3.53.86 \(\int \frac {e^x (-x-x^2)+4 e^{5 x} x \log ^3(1+x)+e^{5 x} (4 x+4 x^2) \log ^4(1+x)+(e^x (-x-x^2)+e^{5 x} (1+x) \log ^4(1+x)) \log (\frac {1}{3} (2 x-2 e^{4 x} \log ^4(1+x)))+(e^x (-x^2-x^3)+e^{5 x} (x+x^2) \log ^4(1+x)) \log (\frac {1}{3} (2 x-2 e^{4 x} \log ^4(1+x))) \log (x \log (\frac {1}{3} (2 x-2 e^{4 x} \log ^4(1+x))))}{(-x^2-x^3+e^{4 x} (x+x^2) \log ^4(1+x)) \log (\frac {1}{3} (2 x-2 e^{4 x} \log ^4(1+x)))} \, dx\)

Optimal. Leaf size=27 \[ e^x \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right ) \]

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Rubi [A]  time = 39.58, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 41, number of rules used = 6, integrand size = 237, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6741, 6742, 6688, 2178, 2194, 2555} \begin {gather*} e^x \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(x+1)\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(-x - x^2) + 4*E^(5*x)*x*Log[1 + x]^3 + E^(5*x)*(4*x + 4*x^2)*Log[1 + x]^4 + (E^x*(-x - x^2) + E^(5*x
)*(1 + x)*Log[1 + x]^4)*Log[(2*x - 2*E^(4*x)*Log[1 + x]^4)/3] + (E^x*(-x^2 - x^3) + E^(5*x)*(x + x^2)*Log[1 +
x]^4)*Log[(2*x - 2*E^(4*x)*Log[1 + x]^4)/3]*Log[x*Log[(2*x - 2*E^(4*x)*Log[1 + x]^4)/3]])/((-x^2 - x^3 + E^(4*
x)*(x + x^2)*Log[1 + x]^4)*Log[(2*x - 2*E^(4*x)*Log[1 + x]^4)/3]),x]

[Out]

E^x*Log[x*Log[(2*(x - E^(4*x)*Log[1 + x]^4))/3]]

Rule 2178

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

Rule 2194

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 2555

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 6688

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

Rule 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^x \left (-x-x^2\right )-4 e^{5 x} x \log ^3(1+x)-e^{5 x} \left (4 x+4 x^2\right ) \log ^4(1+x)-\left (e^x \left (-x-x^2\right )+e^{5 x} (1+x) \log ^4(1+x)\right ) \log \left (\frac {1}{3} \left (2 x-2 e^{4 x} \log ^4(1+x)\right )\right )-\left (e^x \left (-x^2-x^3\right )+e^{5 x} \left (x+x^2\right ) \log ^4(1+x)\right ) \log \left (\frac {1}{3} \left (2 x-2 e^{4 x} \log ^4(1+x)\right )\right ) \log \left (x \log \left (\frac {1}{3} \left (2 x-2 e^{4 x} \log ^4(1+x)\right )\right )\right )}{x (1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx\\ &=\int \left (\frac {e^x \left (4 x-\log (1+x)+3 x \log (1+x)+4 x^2 \log (1+x)\right )}{(1+x) \log (1+x) \left (-x+e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}+\frac {e^x \left (4 x+4 x \log (1+x)+4 x^2 \log (1+x)+\log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )+x \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )+x \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right ) \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )+x^2 \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right ) \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )\right )}{x (1+x) \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}\right ) \, dx\\ &=\int \frac {e^x \left (4 x-\log (1+x)+3 x \log (1+x)+4 x^2 \log (1+x)\right )}{(1+x) \log (1+x) \left (-x+e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+\int \frac {e^x \left (4 x+4 x \log (1+x)+4 x^2 \log (1+x)+\log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )+x \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )+x \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right ) \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )+x^2 \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right ) \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )\right )}{x (1+x) \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx\\ &=\int \left (\frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}-\frac {3 e^x x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}-\frac {4 e^x x^2}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}-\frac {4 e^x x}{(1+x) \log (1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}\right ) \, dx+\int \frac {e^x \left (4 x+(1+x) \log (1+x) \left (4 x+\log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right ) \left (1+x \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )\right )\right )\right )}{x (1+x) \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx\\ &=-\left (3 \int \frac {e^x x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx\right )-4 \int \frac {e^x x^2}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx-4 \int \frac {e^x x}{(1+x) \log (1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+\int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+\int \left (\frac {e^x \left (4 x+4 x \log (1+x)+4 x^2 \log (1+x)+\log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )+x \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )}{x (1+x) \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}+e^x \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )\right ) \, dx\\ &=-\left (3 \int \left (\frac {e^x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}-\frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}\right ) \, dx\right )-4 \int \left (-\frac {e^x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}+\frac {e^x x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}+\frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}\right ) \, dx-4 \int \left (-\frac {e^x}{(1+x) \log (1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}-\frac {e^x}{\log (1+x) \left (-x+e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}\right ) \, dx+\int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+\int \frac {e^x \left (4 x+4 x \log (1+x)+4 x^2 \log (1+x)+\log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )+x \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )}{x (1+x) \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+\int e^x \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right ) \, dx\\ &=e^x \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )-3 \int \frac {e^x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+3 \int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx-4 \int \frac {e^x x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx-4 \int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{(1+x) \log (1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{\log (1+x) \left (-x+e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx-\int e^x \left (\frac {1}{x}+\frac {1-\frac {4 e^{4 x} \log ^3(1+x)}{1+x}-4 e^{4 x} \log ^4(1+x)}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}\right ) \, dx+\int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+\int \frac {e^x \left (4 x+(1+x) \log (1+x) \left (4 x+\log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )\right )}{x (1+x) \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx\\ &=e^x \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )-3 \int \frac {e^x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+3 \int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx-4 \int \frac {e^x x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx-4 \int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{(1+x) \log (1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{\log (1+x) \left (-x+e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+\int \left (\frac {e^x}{x}+\frac {4 e^x (1+\log (1+x)+x \log (1+x))}{(1+x) \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}\right ) \, dx-\int \left (\frac {e^x}{x}-\frac {e^x \left (-1-x+4 e^{4 x} \log ^3(1+x)+4 e^{4 x} \log ^4(1+x)+4 e^{4 x} x \log ^4(1+x)\right )}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}\right ) \, dx+\int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx\\ &=e^x \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )-3 \int \frac {e^x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+3 \int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x (1+\log (1+x)+x \log (1+x))}{(1+x) \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx-4 \int \frac {e^x x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx-4 \int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{(1+x) \log (1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{\log (1+x) \left (-x+e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+\int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+\int \frac {e^x \left (-1-x+4 e^{4 x} \log ^3(1+x)+4 e^{4 x} \log ^4(1+x)+4 e^{4 x} x \log ^4(1+x)\right )}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx\\ &=e^x \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )-3 \int \frac {e^x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+3 \int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \left (\frac {e^x}{(1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}+\frac {e^x x}{(1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}+\frac {e^x}{(1+x) \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}\right ) \, dx+4 \int \frac {e^x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx-4 \int \frac {e^x x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx-4 \int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{(1+x) \log (1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{\log (1+x) \left (-x+e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+\int \left (-\frac {4 e^x (1+\log (1+x)+x \log (1+x))}{(1+x) \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}+\frac {e^x \left (4 x-\log (1+x)+3 x \log (1+x)+4 x^2 \log (1+x)\right )}{(1+x) \log (1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )}\right ) \, dx+\int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx\\ &=e^x \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right )-3 \int \frac {e^x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+3 \int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{(1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x x}{(1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{(1+x) \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx-4 \int \frac {e^x (1+\log (1+x)+x \log (1+x))}{(1+x) \log (1+x) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx-4 \int \frac {e^x x}{\left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx-4 \int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{(1+x) \log (1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+4 \int \frac {e^x}{\log (1+x) \left (-x+e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+\int \frac {e^x}{(1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx+\int \frac {e^x \left (4 x-\log (1+x)+3 x \log (1+x)+4 x^2 \log (1+x)\right )}{(1+x) \log (1+x) \left (x-e^{4 x} \log ^4(1+x)\right ) \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.27, size = 27, normalized size = 1.00 \begin {gather*} e^x \log \left (x \log \left (\frac {2}{3} \left (x-e^{4 x} \log ^4(1+x)\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-x - x^2) + 4*E^(5*x)*x*Log[1 + x]^3 + E^(5*x)*(4*x + 4*x^2)*Log[1 + x]^4 + (E^x*(-x - x^2) +
E^(5*x)*(1 + x)*Log[1 + x]^4)*Log[(2*x - 2*E^(4*x)*Log[1 + x]^4)/3] + (E^x*(-x^2 - x^3) + E^(5*x)*(x + x^2)*Lo
g[1 + x]^4)*Log[(2*x - 2*E^(4*x)*Log[1 + x]^4)/3]*Log[x*Log[(2*x - 2*E^(4*x)*Log[1 + x]^4)/3]])/((-x^2 - x^3 +
E^(4*x)*(x + x^2)*Log[1 + x]^4)*Log[(2*x - 2*E^(4*x)*Log[1 + x]^4)/3]),x]

[Out]

E^x*Log[x*Log[(2*(x - E^(4*x)*Log[1 + x]^4))/3]]

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fricas [A]  time = 0.71, size = 23, normalized size = 0.85 \begin {gather*} e^{x} \log \left (x \log \left (-\frac {2}{3} \, e^{\left (4 \, x\right )} \log \left (x + 1\right )^{4} + \frac {2}{3} \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+x)*exp(x)^5*log(x+1)^4+(-x^3-x^2)*exp(x))*log(-2/3*exp(x)^4*log(x+1)^4+2/3*x)*log(x*log(-2/3*
exp(x)^4*log(x+1)^4+2/3*x))+((x+1)*exp(x)^5*log(x+1)^4+(-x^2-x)*exp(x))*log(-2/3*exp(x)^4*log(x+1)^4+2/3*x)+(4
*x^2+4*x)*exp(x)^5*log(x+1)^4+4*x*exp(x)^5*log(x+1)^3+(-x^2-x)*exp(x))/((x^2+x)*exp(x)^4*log(x+1)^4-x^3-x^2)/l
og(-2/3*exp(x)^4*log(x+1)^4+2/3*x),x, algorithm="fricas")

[Out]

e^x*log(x*log(-2/3*e^(4*x)*log(x + 1)^4 + 2/3*x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, {\left (x^{2} + x\right )} e^{\left (5 \, x\right )} \log \left (x + 1\right )^{4} + 4 \, x e^{\left (5 \, x\right )} \log \left (x + 1\right )^{3} + {\left ({\left (x^{2} + x\right )} e^{\left (5 \, x\right )} \log \left (x + 1\right )^{4} - {\left (x^{3} + x^{2}\right )} e^{x}\right )} \log \left (-\frac {2}{3} \, e^{\left (4 \, x\right )} \log \left (x + 1\right )^{4} + \frac {2}{3} \, x\right ) \log \left (x \log \left (-\frac {2}{3} \, e^{\left (4 \, x\right )} \log \left (x + 1\right )^{4} + \frac {2}{3} \, x\right )\right ) - {\left (x^{2} + x\right )} e^{x} + {\left ({\left (x + 1\right )} e^{\left (5 \, x\right )} \log \left (x + 1\right )^{4} - {\left (x^{2} + x\right )} e^{x}\right )} \log \left (-\frac {2}{3} \, e^{\left (4 \, x\right )} \log \left (x + 1\right )^{4} + \frac {2}{3} \, x\right )}{{\left ({\left (x^{2} + x\right )} e^{\left (4 \, x\right )} \log \left (x + 1\right )^{4} - x^{3} - x^{2}\right )} \log \left (-\frac {2}{3} \, e^{\left (4 \, x\right )} \log \left (x + 1\right )^{4} + \frac {2}{3} \, x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+x)*exp(x)^5*log(x+1)^4+(-x^3-x^2)*exp(x))*log(-2/3*exp(x)^4*log(x+1)^4+2/3*x)*log(x*log(-2/3*
exp(x)^4*log(x+1)^4+2/3*x))+((x+1)*exp(x)^5*log(x+1)^4+(-x^2-x)*exp(x))*log(-2/3*exp(x)^4*log(x+1)^4+2/3*x)+(4
*x^2+4*x)*exp(x)^5*log(x+1)^4+4*x*exp(x)^5*log(x+1)^3+(-x^2-x)*exp(x))/((x^2+x)*exp(x)^4*log(x+1)^4-x^3-x^2)/l
og(-2/3*exp(x)^4*log(x+1)^4+2/3*x),x, algorithm="giac")

[Out]

integrate((4*(x^2 + x)*e^(5*x)*log(x + 1)^4 + 4*x*e^(5*x)*log(x + 1)^3 + ((x^2 + x)*e^(5*x)*log(x + 1)^4 - (x^
3 + x^2)*e^x)*log(-2/3*e^(4*x)*log(x + 1)^4 + 2/3*x)*log(x*log(-2/3*e^(4*x)*log(x + 1)^4 + 2/3*x)) - (x^2 + x)
*e^x + ((x + 1)*e^(5*x)*log(x + 1)^4 - (x^2 + x)*e^x)*log(-2/3*e^(4*x)*log(x + 1)^4 + 2/3*x))/(((x^2 + x)*e^(4
*x)*log(x + 1)^4 - x^3 - x^2)*log(-2/3*e^(4*x)*log(x + 1)^4 + 2/3*x)), x)

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maple [C]  time = 0.30, size = 198, normalized size = 7.33

method	result	size
risch	\({\mathrm e}^{x} \ln \left (\ln \left (-\frac {2 \,{\mathrm e}^{4 x} \ln \left (x +1\right )^{4}}{3}+\frac {2 x}{3}\right )\right )+{\mathrm e}^{x} \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (-\frac {2 \,{\mathrm e}^{4 x} \ln \left (x +1\right )^{4}}{3}+\frac {2 x}{3}\right )\right ) \mathrm {csgn}\left (i x \ln \left (-\frac {2 \,{\mathrm e}^{4 x} \ln \left (x +1\right )^{4}}{3}+\frac {2 x}{3}\right )\right ) {\mathrm e}^{x}}{2}+\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (-\frac {2 \,{\mathrm e}^{4 x} \ln \left (x +1\right )^{4}}{3}+\frac {2 x}{3}\right )\right )^{2} {\mathrm e}^{x}}{2}+\frac {i \pi \,\mathrm {csgn}\left (i \ln \left (-\frac {2 \,{\mathrm e}^{4 x} \ln \left (x +1\right )^{4}}{3}+\frac {2 x}{3}\right )\right ) \mathrm {csgn}\left (i x \ln \left (-\frac {2 \,{\mathrm e}^{4 x} \ln \left (x +1\right )^{4}}{3}+\frac {2 x}{3}\right )\right )^{2} {\mathrm e}^{x}}{2}-\frac {i \pi \mathrm {csgn}\left (i x \ln \left (-\frac {2 \,{\mathrm e}^{4 x} \ln \left (x +1\right )^{4}}{3}+\frac {2 x}{3}\right )\right )^{3} {\mathrm e}^{x}}{2}\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2+x)*exp(x)^5*ln(x+1)^4+(-x^3-x^2)*exp(x))*ln(-2/3*exp(x)^4*ln(x+1)^4+2/3*x)*ln(x*ln(-2/3*exp(x)^4*ln
(x+1)^4+2/3*x))+((x+1)*exp(x)^5*ln(x+1)^4+(-x^2-x)*exp(x))*ln(-2/3*exp(x)^4*ln(x+1)^4+2/3*x)+(4*x^2+4*x)*exp(x
)^5*ln(x+1)^4+4*x*exp(x)^5*ln(x+1)^3+(-x^2-x)*exp(x))/((x^2+x)*exp(x)^4*ln(x+1)^4-x^3-x^2)/ln(-2/3*exp(x)^4*ln
(x+1)^4+2/3*x),x,method=_RETURNVERBOSE)

[Out]

exp(x)*ln(ln(-2/3*exp(4*x)*ln(x+1)^4+2/3*x))+exp(x)*ln(x)-1/2*I*Pi*csgn(I*x)*csgn(I*ln(-2/3*exp(4*x)*ln(x+1)^4
+2/3*x))*csgn(I*x*ln(-2/3*exp(4*x)*ln(x+1)^4+2/3*x))*exp(x)+1/2*I*Pi*csgn(I*x)*csgn(I*x*ln(-2/3*exp(4*x)*ln(x+
1)^4+2/3*x))^2*exp(x)+1/2*I*Pi*csgn(I*ln(-2/3*exp(4*x)*ln(x+1)^4+2/3*x))*csgn(I*x*ln(-2/3*exp(4*x)*ln(x+1)^4+2
/3*x))^2*exp(x)-1/2*I*Pi*csgn(I*x*ln(-2/3*exp(4*x)*ln(x+1)^4+2/3*x))^3*exp(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{x} \log \relax (x) + e^{x} \log \left (-\log \relax (3) + \log \relax (2) + \log \left (-e^{\left (4 \, x\right )} \log \left (x + 1\right )^{4} + x\right )\right ) + \int \frac {4 \, {\left (x + 1\right )} e^{\left (5 \, x\right )} \log \left (x + 1\right )^{4} + 4 \, e^{\left (5 \, x\right )} \log \left (x + 1\right )^{3} - {\left (x + 1\right )} e^{x}}{{\left (i \, \pi + {\left (i \, \pi - \log \relax (3) + \log \relax (2)\right )} x - \log \relax (3) + \log \relax (2)\right )} e^{\left (4 \, x\right )} \log \left (x + 1\right )^{4} + {\left (-i \, \pi + \log \relax (3) - \log \relax (2)\right )} x^{2} + {\left (-i \, \pi + \log \relax (3) - \log \relax (2)\right )} x + {\left ({\left (x + 1\right )} e^{\left (4 \, x\right )} \log \left (x + 1\right )^{4} - x^{2} - x\right )} \log \left (e^{\left (4 \, x\right )} \log \left (x + 1\right )^{4} - x\right )}\,{d x} - \int -\frac {4 \, {\left (x + 1\right )} e^{\left (5 \, x\right )} \log \left (x + 1\right )^{4} + 4 \, e^{\left (5 \, x\right )} \log \left (x + 1\right )^{3} - {\left (x + 1\right )} e^{x}}{{\left (x {\left (\log \relax (3) - \log \relax (2)\right )} + \log \relax (3) - \log \relax (2)\right )} e^{\left (4 \, x\right )} \log \left (x + 1\right )^{4} - x^{2} {\left (\log \relax (3) - \log \relax (2)\right )} - x {\left (\log \relax (3) - \log \relax (2)\right )} - {\left ({\left (x + 1\right )} e^{\left (4 \, x\right )} \log \left (x + 1\right )^{4} - x^{2} - x\right )} \log \left (-e^{\left (4 \, x\right )} \log \left (x + 1\right )^{4} + x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+x)*exp(x)^5*log(x+1)^4+(-x^3-x^2)*exp(x))*log(-2/3*exp(x)^4*log(x+1)^4+2/3*x)*log(x*log(-2/3*
exp(x)^4*log(x+1)^4+2/3*x))+((x+1)*exp(x)^5*log(x+1)^4+(-x^2-x)*exp(x))*log(-2/3*exp(x)^4*log(x+1)^4+2/3*x)+(4
*x^2+4*x)*exp(x)^5*log(x+1)^4+4*x*exp(x)^5*log(x+1)^3+(-x^2-x)*exp(x))/((x^2+x)*exp(x)^4*log(x+1)^4-x^3-x^2)/l
og(-2/3*exp(x)^4*log(x+1)^4+2/3*x),x, algorithm="maxima")

[Out]

e^x*log(x) + e^x*log(-log(3) + log(2) + log(-e^(4*x)*log(x + 1)^4 + x)) + integrate((4*(x + 1)*e^(5*x)*log(x +
1)^4 + 4*e^(5*x)*log(x + 1)^3 - (x + 1)*e^x)/((I*pi + (I*pi - log(3) + log(2))*x - log(3) + log(2))*e^(4*x)*l
og(x + 1)^4 + (-I*pi + log(3) - log(2))*x^2 + (-I*pi + log(3) - log(2))*x + ((x + 1)*e^(4*x)*log(x + 1)^4 - x^
2 - x)*log(e^(4*x)*log(x + 1)^4 - x)), x) - integrate(-(4*(x + 1)*e^(5*x)*log(x + 1)^4 + 4*e^(5*x)*log(x + 1)^
3 - (x + 1)*e^x)/((x*(log(3) - log(2)) + log(3) - log(2))*e^(4*x)*log(x + 1)^4 - x^2*(log(3) - log(2)) - x*(lo
g(3) - log(2)) - ((x + 1)*e^(4*x)*log(x + 1)^4 - x^2 - x)*log(-e^(4*x)*log(x + 1)^4 + x)), x)

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mupad [B]  time = 4.27, size = 23, normalized size = 0.85 \begin {gather*} \ln \left (x\,\ln \left (\frac {2\,x}{3}-\frac {2\,{\ln \left (x+1\right )}^4\,{\mathrm {e}}^{4\,x}}{3}\right )\right )\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((2*x)/3 - (2*log(x + 1)^4*exp(4*x))/3)*(exp(x)*(x + x^2) - log(x + 1)^4*exp(5*x)*(x + 1)) + exp(x)*(x
+ x^2) + log(x*log((2*x)/3 - (2*log(x + 1)^4*exp(4*x))/3))*log((2*x)/3 - (2*log(x + 1)^4*exp(4*x))/3)*(exp(x)
*(x^2 + x^3) - log(x + 1)^4*exp(5*x)*(x + x^2)) - log(x + 1)^4*exp(5*x)*(4*x + 4*x^2) - 4*x*log(x + 1)^3*exp(5
*x))/(log((2*x)/3 - (2*log(x + 1)^4*exp(4*x))/3)*(x^2 + x^3 - log(x + 1)^4*exp(4*x)*(x + x^2))),x)

[Out]

log(x*log((2*x)/3 - (2*log(x + 1)^4*exp(4*x))/3))*exp(x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2+x)*exp(x)**5*ln(x+1)**4+(-x**3-x**2)*exp(x))*ln(-2/3*exp(x)**4*ln(x+1)**4+2/3*x)*ln(x*ln(-2/
3*exp(x)**4*ln(x+1)**4+2/3*x))+((x+1)*exp(x)**5*ln(x+1)**4+(-x**2-x)*exp(x))*ln(-2/3*exp(x)**4*ln(x+1)**4+2/3*
x)+(4*x**2+4*x)*exp(x)**5*ln(x+1)**4+4*x*exp(x)**5*ln(x+1)**3+(-x**2-x)*exp(x))/((x**2+x)*exp(x)**4*ln(x+1)**4
-x**3-x**2)/ln(-2/3*exp(x)**4*ln(x+1)**4+2/3*x),x)

[Out]

Timed out

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