\(\int \frac {x+e^x (x+x^2)+(1+e^x (1+x)) \log (2 x)+(2+3 x+e^x (2+3 x)+(1+e^x) \log (2 x)) \log (x+e^x x)}{(x^2+e^x x^2+(x+e^x x) \log (2 x)) \log (x+e^x x) \log (\frac {1}{5} (3 x^3+6 x^2 \log (2 x)+3 x \log ^2(2 x)) \log (x+e^x x))} \, dx\) [6599]

Optimal result

Integrand size = 133, antiderivative size = 23 \[ \int \frac {x+e^x \left (x+x^2\right )+\left (1+e^x (1+x)\right ) \log (2 x)+\left (2+3 x+e^x (2+3 x)+\left (1+e^x\right ) \log (2 x)\right ) \log \left (x+e^x x\right )}{\left (x^2+e^x x^2+\left (x+e^x x\right ) \log (2 x)\right ) \log \left (x+e^x x\right ) \log \left (\frac {1}{5} \left (3 x^3+6 x^2 \log (2 x)+3 x \log ^2(2 x)\right ) \log \left (x+e^x x\right )\right )} \, dx=\log \left (\log \left (\frac {3}{5} x (x+\log (2 x))^2 \log \left (x+e^x x\right )\right )\right ) \]

[Out]

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

Rubi [F]

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

[In]

Int[(x + E^x*(x + x^2) + (1 + E^x*(1 + x))*Log[2*x] + (2 + 3*x + E^x*(2 + 3*x) + (1 + E^x)*Log[2*x])*Log[x + E
^x*x])/((x^2 + E^x*x^2 + (x + E^x*x)*Log[2*x])*Log[x + E^x*x]*Log[((3*x^3 + 6*x^2*Log[2*x] + 3*x*Log[2*x]^2)*L
og[x + E^x*x])/5]),x]

[Out]

3*Defer[Int][1/((x + Log[2*x])*Log[(3*x*(x + Log[2*x])^2*Log[x + E^x*x])/5]), x] + 2*Defer[Int][1/(x*(x + Log[
2*x])*Log[(3*x*(x + Log[2*x])^2*Log[x + E^x*x])/5]), x] + Defer[Int][Log[2*x]/(x*(x + Log[2*x])*Log[(3*x*(x +
Log[2*x])^2*Log[x + E^x*x])/5]), x] + Defer[Int][1/((-1 - E^x)*Log[x + E^x*x]*Log[(3*x*(x + Log[2*x])^2*Log[x
+ E^x*x])/5]), x] + Defer[Int][1/((x + Log[2*x])*Log[x + E^x*x]*Log[(3*x*(x + Log[2*x])^2*Log[x + E^x*x])/5]),
x] + Defer[Int][x/((x + Log[2*x])*Log[x + E^x*x]*Log[(3*x*(x + Log[2*x])^2*Log[x + E^x*x])/5]), x] + Defer[In
t][Log[2*x]/((x + Log[2*x])*Log[x + E^x*x]*Log[(3*x*(x + Log[2*x])^2*Log[x + E^x*x])/5]), x] + Defer[Int][Log[
2*x]/(x*(x + Log[2*x])*Log[x + E^x*x]*Log[(3*x*(x + Log[2*x])^2*Log[x + E^x*x])/5]), x]

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x+e^x \left (x+x^2\right )+\left (1+e^x (1+x)\right ) \log (2 x)+\left (2+3 x+e^x (2+3 x)+\left (1+e^x\right ) \log (2 x)\right ) \log \left (x+e^x x\right )}{\left (x^2+e^x x^2+\left (x+e^x x\right ) \log (2 x)\right ) \log \left (x+e^x x\right ) \log \left (\frac {1}{5} \left (3 x^3+6 x^2 \log (2 x)+3 x \log ^2(2 x)\right ) \log \left (x+e^x x\right )\right )} \, dx=\log \left (\log \left (\frac {3}{5} x (x+\log (2 x))^2 \log \left (\left (1+e^x\right ) x\right )\right )\right ) \]

[In]

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

[Out]

Log[Log[(3*x*(x + Log[2*x])^2*Log[(1 + E^x)*x])/5]]

Maple [A] (verified)

Time = 121.51 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30

method	result	size
parallelrisch	\(\ln \left (\ln \left (\frac {3 x \left (\ln \left (2 x \right )^{2}+2 x \ln \left (2 x \right )+x^{2}\right ) \ln \left (x \left ({\mathrm e}^{x}+1\right )\right )}{5}\right )\right )\)	\(30\)

[In]

int((((exp(x)+1)*ln(2*x)+(2+3*x)*exp(x)+3*x+2)*ln(exp(x)*x+x)+((1+x)*exp(x)+1)*ln(2*x)+(x^2+x)*exp(x)+x)/((exp
(x)*x+x)*ln(2*x)+exp(x)*x^2+x^2)/ln(exp(x)*x+x)/ln(1/5*(3*x*ln(2*x)^2+6*x^2*ln(2*x)+3*x^3)*ln(exp(x)*x+x)),x,m
ethod=_RETURNVERBOSE)

[Out]

ln(ln(3/5*x*(ln(2*x)^2+2*x*ln(2*x)+x^2)*ln(x*(exp(x)+1))))

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {x+e^x \left (x+x^2\right )+\left (1+e^x (1+x)\right ) \log (2 x)+\left (2+3 x+e^x (2+3 x)+\left (1+e^x\right ) \log (2 x)\right ) \log \left (x+e^x x\right )}{\left (x^2+e^x x^2+\left (x+e^x x\right ) \log (2 x)\right ) \log \left (x+e^x x\right ) \log \left (\frac {1}{5} \left (3 x^3+6 x^2 \log (2 x)+3 x \log ^2(2 x)\right ) \log \left (x+e^x x\right )\right )} \, dx=\log \left (\log \left (\frac {3}{5} \, {\left (x^{3} + 2 \, x^{2} \log \left (2 \, x\right ) + x \log \left (2 \, x\right )^{2}\right )} \log \left (x e^{x} + x\right )\right )\right ) \]

[In]

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

[Out]

log(log(3/5*(x^3 + 2*x^2*log(2*x) + x*log(2*x)^2)*log(x*e^x + x)))

Sympy [A] (verification not implemented)

Time = 13.33 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78 \[ \int \frac {x+e^x \left (x+x^2\right )+\left (1+e^x (1+x)\right ) \log (2 x)+\left (2+3 x+e^x (2+3 x)+\left (1+e^x\right ) \log (2 x)\right ) \log \left (x+e^x x\right )}{\left (x^2+e^x x^2+\left (x+e^x x\right ) \log (2 x)\right ) \log \left (x+e^x x\right ) \log \left (\frac {1}{5} \left (3 x^3+6 x^2 \log (2 x)+3 x \log ^2(2 x)\right ) \log \left (x+e^x x\right )\right )} \, dx=\log {\left (\log {\left (\left (\frac {3 x^{3}}{5} + \frac {6 x^{2} \log {\left (2 x \right )}}{5} + \frac {3 x \log {\left (2 x \right )}^{2}}{5}\right ) \log {\left (x e^{x} + x \right )} \right )} \right )} \]

[In]

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

[Out]

log(log((3*x**3/5 + 6*x**2*log(2*x)/5 + 3*x*log(2*x)**2/5)*log(x*exp(x) + x)))

Maxima [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {x+e^x \left (x+x^2\right )+\left (1+e^x (1+x)\right ) \log (2 x)+\left (2+3 x+e^x (2+3 x)+\left (1+e^x\right ) \log (2 x)\right ) \log \left (x+e^x x\right )}{\left (x^2+e^x x^2+\left (x+e^x x\right ) \log (2 x)\right ) \log \left (x+e^x x\right ) \log \left (\frac {1}{5} \left (3 x^3+6 x^2 \log (2 x)+3 x \log ^2(2 x)\right ) \log \left (x+e^x x\right )\right )} \, dx=\log \left (-\log \left (5\right ) + \log \left (3\right ) + 2 \, \log \left (x + \log \left (2\right ) + \log \left (x\right )\right ) + \log \left (x\right ) + \log \left (\log \left (x\right ) + \log \left (e^{x} + 1\right )\right )\right ) \]

[In]

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

[Out]

log(-log(5) + log(3) + 2*log(x + log(2) + log(x)) + log(x) + log(log(x) + log(e^x + 1)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (20) = 40\).

Time = 1.05 (sec) , antiderivative size = 116, normalized size of antiderivative = 5.04 \[ \int \frac {x+e^x \left (x+x^2\right )+\left (1+e^x (1+x)\right ) \log (2 x)+\left (2+3 x+e^x (2+3 x)+\left (1+e^x\right ) \log (2 x)\right ) \log \left (x+e^x x\right )}{\left (x^2+e^x x^2+\left (x+e^x x\right ) \log (2 x)\right ) \log \left (x+e^x x\right ) \log \left (\frac {1}{5} \left (3 x^3+6 x^2 \log (2 x)+3 x \log ^2(2 x)\right ) \log \left (x+e^x x\right )\right )} \, dx=\log \left (-\log \left (5\right ) + \log \left (3 \, x^{2} \log \left (x\right ) + 6 \, x \log \left (2\right ) \log \left (x\right ) + 3 \, \log \left (2\right )^{2} \log \left (x\right ) + 6 \, x \log \left (x\right )^{2} + 6 \, \log \left (2\right ) \log \left (x\right )^{2} + 3 \, \log \left (x\right )^{3} + 3 \, x^{2} \log \left (e^{x} + 1\right ) + 6 \, x \log \left (2\right ) \log \left (e^{x} + 1\right ) + 3 \, \log \left (2\right )^{2} \log \left (e^{x} + 1\right ) + 6 \, x \log \left (x\right ) \log \left (e^{x} + 1\right ) + 6 \, \log \left (2\right ) \log \left (x\right ) \log \left (e^{x} + 1\right ) + 3 \, \log \left (x\right )^{2} \log \left (e^{x} + 1\right )\right ) + \log \left (x\right )\right ) \]

[In]

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

[Out]

log(-log(5) + log(3*x^2*log(x) + 6*x*log(2)*log(x) + 3*log(2)^2*log(x) + 6*x*log(x)^2 + 6*log(2)*log(x)^2 + 3*
log(x)^3 + 3*x^2*log(e^x + 1) + 6*x*log(2)*log(e^x + 1) + 3*log(2)^2*log(e^x + 1) + 6*x*log(x)*log(e^x + 1) +
6*log(2)*log(x)*log(e^x + 1) + 3*log(x)^2*log(e^x + 1)) + log(x))

Mupad [B] (verification not implemented)

Time = 11.56 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52 \[ \int \frac {x+e^x \left (x+x^2\right )+\left (1+e^x (1+x)\right ) \log (2 x)+\left (2+3 x+e^x (2+3 x)+\left (1+e^x\right ) \log (2 x)\right ) \log \left (x+e^x x\right )}{\left (x^2+e^x x^2+\left (x+e^x x\right ) \log (2 x)\right ) \log \left (x+e^x x\right ) \log \left (\frac {1}{5} \left (3 x^3+6 x^2 \log (2 x)+3 x \log ^2(2 x)\right ) \log \left (x+e^x x\right )\right )} \, dx=\ln \left (\ln \left (\frac {\ln \left (x+x\,{\mathrm {e}}^x\right )\,\left (3\,x^3+6\,x^2\,\ln \left (2\,x\right )+3\,x\,{\ln \left (2\,x\right )}^2\right )}{5}\right )\right ) \]

[In]

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

[Out]

log(log((log(x + x*exp(x))*(3*x*log(2*x)^2 + 6*x^2*log(2*x) + 3*x^3))/5))