\(\int \frac {(4 e^2 x+8 x^2+e^x (e^2 x+2 x^2)) \log (4 x+e^x x) \log (x \log (4 x+e^x x))+e^{4-x} \log (x \log (4 x+e^x x)) (4 e^2+4 x+e^x (x+x^2+e^2 (1+x))+(4 e^2+4 x+e^x (e^2+x)) \log (4 x+e^x x)+(4 x-4 e^2 x-4 x^2+e^x (x-e^2 x-x^2)) \log (4 x+e^x x) \log (x \log (4 x+e^x x)))}{(4 x+e^x x) \log (4 x+e^x x) \log (x \log (4 x+e^x x))} \, dx\) [5657]

Optimal result

Integrand size = 222, antiderivative size = 29 \[ \int \frac {\left (4 e^2 x+8 x^2+e^x \left (e^2 x+2 x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right ) \left (4 e^2+4 x+e^x \left (x+x^2+e^2 (1+x)\right )+\left (4 e^2+4 x+e^x \left (e^2+x\right )\right ) \log \left (4 x+e^x x\right )+\left (4 x-4 e^2 x-4 x^2+e^x \left (x-e^2 x-x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )\right )}{\left (4 x+e^x x\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )} \, dx=4+\left (e^2+x\right ) \left (x+e^{4-x} \log \left (x \log \left (\left (4+e^x\right ) x\right )\right )\right ) \]

[Out]

4+(x+exp(2))*(exp(ln(ln(ln((exp(x)+4)*x)*x))-x+4)+x)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(29)=58\).

Time = 13.14 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.14, number of steps used = 29, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {6820, 6874, 2225, 2209, 2207, 2635} \[ \int \frac {\left (4 e^2 x+8 x^2+e^x \left (e^2 x+2 x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right ) \left (4 e^2+4 x+e^x \left (x+x^2+e^2 (1+x)\right )+\left (4 e^2+4 x+e^x \left (e^2+x\right )\right ) \log \left (4 x+e^x x\right )+\left (4 x-4 e^2 x-4 x^2+e^x \left (x-e^2 x-x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )\right )}{\left (4 x+e^x x\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )} \, dx=x^2+e^2 x+e^{4-x} \log \left (x \log \left (e^x x+4 x\right )\right )-e^{4-x} \left (-x-e^2+1\right ) \log \left (x \log \left (e^x x+4 x\right )\right ) \]

[In]

Int[((4*E^2*x + 8*x^2 + E^x*(E^2*x + 2*x^2))*Log[4*x + E^x*x]*Log[x*Log[4*x + E^x*x]] + E^(4 - x)*Log[x*Log[4*
x + E^x*x]]*(4*E^2 + 4*x + E^x*(x + x^2 + E^2*(1 + x)) + (4*E^2 + 4*x + E^x*(E^2 + x))*Log[4*x + E^x*x] + (4*x
- 4*E^2*x - 4*x^2 + E^x*(x - E^2*x - x^2))*Log[4*x + E^x*x]*Log[x*Log[4*x + E^x*x]]))/((4*x + E^x*x)*Log[4*x
+ E^x*x]*Log[x*Log[4*x + E^x*x]]),x]

[Out]

E^2*x + x^2 + E^(4 - x)*Log[x*Log[4*x + E^x*x]] - E^(4 - x)*(1 - E^2 - x)*Log[x*Log[4*x + E^x*x]]

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2635

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*Simplify
[D[u, x]/u], x], x] /; InverseFunctionFreeQ[w, x]] /; ProductQ[u]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (e^6+e^4 x+e^{2+x} x+2 e^x x^2+\frac {e^4 \left (e^2+x\right ) \left (4+e^x (1+x)\right )}{\left (4+e^x\right ) \log \left (\left (4+e^x\right ) x\right )}-e^4 x \left (-1+e^2+x\right ) \log \left (x \log \left (\left (4+e^x\right ) x\right )\right )\right )}{x} \, dx \\ & = \int \left (e^2+2 x+\frac {4 e^{4-x} \left (-e^2-x\right )}{\left (4+e^x\right ) \log \left (4 x+e^x x\right )}+\frac {e^{4-x} \left (e^2+\left (1+e^2\right ) x+x^2+e^2 \log \left (\left (4+e^x\right ) x\right )+x \log \left (\left (4+e^x\right ) x\right )+\left (1-e^2\right ) x \log \left (\left (4+e^x\right ) x\right ) \log \left (x \log \left (\left (4+e^x\right ) x\right )\right )-x^2 \log \left (\left (4+e^x\right ) x\right ) \log \left (x \log \left (\left (4+e^x\right ) x\right )\right )\right )}{x \log \left (4 x+e^x x\right )}\right ) \, dx \\ & = e^2 x+x^2+4 \int \frac {e^{4-x} \left (-e^2-x\right )}{\left (4+e^x\right ) \log \left (4 x+e^x x\right )} \, dx+\int \frac {e^{4-x} \left (e^2+\left (1+e^2\right ) x+x^2+e^2 \log \left (\left (4+e^x\right ) x\right )+x \log \left (\left (4+e^x\right ) x\right )+\left (1-e^2\right ) x \log \left (\left (4+e^x\right ) x\right ) \log \left (x \log \left (\left (4+e^x\right ) x\right )\right )-x^2 \log \left (\left (4+e^x\right ) x\right ) \log \left (x \log \left (\left (4+e^x\right ) x\right )\right )\right )}{x \log \left (4 x+e^x x\right )} \, dx \\ & = e^2 x+x^2+4 \int \left (\frac {e^{6-x}}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )}+\frac {e^{4-x} x}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )}\right ) \, dx+\int \frac {e^{4-x} \left ((1+x) \left (e^2+x\right )-\log \left (\left (4+e^x\right ) x\right ) \left (-e^2-x+x \left (-1+e^2+x\right ) \log \left (x \log \left (\left (4+e^x\right ) x\right )\right )\right )\right )}{x \log \left (4 x+e^x x\right )} \, dx \\ & = e^2 x+x^2+4 \int \frac {e^{6-x}}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )} \, dx+4 \int \frac {e^{4-x} x}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )} \, dx+\int \left (\frac {e^{4-x} \left (e^2+x\right ) \left (1+x+\log \left (\left (4+e^x\right ) x\right )\right )}{x \log \left (4 x+e^x x\right )}+e^{4-x} \left (1-e^2-x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )\right ) \, dx \\ & = e^2 x+x^2+4 \int \frac {e^{6-x}}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )} \, dx+4 \int \frac {e^{4-x} x}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )} \, dx+\int \frac {e^{4-x} \left (e^2+x\right ) \left (1+x+\log \left (\left (4+e^x\right ) x\right )\right )}{x \log \left (4 x+e^x x\right )} \, dx+\int e^{4-x} \left (1-e^2-x\right ) \log \left (x \log \left (4 x+e^x x\right )\right ) \, dx \\ & = e^2 x+x^2+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right )-e^{4-x} \left (1-e^2-x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )+4 \int \frac {e^{6-x}}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )} \, dx+4 \int \frac {e^{4-x} x}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )} \, dx+\int \left (\frac {e^{4-x} \left (1+x+\log \left (\left (4+e^x\right ) x\right )\right )}{\log \left (4 x+e^x x\right )}+\frac {e^{6-x} \left (1+x+\log \left (\left (4+e^x\right ) x\right )\right )}{x \log \left (4 x+e^x x\right )}\right ) \, dx-\int \frac {e^{4-x} \left (e^2+x\right ) \left (4+e^x (1+x)+\left (4+e^x\right ) \log \left (\left (4+e^x\right ) x\right )\right )}{\left (4+e^x\right ) x \log \left (4 x+e^x x\right )} \, dx \\ & = e^2 x+x^2+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right )-e^{4-x} \left (1-e^2-x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )+4 \int \frac {e^{6-x}}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )} \, dx+4 \int \frac {e^{4-x} x}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )} \, dx-\int \left (\frac {4 e^{4-x} \left (-e^2-x\right )}{\left (4+e^x\right ) \log \left (4 x+e^x x\right )}+\frac {e^{4-x} \left (e^2+x\right ) \left (1+x+\log \left (\left (4+e^x\right ) x\right )\right )}{x \log \left (4 x+e^x x\right )}\right ) \, dx+\int \frac {e^{4-x} \left (1+x+\log \left (\left (4+e^x\right ) x\right )\right )}{\log \left (4 x+e^x x\right )} \, dx+\int \frac {e^{6-x} \left (1+x+\log \left (\left (4+e^x\right ) x\right )\right )}{x \log \left (4 x+e^x x\right )} \, dx \\ & = e^2 x+x^2+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right )-e^{4-x} \left (1-e^2-x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )+4 \int \frac {e^{6-x}}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )} \, dx-4 \int \frac {e^{4-x} \left (-e^2-x\right )}{\left (4+e^x\right ) \log \left (4 x+e^x x\right )} \, dx+4 \int \frac {e^{4-x} x}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )} \, dx+\int \left (\frac {e^{6-x}}{x}+\frac {e^{6-x}}{\log \left (4 x+e^x x\right )}+\frac {e^{6-x}}{x \log \left (4 x+e^x x\right )}\right ) \, dx+\int \left (e^{4-x}+\frac {e^{4-x}}{\log \left (4 x+e^x x\right )}+\frac {e^{4-x} x}{\log \left (4 x+e^x x\right )}\right ) \, dx-\int \frac {e^{4-x} \left (e^2+x\right ) \left (1+x+\log \left (\left (4+e^x\right ) x\right )\right )}{x \log \left (4 x+e^x x\right )} \, dx \\ & = e^2 x+x^2+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right )-e^{4-x} \left (1-e^2-x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )-4 \int \left (\frac {e^{6-x}}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )}+\frac {e^{4-x} x}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )}\right ) \, dx+4 \int \frac {e^{6-x}}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )} \, dx+4 \int \frac {e^{4-x} x}{\left (-4-e^x\right ) \log \left (4 x+e^x x\right )} \, dx+\int e^{4-x} \, dx+\int \frac {e^{6-x}}{x} \, dx-\int \left (\frac {e^{4-x} \left (1+x+\log \left (\left (4+e^x\right ) x\right )\right )}{\log \left (4 x+e^x x\right )}+\frac {e^{6-x} \left (1+x+\log \left (\left (4+e^x\right ) x\right )\right )}{x \log \left (4 x+e^x x\right )}\right ) \, dx+\int \frac {e^{4-x}}{\log \left (4 x+e^x x\right )} \, dx+\int \frac {e^{6-x}}{\log \left (4 x+e^x x\right )} \, dx+\int \frac {e^{6-x}}{x \log \left (4 x+e^x x\right )} \, dx+\int \frac {e^{4-x} x}{\log \left (4 x+e^x x\right )} \, dx \\ & = -e^{4-x}+e^2 x+x^2+e^6 \text {Ei}(-x)+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right )-e^{4-x} \left (1-e^2-x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )+\int \frac {e^{4-x}}{\log \left (4 x+e^x x\right )} \, dx+\int \frac {e^{6-x}}{\log \left (4 x+e^x x\right )} \, dx+\int \frac {e^{6-x}}{x \log \left (4 x+e^x x\right )} \, dx+\int \frac {e^{4-x} x}{\log \left (4 x+e^x x\right )} \, dx-\int \frac {e^{4-x} \left (1+x+\log \left (\left (4+e^x\right ) x\right )\right )}{\log \left (4 x+e^x x\right )} \, dx-\int \frac {e^{6-x} \left (1+x+\log \left (\left (4+e^x\right ) x\right )\right )}{x \log \left (4 x+e^x x\right )} \, dx \\ & = -e^{4-x}+e^2 x+x^2+e^6 \text {Ei}(-x)+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right )-e^{4-x} \left (1-e^2-x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )-\int \left (\frac {e^{6-x}}{x}+\frac {e^{6-x}}{\log \left (4 x+e^x x\right )}+\frac {e^{6-x}}{x \log \left (4 x+e^x x\right )}\right ) \, dx-\int \left (e^{4-x}+\frac {e^{4-x}}{\log \left (4 x+e^x x\right )}+\frac {e^{4-x} x}{\log \left (4 x+e^x x\right )}\right ) \, dx+\int \frac {e^{4-x}}{\log \left (4 x+e^x x\right )} \, dx+\int \frac {e^{6-x}}{\log \left (4 x+e^x x\right )} \, dx+\int \frac {e^{6-x}}{x \log \left (4 x+e^x x\right )} \, dx+\int \frac {e^{4-x} x}{\log \left (4 x+e^x x\right )} \, dx \\ & = -e^{4-x}+e^2 x+x^2+e^6 \text {Ei}(-x)+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right )-e^{4-x} \left (1-e^2-x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )-\int e^{4-x} \, dx-\int \frac {e^{6-x}}{x} \, dx \\ & = e^2 x+x^2+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right )-e^{4-x} \left (1-e^2-x\right ) \log \left (x \log \left (4 x+e^x x\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {\left (4 e^2 x+8 x^2+e^x \left (e^2 x+2 x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right ) \left (4 e^2+4 x+e^x \left (x+x^2+e^2 (1+x)\right )+\left (4 e^2+4 x+e^x \left (e^2+x\right )\right ) \log \left (4 x+e^x x\right )+\left (4 x-4 e^2 x-4 x^2+e^x \left (x-e^2 x-x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )\right )}{\left (4 x+e^x x\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )} \, dx=e^2 x+x^2+e^{4-x} \left (e^2+x\right ) \log \left (x \log \left (\left (4+e^x\right ) x\right )\right ) \]

[In]

Integrate[((4*E^2*x + 8*x^2 + E^x*(E^2*x + 2*x^2))*Log[4*x + E^x*x]*Log[x*Log[4*x + E^x*x]] + E^(4 - x)*Log[x*
Log[4*x + E^x*x]]*(4*E^2 + 4*x + E^x*(x + x^2 + E^2*(1 + x)) + (4*E^2 + 4*x + E^x*(E^2 + x))*Log[4*x + E^x*x]
+ (4*x - 4*E^2*x - 4*x^2 + E^x*(x - E^2*x - x^2))*Log[4*x + E^x*x]*Log[x*Log[4*x + E^x*x]]))/((4*x + E^x*x)*Lo
g[4*x + E^x*x]*Log[x*Log[4*x + E^x*x]]),x]

[Out]

E^2*x + x^2 + E^(4 - x)*(E^2 + x)*Log[x*Log[(4 + E^x)*x]]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.10 (sec) , antiderivative size = 2626, normalized size of antiderivative = 90.55

\[\text {Expression too large to display}\]

[In]

int(((((-exp(2)*x-x^2+x)*exp(x)-4*exp(2)*x-4*x^2+4*x)*ln(exp(x)*x+4*x)*ln(x*ln(exp(x)*x+4*x))+((x+exp(2))*exp(
x)+4*exp(2)+4*x)*ln(exp(x)*x+4*x)+((1+x)*exp(2)+x^2+x)*exp(x)+4*exp(2)+4*x)*exp(ln(ln(x*ln(exp(x)*x+4*x)))-x+4
)+((exp(2)*x+2*x^2)*exp(x)+4*exp(2)*x+8*x^2)*ln(exp(x)*x+4*x)*ln(x*ln(exp(x)*x+4*x)))/(exp(x)*x+4*x)/ln(exp(x)
*x+4*x)/ln(x*ln(exp(x)*x+4*x)),x)

[Out]

ln(-Pi*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(
x)+4))*csgn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3)*(x+exp(2))*exp(-x+4)+1/2*(I*
exp(6)*Pi*csgn(I*x*(-Pi*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(
I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3))*csgn(x*(-P
i*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4)
)*csgn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3))^2+I*exp(4)*x*Pi*csgn(I*x*(-Pi*cs
gn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*cs
gn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3))*csgn(x*(-Pi*csgn(I*x)*csgn(I*(exp(x)
+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*I
*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3))^2-I*exp(6)*Pi+I*exp(6)*Pi*csgn(x*(-Pi*csgn(I*x)*csgn(I*(ex
p(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)
+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3))^2-I*exp(6)*Pi*csgn(x*(-Pi*csgn(I*x)*csgn(I*(exp(x)+4)*
x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*I*ln(
x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3))^3+I*exp(6)*Pi*csgn(I*x)*csgn(I*Pi*csgn(I*x)*csgn(I*(exp(x)+4)*
x)^2+I*Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2-I*Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*l
n(x)+2*ln(exp(x)+4)-I*Pi*csgn(I*(exp(x)+4)*x)^3)*csgn(I*x*(-Pi*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp
(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)
+Pi*csgn(I*(exp(x)+4)*x)^3))+I*exp(4)*x*Pi*csgn(I*x)*csgn(I*Pi*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2+I*Pi*csgn(I*(e
xp(x)+4))*csgn(I*(exp(x)+4)*x)^2-I*Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*ln(x)+2*ln(exp(x)+4)
-I*Pi*csgn(I*(exp(x)+4)*x)^3)*csgn(I*x*(-Pi*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp
(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+
4)*x)^3))-I*exp(6)*Pi*csgn(I*x*(-Pi*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x
)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3)
)*csgn(x*(-Pi*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(
I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3))+I*exp(4)*x*Pi*csgn(x
*(-Pi*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x
)+4))*csgn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3))^2-I*exp(6)*Pi*csgn(I*Pi*csgn
(I*x)*csgn(I*(exp(x)+4)*x)^2+I*Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2-I*Pi*csgn(I*x)*csgn(I*(exp(x)+4))*
csgn(I*(exp(x)+4)*x)+2*ln(x)+2*ln(exp(x)+4)-I*Pi*csgn(I*(exp(x)+4)*x)^3)*csgn(I*x*(-Pi*csgn(I*x)*csgn(I*(exp(x
)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*
I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3))^2+I*exp(6)*Pi*csgn(I*x)*csgn(I*x*(-Pi*csgn(I*x)*csgn(I*(e
xp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x
)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3))^2+I*exp(4)*x*Pi*csgn(I*x)*csgn(I*x*(-Pi*csgn(I*x)*csg
n(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x
)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3))^2-I*exp(4)*x*Pi*csgn(I*x*(-Pi*csgn(I*x)*csgn(I*
(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)
*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3))^3-I*exp(4)*x*Pi*csgn(I*Pi*csgn(I*x)*csgn(I*(exp(x)+
4)*x)^2+I*Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2-I*Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+
2*ln(x)+2*ln(exp(x)+4)-I*Pi*csgn(I*(exp(x)+4)*x)^3)*csgn(I*x*(-Pi*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(
exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)
+4)+Pi*csgn(I*(exp(x)+4)*x)^3))^2-I*exp(4)*x*Pi*csgn(x*(-Pi*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)
+4))*csgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi
*csgn(I*(exp(x)+4)*x)^3))^3-I*exp(6)*Pi*csgn(I*x*(-Pi*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*c
sgn(I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(
I*(exp(x)+4)*x)^3))^3-I*exp(4)*x*Pi*csgn(I*x*(-Pi*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(
I*(exp(x)+4)*x)^2+Pi*csgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(e
xp(x)+4)*x)^3))*csgn(x*(-Pi*csgn(I*x)*csgn(I*(exp(x)+4)*x)^2-Pi*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)^2+Pi*c
sgn(I*x)*csgn(I*(exp(x)+4))*csgn(I*(exp(x)+4)*x)+2*I*ln(x)+2*I*ln(exp(x)+4)+Pi*csgn(I*(exp(x)+4)*x)^3))-I*exp(
4)*x*Pi-2*exp(6)*ln(2)+2*exp(6)*ln(x)-2*x*exp(4)*ln(2)+2*x*exp(4)*ln(x)+2*x*exp(2+x)+2*exp(x)*x^2)*exp(-x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {\left (4 e^2 x+8 x^2+e^x \left (e^2 x+2 x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right ) \left (4 e^2+4 x+e^x \left (x+x^2+e^2 (1+x)\right )+\left (4 e^2+4 x+e^x \left (e^2+x\right )\right ) \log \left (4 x+e^x x\right )+\left (4 x-4 e^2 x-4 x^2+e^x \left (x-e^2 x-x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )\right )}{\left (4 x+e^x x\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )} \, dx={\left ({\left (x^{2} + x e^{2}\right )} e^{x} + {\left (x e^{4} + e^{6}\right )} \log \left (x \log \left (x e^{x} + 4 \, x\right )\right )\right )} e^{\left (-x\right )} \]

[In]

integrate(((((-exp(2)*x-x^2+x)*exp(x)-4*exp(2)*x-4*x^2+4*x)*log(exp(x)*x+4*x)*log(x*log(exp(x)*x+4*x))+((x+exp
(2))*exp(x)+4*exp(2)+4*x)*log(exp(x)*x+4*x)+((1+x)*exp(2)+x^2+x)*exp(x)+4*exp(2)+4*x)*exp(log(log(x*log(exp(x)
*x+4*x)))-x+4)+((exp(2)*x+2*x^2)*exp(x)+4*exp(2)*x+8*x^2)*log(exp(x)*x+4*x)*log(x*log(exp(x)*x+4*x)))/(exp(x)*
x+4*x)/log(exp(x)*x+4*x)/log(x*log(exp(x)*x+4*x)),x, algorithm="fricas")

[Out]

((x^2 + x*e^2)*e^x + (x*e^4 + e^6)*log(x*log(x*e^x + 4*x)))*e^(-x)

Sympy [A] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {\left (4 e^2 x+8 x^2+e^x \left (e^2 x+2 x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right ) \left (4 e^2+4 x+e^x \left (x+x^2+e^2 (1+x)\right )+\left (4 e^2+4 x+e^x \left (e^2+x\right )\right ) \log \left (4 x+e^x x\right )+\left (4 x-4 e^2 x-4 x^2+e^x \left (x-e^2 x-x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )\right )}{\left (4 x+e^x x\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )} \, dx=x^{2} + x e^{2} + \left (x e^{4} + e^{6}\right ) e^{- x} \log {\left (x \log {\left (x e^{x} + 4 x \right )} \right )} \]

[In]

integrate(((((-exp(2)*x-x**2+x)*exp(x)-4*exp(2)*x-4*x**2+4*x)*ln(exp(x)*x+4*x)*ln(x*ln(exp(x)*x+4*x))+((x+exp(
2))*exp(x)+4*exp(2)+4*x)*ln(exp(x)*x+4*x)+((1+x)*exp(2)+x**2+x)*exp(x)+4*exp(2)+4*x)*exp(ln(ln(x*ln(exp(x)*x+4
*x)))-x+4)+((exp(2)*x+2*x**2)*exp(x)+4*exp(2)*x+8*x**2)*ln(exp(x)*x+4*x)*ln(x*ln(exp(x)*x+4*x)))/(exp(x)*x+4*x
)/ln(exp(x)*x+4*x)/ln(x*ln(exp(x)*x+4*x)),x)

[Out]

x**2 + x*exp(2) + (x*exp(4) + exp(6))*exp(-x)*log(x*log(x*exp(x) + 4*x))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {\left (4 e^2 x+8 x^2+e^x \left (e^2 x+2 x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right ) \left (4 e^2+4 x+e^x \left (x+x^2+e^2 (1+x)\right )+\left (4 e^2+4 x+e^x \left (e^2+x\right )\right ) \log \left (4 x+e^x x\right )+\left (4 x-4 e^2 x-4 x^2+e^x \left (x-e^2 x-x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )\right )}{\left (4 x+e^x x\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )} \, dx={\left (x e^{4} + e^{6}\right )} e^{\left (-x\right )} \log \left (x\right ) + {\left (x e^{4} + e^{6}\right )} e^{\left (-x\right )} \log \left (\log \left (x\right ) + \log \left (e^{x} + 4\right )\right ) + x^{2} + x e^{2} \]

[In]

integrate(((((-exp(2)*x-x^2+x)*exp(x)-4*exp(2)*x-4*x^2+4*x)*log(exp(x)*x+4*x)*log(x*log(exp(x)*x+4*x))+((x+exp
(2))*exp(x)+4*exp(2)+4*x)*log(exp(x)*x+4*x)+((1+x)*exp(2)+x^2+x)*exp(x)+4*exp(2)+4*x)*exp(log(log(x*log(exp(x)
*x+4*x)))-x+4)+((exp(2)*x+2*x^2)*exp(x)+4*exp(2)*x+8*x^2)*log(exp(x)*x+4*x)*log(x*log(exp(x)*x+4*x)))/(exp(x)*
x+4*x)/log(exp(x)*x+4*x)/log(x*log(exp(x)*x+4*x)),x, algorithm="maxima")

[Out]

(x*e^4 + e^6)*e^(-x)*log(x) + (x*e^4 + e^6)*e^(-x)*log(log(x) + log(e^x + 4)) + x^2 + x*e^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).

Time = 0.44 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {\left (4 e^2 x+8 x^2+e^x \left (e^2 x+2 x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right ) \left (4 e^2+4 x+e^x \left (x+x^2+e^2 (1+x)\right )+\left (4 e^2+4 x+e^x \left (e^2+x\right )\right ) \log \left (4 x+e^x x\right )+\left (4 x-4 e^2 x-4 x^2+e^x \left (x-e^2 x-x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )\right )}{\left (4 x+e^x x\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )} \, dx={\left (x^{2} e^{x} + x e^{4} \log \left (x\right ) + x e^{4} \log \left (\log \left (x\right ) + \log \left (e^{x} + 4\right )\right ) + x e^{\left (x + 2\right )} + e^{6} \log \left (x\right ) + e^{6} \log \left (\log \left (x\right ) + \log \left (e^{x} + 4\right )\right )\right )} e^{\left (-x\right )} \]

[In]

integrate(((((-exp(2)*x-x^2+x)*exp(x)-4*exp(2)*x-4*x^2+4*x)*log(exp(x)*x+4*x)*log(x*log(exp(x)*x+4*x))+((x+exp
(2))*exp(x)+4*exp(2)+4*x)*log(exp(x)*x+4*x)+((1+x)*exp(2)+x^2+x)*exp(x)+4*exp(2)+4*x)*exp(log(log(x*log(exp(x)
*x+4*x)))-x+4)+((exp(2)*x+2*x^2)*exp(x)+4*exp(2)*x+8*x^2)*log(exp(x)*x+4*x)*log(x*log(exp(x)*x+4*x)))/(exp(x)*
x+4*x)/log(exp(x)*x+4*x)/log(x*log(exp(x)*x+4*x)),x, algorithm="giac")

[Out]

(x^2*e^x + x*e^4*log(x) + x*e^4*log(log(x) + log(e^x + 4)) + x*e^(x + 2) + e^6*log(x) + e^6*log(log(x) + log(e
^x + 4)))*e^(-x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (4 e^2 x+8 x^2+e^x \left (e^2 x+2 x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )+e^{4-x} \log \left (x \log \left (4 x+e^x x\right )\right ) \left (4 e^2+4 x+e^x \left (x+x^2+e^2 (1+x)\right )+\left (4 e^2+4 x+e^x \left (e^2+x\right )\right ) \log \left (4 x+e^x x\right )+\left (4 x-4 e^2 x-4 x^2+e^x \left (x-e^2 x-x^2\right )\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )\right )}{\left (4 x+e^x x\right ) \log \left (4 x+e^x x\right ) \log \left (x \log \left (4 x+e^x x\right )\right )} \, dx=\int \frac {{\mathrm {e}}^{\ln \left (\ln \left (x\,\ln \left (4\,x+x\,{\mathrm {e}}^x\right )\right )\right )-x+4}\,\left (4\,x+4\,{\mathrm {e}}^2+\ln \left (4\,x+x\,{\mathrm {e}}^x\right )\,\left (4\,x+4\,{\mathrm {e}}^2+{\mathrm {e}}^x\,\left (x+{\mathrm {e}}^2\right )\right )+{\mathrm {e}}^x\,\left (x+{\mathrm {e}}^2\,\left (x+1\right )+x^2\right )-\ln \left (4\,x+x\,{\mathrm {e}}^x\right )\,\ln \left (x\,\ln \left (4\,x+x\,{\mathrm {e}}^x\right )\right )\,\left (4\,x\,{\mathrm {e}}^2-4\,x+{\mathrm {e}}^x\,\left (x\,{\mathrm {e}}^2-x+x^2\right )+4\,x^2\right )\right )+\ln \left (4\,x+x\,{\mathrm {e}}^x\right )\,\ln \left (x\,\ln \left (4\,x+x\,{\mathrm {e}}^x\right )\right )\,\left ({\mathrm {e}}^x\,\left (2\,x^2+{\mathrm {e}}^2\,x\right )+4\,x\,{\mathrm {e}}^2+8\,x^2\right )}{\ln \left (4\,x+x\,{\mathrm {e}}^x\right )\,\ln \left (x\,\ln \left (4\,x+x\,{\mathrm {e}}^x\right )\right )\,\left (4\,x+x\,{\mathrm {e}}^x\right )} \,d x \]

[In]

int((exp(log(log(x*log(4*x + x*exp(x)))) - x + 4)*(4*x + 4*exp(2) + log(4*x + x*exp(x))*(4*x + 4*exp(2) + exp(
x)*(x + exp(2))) + exp(x)*(x + exp(2)*(x + 1) + x^2) - log(4*x + x*exp(x))*log(x*log(4*x + x*exp(x)))*(4*x*exp
(2) - 4*x + exp(x)*(x*exp(2) - x + x^2) + 4*x^2)) + log(4*x + x*exp(x))*log(x*log(4*x + x*exp(x)))*(exp(x)*(x*
exp(2) + 2*x^2) + 4*x*exp(2) + 8*x^2))/(log(4*x + x*exp(x))*log(x*log(4*x + x*exp(x)))*(4*x + x*exp(x))),x)

[Out]

int((exp(log(log(x*log(4*x + x*exp(x)))) - x + 4)*(4*x + 4*exp(2) + log(4*x + x*exp(x))*(4*x + 4*exp(2) + exp(
x)*(x + exp(2))) + exp(x)*(x + exp(2)*(x + 1) + x^2) - log(4*x + x*exp(x))*log(x*log(4*x + x*exp(x)))*(4*x*exp
(2) - 4*x + exp(x)*(x*exp(2) - x + x^2) + 4*x^2)) + log(4*x + x*exp(x))*log(x*log(4*x + x*exp(x)))*(exp(x)*(x*
exp(2) + 2*x^2) + 4*x*exp(2) + 8*x^2))/(log(4*x + x*exp(x))*log(x*log(4*x + x*exp(x)))*(4*x + x*exp(x))), x)