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

Optimal. Leaf size=23 \[ \log \left (\log \left (\frac {3}{5} x (x+\log (2 x))^2 \log \left (x+e^x x\right )\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 7.17, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 23, normalized size = 1.00 \begin {gather*} \log \left (\log \left (\frac {3}{5} x (x+\log (2 x))^2 \log \left (\left (1+e^x\right ) x\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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]]

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 32, normalized size = 1.39 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(x)+1)*log(2*x)+(3*x+2)*exp(x)+3*x+2)*log(exp(x)*x+x)+((x+1)*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)))

________________________________________________________________________________________

giac [B]  time = 4.72, size = 116, normalized size = 5.04 \begin {gather*} \log \left (-\log \relax (5) + \log \left (3 \, x^{2} \log \relax (x) + 6 \, x \log \relax (2) \log \relax (x) + 3 \, \log \relax (2)^{2} \log \relax (x) + 6 \, x \log \relax (x)^{2} + 6 \, \log \relax (2) \log \relax (x)^{2} + 3 \, \log \relax (x)^{3} + 3 \, x^{2} \log \left (e^{x} + 1\right ) + 6 \, x \log \relax (2) \log \left (e^{x} + 1\right ) + 3 \, \log \relax (2)^{2} \log \left (e^{x} + 1\right ) + 6 \, x \log \relax (x) \log \left (e^{x} + 1\right ) + 6 \, \log \relax (2) \log \relax (x) \log \left (e^{x} + 1\right ) + 3 \, \log \relax (x)^{2} \log \left (e^{x} + 1\right )\right ) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(x)+1)*log(2*x)+(3*x+2)*exp(x)+3*x+2)*log(exp(x)*x+x)+((x+1)*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))

________________________________________________________________________________________

maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (\left ({\mathrm e}^{x}+1\right ) \ln \left (2 x \right )+\left (3 x +2\right ) {\mathrm e}^{x}+3 x +2\right ) \ln \left ({\mathrm e}^{x} x +x \right )+\left (\left (x +1\right ) {\mathrm e}^{x}+1\right ) \ln \left (2 x \right )+\left (x^{2}+x \right ) {\mathrm e}^{x}+x}{\left (\left ({\mathrm e}^{x} x +x \right ) \ln \left (2 x \right )+{\mathrm e}^{x} x^{2}+x^{2}\right ) \ln \left ({\mathrm e}^{x} x +x \right ) \ln \left (\frac {\left (3 x \ln \left (2 x \right )^{2}+6 x^{2} \ln \left (2 x \right )+3 x^{3}\right ) \ln \left ({\mathrm e}^{x} x +x \right )}{5}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((exp(x)+1)*ln(2*x)+(3*x+2)*exp(x)+3*x+2)*ln(exp(x)*x+x)+((x+1)*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]

int((((exp(x)+1)*ln(2*x)+(3*x+2)*exp(x)+3*x+2)*ln(exp(x)*x+x)+((x+1)*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)

________________________________________________________________________________________

maxima [A]  time = 0.66, size = 28, normalized size = 1.22 \begin {gather*} \log \left (-\log \relax (5) + \log \relax (3) + 2 \, \log \left (x + \log \relax (2) + \log \relax (x)\right ) + \log \relax (x) + \log \left (\log \relax (x) + \log \left (e^{x} + 1\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(x)+1)*log(2*x)+(3*x+2)*exp(x)+3*x+2)*log(exp(x)*x+x)+((x+1)*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)))

________________________________________________________________________________________

mupad [B]  time = 4.78, size = 35, normalized size = 1.52 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

sympy [A]  time = 70.60, size = 41, normalized size = 1.78 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(x)+1)*ln(2*x)+(3*x+2)*exp(x)+3*x+2)*ln(exp(x)*x+x)+((x+1)*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)))

________________________________________________________________________________________