\(\int \frac {1764 x+e^x (3+588 x-3 x^2)+(-9 x-3 e^x x) \log (x)}{e^{2 x} x+6 e^x x^2+9 x^3} \, dx\) [8845]

Optimal result

Integrand size = 55, antiderivative size = 17 \[ \int \frac {1764 x+e^x \left (3+588 x-3 x^2\right )+\left (-9 x-3 e^x x\right ) \log (x)}{e^{2 x} x+6 e^x x^2+9 x^3} \, dx=\frac {-195+x+\log (x)}{\frac {e^x}{3}+x} \]

[Out]

1/(x+1/3*exp(x))*(x+ln(x)-195)

Rubi [F]

\[ \int \frac {1764 x+e^x \left (3+588 x-3 x^2\right )+\left (-9 x-3 e^x x\right ) \log (x)}{e^{2 x} x+6 e^x x^2+9 x^3} \, dx=\int \frac {1764 x+e^x \left (3+588 x-3 x^2\right )+\left (-9 x-3 e^x x\right ) \log (x)}{e^{2 x} x+6 e^x x^2+9 x^3} \, dx \]

[In]

Int[(1764*x + E^x*(3 + 588*x - 3*x^2) + (-9*x - 3*E^x*x)*Log[x])/(E^(2*x)*x + 6*E^x*x^2 + 9*x^3),x]

[Out]

1755*Defer[Int][(E^x + 3*x)^(-2), x] - 9*Log[x]*Defer[Int][(E^x + 3*x)^(-2), x] - 1764*Defer[Int][x/(E^x + 3*x
)^2, x] + 9*Log[x]*Defer[Int][x/(E^x + 3*x)^2, x] + 9*Defer[Int][x^2/(E^x + 3*x)^2, x] + 588*Defer[Int][(E^x +
3*x)^(-1), x] - 3*Log[x]*Defer[Int][(E^x + 3*x)^(-1), x] + 3*Defer[Int][1/(x*(E^x + 3*x)), x] - 3*Defer[Int][
x/(E^x + 3*x), x] + 9*Defer[Int][Defer[Int][(E^x + 3*x)^(-2), x]/x, x] - 9*Defer[Int][Defer[Int][x/(E^x + 3*x)
^2, x]/x, x] + 3*Defer[Int][Defer[Int][(E^x + 3*x)^(-1), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {3 \left (588 x-e^x \left (-1-196 x+x^2\right )-\left (3+e^x\right ) x \log (x)\right )}{x \left (e^x+3 x\right )^2} \, dx \\ & = 3 \int \frac {588 x-e^x \left (-1-196 x+x^2\right )-\left (3+e^x\right ) x \log (x)}{x \left (e^x+3 x\right )^2} \, dx \\ & = 3 \int \left (\frac {3 (-1+x) (-195+x+\log (x))}{\left (e^x+3 x\right )^2}-\frac {-1-196 x+x^2+x \log (x)}{x \left (e^x+3 x\right )}\right ) \, dx \\ & = -\left (3 \int \frac {-1-196 x+x^2+x \log (x)}{x \left (e^x+3 x\right )} \, dx\right )+9 \int \frac {(-1+x) (-195+x+\log (x))}{\left (e^x+3 x\right )^2} \, dx \\ & = -\left (3 \int \left (-\frac {196}{e^x+3 x}-\frac {1}{x \left (e^x+3 x\right )}+\frac {x}{e^x+3 x}+\frac {\log (x)}{e^x+3 x}\right ) \, dx\right )+9 \int \left (-\frac {-195+x+\log (x)}{\left (e^x+3 x\right )^2}+\frac {x (-195+x+\log (x))}{\left (e^x+3 x\right )^2}\right ) \, dx \\ & = 3 \int \frac {1}{x \left (e^x+3 x\right )} \, dx-3 \int \frac {x}{e^x+3 x} \, dx-3 \int \frac {\log (x)}{e^x+3 x} \, dx-9 \int \frac {-195+x+\log (x)}{\left (e^x+3 x\right )^2} \, dx+9 \int \frac {x (-195+x+\log (x))}{\left (e^x+3 x\right )^2} \, dx+588 \int \frac {1}{e^x+3 x} \, dx \\ & = 3 \int \frac {1}{x \left (e^x+3 x\right )} \, dx-3 \int \frac {x}{e^x+3 x} \, dx+3 \int \frac {\int \frac {1}{e^x+3 x} \, dx}{x} \, dx-9 \int \left (-\frac {195}{\left (e^x+3 x\right )^2}+\frac {x}{\left (e^x+3 x\right )^2}+\frac {\log (x)}{\left (e^x+3 x\right )^2}\right ) \, dx+9 \int \left (-\frac {195 x}{\left (e^x+3 x\right )^2}+\frac {x^2}{\left (e^x+3 x\right )^2}+\frac {x \log (x)}{\left (e^x+3 x\right )^2}\right ) \, dx+588 \int \frac {1}{e^x+3 x} \, dx-(3 \log (x)) \int \frac {1}{e^x+3 x} \, dx \\ & = 3 \int \frac {1}{x \left (e^x+3 x\right )} \, dx-3 \int \frac {x}{e^x+3 x} \, dx+3 \int \frac {\int \frac {1}{e^x+3 x} \, dx}{x} \, dx-9 \int \frac {x}{\left (e^x+3 x\right )^2} \, dx+9 \int \frac {x^2}{\left (e^x+3 x\right )^2} \, dx-9 \int \frac {\log (x)}{\left (e^x+3 x\right )^2} \, dx+9 \int \frac {x \log (x)}{\left (e^x+3 x\right )^2} \, dx+588 \int \frac {1}{e^x+3 x} \, dx+1755 \int \frac {1}{\left (e^x+3 x\right )^2} \, dx-1755 \int \frac {x}{\left (e^x+3 x\right )^2} \, dx-(3 \log (x)) \int \frac {1}{e^x+3 x} \, dx \\ & = 3 \int \frac {1}{x \left (e^x+3 x\right )} \, dx-3 \int \frac {x}{e^x+3 x} \, dx+3 \int \frac {\int \frac {1}{e^x+3 x} \, dx}{x} \, dx-9 \int \frac {x}{\left (e^x+3 x\right )^2} \, dx+9 \int \frac {x^2}{\left (e^x+3 x\right )^2} \, dx+9 \int \frac {\int \frac {1}{\left (e^x+3 x\right )^2} \, dx}{x} \, dx-9 \int \frac {\int \frac {x}{\left (e^x+3 x\right )^2} \, dx}{x} \, dx+588 \int \frac {1}{e^x+3 x} \, dx+1755 \int \frac {1}{\left (e^x+3 x\right )^2} \, dx-1755 \int \frac {x}{\left (e^x+3 x\right )^2} \, dx-(3 \log (x)) \int \frac {1}{e^x+3 x} \, dx-(9 \log (x)) \int \frac {1}{\left (e^x+3 x\right )^2} \, dx+(9 \log (x)) \int \frac {x}{\left (e^x+3 x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {1764 x+e^x \left (3+588 x-3 x^2\right )+\left (-9 x-3 e^x x\right ) \log (x)}{e^{2 x} x+6 e^x x^2+9 x^3} \, dx=\frac {3 (-195+x+\log (x))}{e^x+3 x} \]

[In]

Integrate[(1764*x + E^x*(3 + 588*x - 3*x^2) + (-9*x - 3*E^x*x)*Log[x])/(E^(2*x)*x + 6*E^x*x^2 + 9*x^3),x]

[Out]

(3*(-195 + x + Log[x]))/(E^x + 3*x)

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12

method	result	size
parallelrisch	\(\frac {-585+3 x +3 \ln \left (x \right )}{3 x +{\mathrm e}^{x}}\)	\(19\)
risch	\(\frac {3 \ln \left (x \right )}{3 x +{\mathrm e}^{x}}+\frac {3 x -585}{3 x +{\mathrm e}^{x}}\)	\(27\)

[In]

int(((-3*exp(x)*x-9*x)*ln(x)+(-3*x^2+588*x+3)*exp(x)+1764*x)/(x*exp(x)^2+6*exp(x)*x^2+9*x^3),x,method=_RETURNV
ERBOSE)

[Out]

(-585+3*x+3*ln(x))/(3*x+exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {1764 x+e^x \left (3+588 x-3 x^2\right )+\left (-9 x-3 e^x x\right ) \log (x)}{e^{2 x} x+6 e^x x^2+9 x^3} \, dx=\frac {3 \, {\left (x + \log \left (x\right ) - 195\right )}}{3 \, x + e^{x}} \]

[In]

integrate(((-3*exp(x)*x-9*x)*log(x)+(-3*x^2+588*x+3)*exp(x)+1764*x)/(x*exp(x)^2+6*exp(x)*x^2+9*x^3),x, algorit
hm="fricas")

[Out]

3*(x + log(x) - 195)/(3*x + e^x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {1764 x+e^x \left (3+588 x-3 x^2\right )+\left (-9 x-3 e^x x\right ) \log (x)}{e^{2 x} x+6 e^x x^2+9 x^3} \, dx=\frac {3 x + 3 \log {\left (x \right )} - 585}{3 x + e^{x}} \]

[In]

integrate(((-3*exp(x)*x-9*x)*ln(x)+(-3*x**2+588*x+3)*exp(x)+1764*x)/(x*exp(x)**2+6*exp(x)*x**2+9*x**3),x)

[Out]

(3*x + 3*log(x) - 585)/(3*x + exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {1764 x+e^x \left (3+588 x-3 x^2\right )+\left (-9 x-3 e^x x\right ) \log (x)}{e^{2 x} x+6 e^x x^2+9 x^3} \, dx=\frac {3 \, {\left (x + \log \left (x\right ) - 195\right )}}{3 \, x + e^{x}} \]

[In]

integrate(((-3*exp(x)*x-9*x)*log(x)+(-3*x^2+588*x+3)*exp(x)+1764*x)/(x*exp(x)^2+6*exp(x)*x^2+9*x^3),x, algorit
hm="maxima")

[Out]

3*(x + log(x) - 195)/(3*x + e^x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {1764 x+e^x \left (3+588 x-3 x^2\right )+\left (-9 x-3 e^x x\right ) \log (x)}{e^{2 x} x+6 e^x x^2+9 x^3} \, dx=\frac {3 \, {\left (x + \log \left (x\right ) - 195\right )}}{3 \, x + e^{x}} \]

[In]

integrate(((-3*exp(x)*x-9*x)*log(x)+(-3*x^2+588*x+3)*exp(x)+1764*x)/(x*exp(x)^2+6*exp(x)*x^2+9*x^3),x, algorit
hm="giac")

[Out]

3*(x + log(x) - 195)/(3*x + e^x)

Mupad [B] (verification not implemented)

Time = 14.38 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {1764 x+e^x \left (3+588 x-3 x^2\right )+\left (-9 x-3 e^x x\right ) \log (x)}{e^{2 x} x+6 e^x x^2+9 x^3} \, dx=\frac {3\,\left (x+\ln \left (x\right )-195\right )}{3\,x+{\mathrm {e}}^x} \]

[In]

int((1764*x - log(x)*(9*x + 3*x*exp(x)) + exp(x)*(588*x - 3*x^2 + 3))/(x*exp(2*x) + 6*x^2*exp(x) + 9*x^3),x)

[Out]

(3*(x + log(x) - 195))/(3*x + exp(x))