exp(x) + 2*x^2) + log
(x)^2*(15*x + 3*x*exp(x)) + 12*x^2 + exp(x)*(51*x + 14*x^2 - 2*x^3) + 18)/(125*x + 15*x*exp(2*x) + x*exp(3*x)
- x*log(x)^3 - log(x)*(75*x + 3*x*exp(2*x) + 30*x*exp(x)) + log(x)^2*(15*x + 3*x*exp(x)) + 75*x*exp(x)), x)