\(\int \frac {-5+11 x+3 x^2+(1-3 x) \log (\frac {e^x x^2}{-2+6 x})+(3-9 x+(-1+3 x) \log (\frac {e^x x^2}{-2+6 x})) \log (-3+\log (\frac {e^x x^2}{-2+6 x}))}{3-9 x+(-1+3 x) \log (\frac {e^x x^2}{-2+6 x})} \, dx\) [7999]

Optimal result

Integrand size = 105, antiderivative size = 29 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=-x+x \log \left (-3+\log \left (\frac {e^x x}{5-\frac {2-x}{x}}\right )\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=\int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx \]

[In]

Int[(-5 + 11*x + 3*x^2 + (1 - 3*x)*Log[(E^x*x^2)/(-2 + 6*x)] + (3 - 9*x + (-1 + 3*x)*Log[(E^x*x^2)/(-2 + 6*x)]
)*Log[-3 + Log[(E^x*x^2)/(-2 + 6*x)]])/(3 - 9*x + (-1 + 3*x)*Log[(E^x*x^2)/(-2 + 6*x)]),x]

[Out]

-x + Defer[Int][(-3 + Log[(E^x*x^2)/(-2 + 6*x)])^(-1), x] + Defer[Int][x/(-3 + Log[(E^x*x^2)/(-2 + 6*x)]), x]
- Defer[Int][1/((-1 + 3*x)*(-3 + Log[(E^x*x^2)/(-2 + 6*x)])), x] + Defer[Int][Log[-3 + Log[(E^x*x^2)/(-2 + 6*x
)]], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{(1-3 x) \left (3-\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )} \, dx \\ & = \int \left (\frac {-5+11 x+3 x^2+\log \left (\frac {e^x x^2}{-2+6 x}\right )-3 x \log \left (\frac {e^x x^2}{-2+6 x}\right )}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}+\log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )\right ) \, dx \\ & = \int \frac {-5+11 x+3 x^2+\log \left (\frac {e^x x^2}{-2+6 x}\right )-3 x \log \left (\frac {e^x x^2}{-2+6 x}\right )}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )} \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx \\ & = \int \left (-1+\frac {-2+2 x+3 x^2}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}\right ) \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx \\ & = -x+\int \frac {-2+2 x+3 x^2}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )} \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx \\ & = -x+\int \left (\frac {1}{-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )}+\frac {x}{-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )}-\frac {1}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}\right ) \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx \\ & = -x+\int \frac {1}{-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx+\int \frac {x}{-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx-\int \frac {1}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )} \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=-x+x \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \]

[In]

Integrate[(-5 + 11*x + 3*x^2 + (1 - 3*x)*Log[(E^x*x^2)/(-2 + 6*x)] + (3 - 9*x + (-1 + 3*x)*Log[(E^x*x^2)/(-2 +
6*x)])*Log[-3 + Log[(E^x*x^2)/(-2 + 6*x)]])/(3 - 9*x + (-1 + 3*x)*Log[(E^x*x^2)/(-2 + 6*x)]),x]

[Out]

-x + x*Log[-3 + Log[(E^x*x^2)/(-2 + 6*x)]]

Maple [A] (verified)

Time = 3.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90

method	result	size
parallelrisch	\(-\frac {1}{6}+\ln \left (\ln \left (\frac {x^{2} {\mathrm e}^{x}}{6 x -2}\right )-3\right ) x -x\)	\(26\)
risch	\(\ln \left (-\ln \left (2\right )-\ln \left (3\right )+2 \ln \left (x \right )+\ln \left ({\mathrm e}^{x}\right )-\ln \left (x -\frac {1}{3}\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right ) \left (-\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\operatorname {csgn}\left (i {\mathrm e}^{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\operatorname {csgn}\left (\frac {i}{x -\frac {1}{3}}\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x^{2} {\mathrm e}^{x}}{x -\frac {1}{3}}\right ) \left (-\operatorname {csgn}\left (\frac {i x^{2} {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\operatorname {csgn}\left (i x^{2}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i x^{2} {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right )\right )}{2}-3\right ) x -x\)	\(187\)

[In]

int((((-1+3*x)*ln(x^2*exp(x)/(6*x-2))-9*x+3)*ln(ln(x^2*exp(x)/(6*x-2))-3)+(1-3*x)*ln(x^2*exp(x)/(6*x-2))+3*x^2
+11*x-5)/((-1+3*x)*ln(x^2*exp(x)/(6*x-2))-9*x+3),x,method=_RETURNVERBOSE)

[Out]

-1/6+ln(ln(1/2*exp(x)*x^2/(-1+3*x))-3)*x-x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=x \log \left (\log \left (\frac {x^{2} e^{x}}{2 \, {\left (3 \, x - 1\right )}}\right ) - 3\right ) - x \]

[In]

integrate((((-1+3*x)*log(x^2*exp(x)/(6*x-2))-9*x+3)*log(log(x^2*exp(x)/(6*x-2))-3)+(1-3*x)*log(x^2*exp(x)/(6*x
-2))+3*x^2+11*x-5)/((-1+3*x)*log(x^2*exp(x)/(6*x-2))-9*x+3),x, algorithm="fricas")

[Out]

x*log(log(1/2*x^2*e^x/(3*x - 1)) - 3) - x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).

Time = 0.73 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=- x + \left (x - \frac {1}{18}\right ) \log {\left (\log {\left (\frac {x^{2} e^{x}}{6 x - 2} \right )} - 3 \right )} + \frac {\log {\left (\log {\left (\frac {x^{2} e^{x}}{6 x - 2} \right )} - 3 \right )}}{18} \]

[In]

integrate((((-1+3*x)*ln(x**2*exp(x)/(6*x-2))-9*x+3)*ln(ln(x**2*exp(x)/(6*x-2))-3)+(1-3*x)*ln(x**2*exp(x)/(6*x-
2))+3*x**2+11*x-5)/((-1+3*x)*ln(x**2*exp(x)/(6*x-2))-9*x+3),x)

[Out]

-x + (x - 1/18)*log(log(x**2*exp(x)/(6*x - 2)) - 3) + log(log(x**2*exp(x)/(6*x - 2)) - 3)/18

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=x \log \left (x - \log \left (2\right ) - \log \left (3 \, x - 1\right ) + 2 \, \log \left (x\right ) - 3\right ) - x \]

[In]

integrate((((-1+3*x)*log(x^2*exp(x)/(6*x-2))-9*x+3)*log(log(x^2*exp(x)/(6*x-2))-3)+(1-3*x)*log(x^2*exp(x)/(6*x
-2))+3*x^2+11*x-5)/((-1+3*x)*log(x^2*exp(x)/(6*x-2))-9*x+3),x, algorithm="maxima")

[Out]

x*log(x - log(2) - log(3*x - 1) + 2*log(x) - 3) - x

Giac [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=x \log \left (x + \log \left (\frac {x^{2}}{2 \, {\left (3 \, x - 1\right )}}\right ) - 3\right ) - x \]

[In]

integrate((((-1+3*x)*log(x^2*exp(x)/(6*x-2))-9*x+3)*log(log(x^2*exp(x)/(6*x-2))-3)+(1-3*x)*log(x^2*exp(x)/(6*x
-2))+3*x^2+11*x-5)/((-1+3*x)*log(x^2*exp(x)/(6*x-2))-9*x+3),x, algorithm="giac")

[Out]

x*log(x + log(1/2*x^2/(3*x - 1)) - 3) - x

Mupad [B] (verification not implemented)

Time = 13.65 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx=x\,\left (\ln \left (\ln \left (\frac {x^2\,{\mathrm {e}}^x}{6\,x-2}\right )-3\right )-1\right ) \]

[In]

int((11*x + log(log((x^2*exp(x))/(6*x - 2)) - 3)*(log((x^2*exp(x))/(6*x - 2))*(3*x - 1) - 9*x + 3) - log((x^2*
exp(x))/(6*x - 2))*(3*x - 1) + 3*x^2 - 5)/(log((x^2*exp(x))/(6*x - 2))*(3*x - 1) - 9*x + 3),x)

[Out]

x*(log(log((x^2*exp(x))/(6*x - 2)) - 3) - 1)