3.46.89 \(\int \frac {-8 x-8 x^2+8 x \log (5)+(16 x+20 x^2+e^5 (4+8 x)+(-4 e^5-16 x) \log (5)) \log (e^5+x)+(e^5 (-12-24 x)-12 x-24 x^2+(12 e^5+12 x) \log (5)) \log ^2(e^5+x)+(9 x+16 x^2-3 x^3+e^5 (9+16 x-3 x^2)+(-9 x+2 x^2+e^5 (-9+2 x)) \log (5)) \log ^3(e^5+x)}{(e^5+x) \log ^3(e^5+x)} \, dx\)

Optimal. Leaf size=32 \[ \left (-x-x^2+x \log (5)\right ) \left (x-\left (-3+\frac {2}{\log \left (e^5+x\right )}\right )^2\right ) \]

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Rubi [C]  time = 0.79, antiderivative size = 267, normalized size of antiderivative = 8.34, number of steps used = 51, number of rules used = 13, integrand size = 164, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {6, 6688, 2418, 2400, 2399, 2389, 2298, 2390, 2309, 2178, 2297, 2302, 30} \begin {gather*} -16 e^5 \text {li}\left (x+e^5\right )-4 \left (1-e^5-\log (5)\right ) \text {li}\left (x+e^5\right )-12 \left (1-2 e^5-\log (5)\right ) \text {li}\left (x+e^5\right )+4 \left (4-3 e^5-4 \log (5)\right ) \text {li}\left (x+e^5\right )-x^3+x^2 (8+\log (5))+\frac {4 \left (x+e^5\right ) x}{\log ^2\left (x+e^5\right )}+\frac {4 \left (x+e^5\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (x+e^5\right )}-\frac {4 e^5 \left (1-e^5-\log (5)\right )}{\log ^2\left (x+e^5\right )}-\frac {12 \left (x+e^5\right ) x}{\log \left (x+e^5\right )}+9 x (1-\log (5))+\frac {4 e^5 \left (x+e^5\right )}{\log \left (x+e^5\right )}-\frac {4 \left (x+e^5\right ) \left (4-3 e^5-\log (625)\right )}{\log \left (x+e^5\right )}+\frac {4 \left (x+e^5\right ) \left (1-e^5-\log (5)\right )}{\log \left (x+e^5\right )}+\frac {12 e^5 \left (1-e^5-\log (5)\right )}{\log \left (x+e^5\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8*x - 8*x^2 + 8*x*Log[5] + (16*x + 20*x^2 + E^5*(4 + 8*x) + (-4*E^5 - 16*x)*Log[5])*Log[E^5 + x] + (E^5*
(-12 - 24*x) - 12*x - 24*x^2 + (12*E^5 + 12*x)*Log[5])*Log[E^5 + x]^2 + (9*x + 16*x^2 - 3*x^3 + E^5*(9 + 16*x
- 3*x^2) + (-9*x + 2*x^2 + E^5*(-9 + 2*x))*Log[5])*Log[E^5 + x]^3)/((E^5 + x)*Log[E^5 + x]^3),x]

[Out]

-x^3 + 9*x*(1 - Log[5]) + x^2*(8 + Log[5]) + (4*x*(E^5 + x))/Log[E^5 + x]^2 - (4*E^5*(1 - E^5 - Log[5]))/Log[E
^5 + x]^2 + (4*(E^5 + x)*(1 - E^5 - Log[5]))/Log[E^5 + x]^2 + (4*E^5*(E^5 + x))/Log[E^5 + x] - (12*x*(E^5 + x)
)/Log[E^5 + x] + (12*E^5*(1 - E^5 - Log[5]))/Log[E^5 + x] + (4*(E^5 + x)*(1 - E^5 - Log[5]))/Log[E^5 + x] - (4
*(E^5 + x)*(4 - 3*E^5 - Log[625]))/Log[E^5 + x] - 16*E^5*LogIntegral[E^5 + x] + 4*(4 - 3*E^5 - 4*Log[5])*LogIn
tegral[E^5 + x] - 12*(1 - 2*E^5 - Log[5])*LogIntegral[E^5 + x] - 4*(1 - E^5 - Log[5])*LogIntegral[E^5 + x]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2399

Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)), x_Symbol] :> Int[ExpandIn
tegrand[(f + g*x)^q/(a + b*Log[c*(d + e*x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && IGtQ[q, 0]

Rule 2400

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
d + e*x)*(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1))/(b*e*n*(p + 1)), x] + (-Dist[(q + 1)/(b*n*(p + 1)), I
nt[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Dist[(q*(e*f - d*g))/(b*e*n*(p + 1)), Int[(f + g*x
)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && LtQ[p, -1] && GtQ[q, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8 x^2+x (-8+8 \log (5))+\left (16 x+20 x^2+e^5 (4+8 x)+\left (-4 e^5-16 x\right ) \log (5)\right ) \log \left (e^5+x\right )+\left (e^5 (-12-24 x)-12 x-24 x^2+\left (12 e^5+12 x\right ) \log (5)\right ) \log ^2\left (e^5+x\right )+\left (9 x+16 x^2-3 x^3+e^5 \left (9+16 x-3 x^2\right )+\left (-9 x+2 x^2+e^5 (-9+2 x)\right ) \log (5)\right ) \log ^3\left (e^5+x\right )}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx\\ &=\int \left (9-3 x^2-9 \log (5)+2 x (8+\log (5))-\frac {8 x (1+x-\log (5))}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )}+\frac {4 \left (5 x^2+2 x \left (2+e^5-2 \log (5)\right )+e^5 (1-\log (5))\right )}{\left (e^5+x\right ) \log ^2\left (e^5+x\right )}-\frac {12 (1+2 x-\log (5))}{\log \left (e^5+x\right )}\right ) \, dx\\ &=-x^3+9 x (1-\log (5))+x^2 (8+\log (5))+4 \int \frac {5 x^2+2 x \left (2+e^5-2 \log (5)\right )+e^5 (1-\log (5))}{\left (e^5+x\right ) \log ^2\left (e^5+x\right )} \, dx-8 \int \frac {x (1+x-\log (5))}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx-12 \int \frac {1+2 x-\log (5)}{\log \left (e^5+x\right )} \, dx\\ &=-x^3+9 x (1-\log (5))+x^2 (8+\log (5))+4 \int \left (\frac {5 x}{\log ^2\left (e^5+x\right )}+\frac {4 \left (1-\frac {3 e^5}{4}-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {3 e^5 \left (-1+e^5+\log (5)\right )}{\left (e^5+x\right ) \log ^2\left (e^5+x\right )}\right ) \, dx-8 \int \left (\frac {x}{\log ^3\left (e^5+x\right )}+\frac {1-e^5-\log (5)}{\log ^3\left (e^5+x\right )}+\frac {e^5 \left (-1+e^5+\log (5)\right )}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )}\right ) \, dx-12 \int \left (\frac {2 \left (e^5+x\right )}{\log \left (e^5+x\right )}+\frac {1-2 e^5-\log (5)}{\log \left (e^5+x\right )}\right ) \, dx\\ &=-x^3+9 x (1-\log (5))+x^2 (8+\log (5))-8 \int \frac {x}{\log ^3\left (e^5+x\right )} \, dx+20 \int \frac {x}{\log ^2\left (e^5+x\right )} \, dx-24 \int \frac {e^5+x}{\log \left (e^5+x\right )} \, dx+\left (4 \left (4-3 e^5-4 \log (5)\right )\right ) \int \frac {1}{\log ^2\left (e^5+x\right )} \, dx-\left (12 \left (1-2 e^5-\log (5)\right )\right ) \int \frac {1}{\log \left (e^5+x\right )} \, dx-\left (8 \left (1-e^5-\log (5)\right )\right ) \int \frac {1}{\log ^3\left (e^5+x\right )} \, dx+\left (8 e^5 \left (1-e^5-\log (5)\right )\right ) \int \frac {1}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx-\left (12 e^5 \left (1-e^5-\log (5)\right )\right ) \int \frac {1}{\left (e^5+x\right ) \log ^2\left (e^5+x\right )} \, dx\\ &=-x^3+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}-\frac {20 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-8 \int \frac {x}{\log ^2\left (e^5+x\right )} \, dx-24 \operatorname {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,e^5+x\right )+40 \int \frac {x}{\log \left (e^5+x\right )} \, dx-\left (4 e^5\right ) \int \frac {1}{\log ^2\left (e^5+x\right )} \, dx+\left (20 e^5\right ) \int \frac {1}{\log \left (e^5+x\right )} \, dx+\left (4 \left (4-3 e^5-4 \log (5)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,e^5+x\right )-\left (12 \left (1-2 e^5-\log (5)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )-\left (8 \left (1-e^5-\log (5)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\log ^3(x)} \, dx,x,e^5+x\right )+\left (8 e^5 \left (1-e^5-\log (5)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \log ^3(x)} \, dx,x,e^5+x\right )-\left (12 e^5 \left (1-e^5-\log (5)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,e^5+x\right )\\ &=-x^3+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}-\frac {12 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {4 \left (e^5+x\right ) \left (4-3 e^5-4 \log (5)\right )}{\log \left (e^5+x\right )}-12 \left (1-2 e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-16 \int \frac {x}{\log \left (e^5+x\right )} \, dx-24 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log \left (e^5+x\right )\right )+40 \int \left (-\frac {e^5}{\log \left (e^5+x\right )}+\frac {e^5+x}{\log \left (e^5+x\right )}\right ) \, dx-\left (4 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,e^5+x\right )-\left (8 e^5\right ) \int \frac {1}{\log \left (e^5+x\right )} \, dx+\left (20 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )+\left (4 \left (4-3 e^5-4 \log (5)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )-\left (4 \left (1-e^5-\log (5)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,e^5+x\right )+\left (8 e^5 \left (1-e^5-\log (5)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log \left (e^5+x\right )\right )-\left (12 e^5 \left (1-e^5-\log (5)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (e^5+x\right )\right )\\ &=-x^3-24 \text {Ei}\left (2 \log \left (e^5+x\right )\right )+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}-\frac {4 e^5 \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 e^5 \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {12 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {4 \left (e^5+x\right ) \left (4-3 e^5-4 \log (5)\right )}{\log \left (e^5+x\right )}+\frac {12 e^5 \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+20 e^5 \text {li}\left (e^5+x\right )+4 \left (4-3 e^5-4 \log (5)\right ) \text {li}\left (e^5+x\right )-12 \left (1-2 e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-16 \int \left (-\frac {e^5}{\log \left (e^5+x\right )}+\frac {e^5+x}{\log \left (e^5+x\right )}\right ) \, dx+40 \int \frac {e^5+x}{\log \left (e^5+x\right )} \, dx-\left (4 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )-\left (8 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )-\left (40 e^5\right ) \int \frac {1}{\log \left (e^5+x\right )} \, dx-\left (4 \left (1-e^5-\log (5)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )\\ &=-x^3-24 \text {Ei}\left (2 \log \left (e^5+x\right )\right )+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}-\frac {4 e^5 \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 e^5 \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {12 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {4 \left (e^5+x\right ) \left (4-3 e^5-4 \log (5)\right )}{\log \left (e^5+x\right )}+\frac {12 e^5 \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+8 e^5 \text {li}\left (e^5+x\right )+4 \left (4-3 e^5-4 \log (5)\right ) \text {li}\left (e^5+x\right )-12 \left (1-2 e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-4 \left (1-e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-16 \int \frac {e^5+x}{\log \left (e^5+x\right )} \, dx+40 \operatorname {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,e^5+x\right )+\left (16 e^5\right ) \int \frac {1}{\log \left (e^5+x\right )} \, dx-\left (40 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )\\ &=-x^3-24 \text {Ei}\left (2 \log \left (e^5+x\right )\right )+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}-\frac {4 e^5 \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 e^5 \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {12 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {4 \left (e^5+x\right ) \left (4-3 e^5-4 \log (5)\right )}{\log \left (e^5+x\right )}+\frac {12 e^5 \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}-32 e^5 \text {li}\left (e^5+x\right )+4 \left (4-3 e^5-4 \log (5)\right ) \text {li}\left (e^5+x\right )-12 \left (1-2 e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-4 \left (1-e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-16 \operatorname {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,e^5+x\right )+40 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log \left (e^5+x\right )\right )+\left (16 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )\\ &=-x^3+16 \text {Ei}\left (2 \log \left (e^5+x\right )\right )+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}-\frac {4 e^5 \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 e^5 \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {12 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {4 \left (e^5+x\right ) \left (4-3 e^5-4 \log (5)\right )}{\log \left (e^5+x\right )}+\frac {12 e^5 \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}-16 e^5 \text {li}\left (e^5+x\right )+4 \left (4-3 e^5-4 \log (5)\right ) \text {li}\left (e^5+x\right )-12 \left (1-2 e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-4 \left (1-e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-16 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log \left (e^5+x\right )\right )\\ &=-x^3+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}-\frac {4 e^5 \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 e^5 \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {12 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {4 \left (e^5+x\right ) \left (4-3 e^5-4 \log (5)\right )}{\log \left (e^5+x\right )}+\frac {12 e^5 \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}-16 e^5 \text {li}\left (e^5+x\right )+4 \left (4-3 e^5-4 \log (5)\right ) \text {li}\left (e^5+x\right )-12 \left (1-2 e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-4 \left (1-e^5-\log (5)\right ) \text {li}\left (e^5+x\right )\\ \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.32, size = 76, normalized size = 2.38 \begin {gather*} 12 \text {Ei}\left (\log \left (e^5+x\right )\right ) \left (-1+2 e^5+\log (5)\right )-\frac {x (1+x-\log (5)) \left (-4+12 \log \left (e^5+x\right )+(-9+x) \log ^2\left (e^5+x\right )\right )}{\log ^2\left (e^5+x\right )}-12 \left (-1+2 e^5+\log (5)\right ) \text {li}\left (e^5+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*x - 8*x^2 + 8*x*Log[5] + (16*x + 20*x^2 + E^5*(4 + 8*x) + (-4*E^5 - 16*x)*Log[5])*Log[E^5 + x] +
(E^5*(-12 - 24*x) - 12*x - 24*x^2 + (12*E^5 + 12*x)*Log[5])*Log[E^5 + x]^2 + (9*x + 16*x^2 - 3*x^3 + E^5*(9 +
16*x - 3*x^2) + (-9*x + 2*x^2 + E^5*(-9 + 2*x))*Log[5])*Log[E^5 + x]^3)/((E^5 + x)*Log[E^5 + x]^3),x]

[Out]

12*ExpIntegralEi[Log[E^5 + x]]*(-1 + 2*E^5 + Log[5]) - (x*(1 + x - Log[5])*(-4 + 12*Log[E^5 + x] + (-9 + x)*Lo
g[E^5 + x]^2))/Log[E^5 + x]^2 - 12*(-1 + 2*E^5 + Log[5])*LogIntegral[E^5 + x]

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fricas [B]  time = 0.98, size = 71, normalized size = 2.22 \begin {gather*} -\frac {{\left (x^{3} - 8 \, x^{2} - {\left (x^{2} - 9 \, x\right )} \log \relax (5) - 9 \, x\right )} \log \left (x + e^{5}\right )^{2} - 4 \, x^{2} + 4 \, x \log \relax (5) + 12 \, {\left (x^{2} - x \log \relax (5) + x\right )} \log \left (x + e^{5}\right ) - 4 \, x}{\log \left (x + e^{5}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x-9)*exp(5)+2*x^2-9*x)*log(5)+(-3*x^2+16*x+9)*exp(5)-3*x^3+16*x^2+9*x)*log(exp(5)+x)^3+((12*ex
p(5)+12*x)*log(5)+(-24*x-12)*exp(5)-24*x^2-12*x)*log(exp(5)+x)^2+((-4*exp(5)-16*x)*log(5)+(8*x+4)*exp(5)+20*x^
2+16*x)*log(exp(5)+x)+8*x*log(5)-8*x^2-8*x)/(exp(5)+x)/log(exp(5)+x)^3,x, algorithm="fricas")

[Out]

-((x^3 - 8*x^2 - (x^2 - 9*x)*log(5) - 9*x)*log(x + e^5)^2 - 4*x^2 + 4*x*log(5) + 12*(x^2 - x*log(5) + x)*log(x
+ e^5) - 4*x)/log(x + e^5)^2

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giac [B]  time = 0.27, size = 110, normalized size = 3.44 \begin {gather*} -\frac {x^{3} \log \left (x + e^{5}\right )^{2} - x^{2} \log \relax (5) \log \left (x + e^{5}\right )^{2} - 8 \, x^{2} \log \left (x + e^{5}\right )^{2} + 9 \, x \log \relax (5) \log \left (x + e^{5}\right )^{2} + 12 \, x^{2} \log \left (x + e^{5}\right ) - 12 \, x \log \relax (5) \log \left (x + e^{5}\right ) - 9 \, x \log \left (x + e^{5}\right )^{2} - 4 \, x^{2} + 4 \, x \log \relax (5) + 12 \, x \log \left (x + e^{5}\right ) - 4 \, x}{\log \left (x + e^{5}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x-9)*exp(5)+2*x^2-9*x)*log(5)+(-3*x^2+16*x+9)*exp(5)-3*x^3+16*x^2+9*x)*log(exp(5)+x)^3+((12*ex
p(5)+12*x)*log(5)+(-24*x-12)*exp(5)-24*x^2-12*x)*log(exp(5)+x)^2+((-4*exp(5)-16*x)*log(5)+(8*x+4)*exp(5)+20*x^
2+16*x)*log(exp(5)+x)+8*x*log(5)-8*x^2-8*x)/(exp(5)+x)/log(exp(5)+x)^3,x, algorithm="giac")

[Out]

-(x^3*log(x + e^5)^2 - x^2*log(5)*log(x + e^5)^2 - 8*x^2*log(x + e^5)^2 + 9*x*log(5)*log(x + e^5)^2 + 12*x^2*l
og(x + e^5) - 12*x*log(5)*log(x + e^5) - 9*x*log(x + e^5)^2 - 4*x^2 + 4*x*log(5) + 12*x*log(x + e^5) - 4*x)/lo
g(x + e^5)^2

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maple [B]  time = 0.16, size = 67, normalized size = 2.09

method	result	size
risch	\(x^{2} \ln \relax (5)-x^{3}-9 x \ln \relax (5)+8 x^{2}+9 x +\frac {4 x \left (3 \ln \left ({\mathrm e}^{5}+x \right ) \ln \relax (5)-3 \ln \left ({\mathrm e}^{5}+x \right ) x -\ln \relax (5)+x -3 \ln \left ({\mathrm e}^{5}+x \right )+1\right )}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}\)	\(67\)
norman	\(\frac {\left (4-4 \ln \relax (5)\right ) x +\left (-12+12 \ln \relax (5)\right ) x \ln \left ({\mathrm e}^{5}+x \right )+\left (8+\ln \relax (5)\right ) x^{2} \ln \left ({\mathrm e}^{5}+x \right )^{2}+\left (-9 \ln \relax (5)+9\right ) x \ln \left ({\mathrm e}^{5}+x \right )^{2}+4 x^{2}-x^{3} \ln \left ({\mathrm e}^{5}+x \right )^{2}-12 \ln \left ({\mathrm e}^{5}+x \right ) x^{2}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}\)	\(88\)
derivativedivides	\(9 x -\left ({\mathrm e}^{5}+x \right )^{3}+9 \,{\mathrm e}^{5}+8 \left ({\mathrm e}^{5}+x \right )^{2}-3 \,{\mathrm e}^{10} \left ({\mathrm e}^{5}+x \right )+\frac {4 \,{\mathrm e}^{10}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {12 \,{\mathrm e}^{10}}{\ln \left ({\mathrm e}^{5}+x \right )}+8 \ln \relax (5) \left (-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )}-\frac {\expIntegralEi \left (1, -\ln \left ({\mathrm e}^{5}+x \right )\right )}{2}\right )+16 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )}-\frac {\expIntegralEi \left (1, -\ln \left ({\mathrm e}^{5}+x \right )\right )}{2}\right )-24 \,{\mathrm e}^{5} \expIntegralEi \left (1, -\ln \left ({\mathrm e}^{5}+x \right )\right )-12 \ln \relax (5) \expIntegralEi \left (1, -\ln \left ({\mathrm e}^{5}+x \right )\right )-32 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5}+x}{\ln \left ({\mathrm e}^{5}+x \right )}-\expIntegralEi \left (1, -\ln \left ({\mathrm e}^{5}+x \right )\right )\right )-16 \ln \relax (5) \left (-\frac {{\mathrm e}^{5}+x}{\ln \left ({\mathrm e}^{5}+x \right )}-\expIntegralEi \left (1, -\ln \left ({\mathrm e}^{5}+x \right )\right )\right )+\ln \relax (5) \left ({\mathrm e}^{5}+x \right )^{2}-9 \left ({\mathrm e}^{5}+x \right ) \ln \relax (5)-16 \,{\mathrm e}^{5} \left ({\mathrm e}^{5}+x \right )-\frac {12 \left ({\mathrm e}^{5}+x \right )}{\ln \left ({\mathrm e}^{5}+x \right )}+3 \,{\mathrm e}^{5} \left ({\mathrm e}^{5}+x \right )^{2}-\frac {12 \left ({\mathrm e}^{5}+x \right )^{2}}{\ln \left ({\mathrm e}^{5}+x \right )}+\frac {4 \left ({\mathrm e}^{5}+x \right )^{2}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}+\frac {12 \,{\mathrm e}^{5}}{\ln \left ({\mathrm e}^{5}+x \right )}-\frac {4 \,{\mathrm e}^{5}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}-2 \,{\mathrm e}^{5} \ln \relax (5) \left ({\mathrm e}^{5}+x \right )+\frac {4 \,{\mathrm e}^{5} \ln \relax (5)}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {12 \,{\mathrm e}^{5} \ln \relax (5)}{\ln \left ({\mathrm e}^{5}+x \right )}+\frac {4 \,{\mathrm e}^{5}+4 x}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}\)	\(378\)
default	\(9 x -\left ({\mathrm e}^{5}+x \right )^{3}+9 \,{\mathrm e}^{5}+8 \left ({\mathrm e}^{5}+x \right )^{2}-3 \,{\mathrm e}^{10} \left ({\mathrm e}^{5}+x \right )+\frac {4 \,{\mathrm e}^{10}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {12 \,{\mathrm e}^{10}}{\ln \left ({\mathrm e}^{5}+x \right )}+8 \ln \relax (5) \left (-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )}-\frac {\expIntegralEi \left (1, -\ln \left ({\mathrm e}^{5}+x \right )\right )}{2}\right )+16 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )}-\frac {\expIntegralEi \left (1, -\ln \left ({\mathrm e}^{5}+x \right )\right )}{2}\right )-24 \,{\mathrm e}^{5} \expIntegralEi \left (1, -\ln \left ({\mathrm e}^{5}+x \right )\right )-12 \ln \relax (5) \expIntegralEi \left (1, -\ln \left ({\mathrm e}^{5}+x \right )\right )-32 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5}+x}{\ln \left ({\mathrm e}^{5}+x \right )}-\expIntegralEi \left (1, -\ln \left ({\mathrm e}^{5}+x \right )\right )\right )-16 \ln \relax (5) \left (-\frac {{\mathrm e}^{5}+x}{\ln \left ({\mathrm e}^{5}+x \right )}-\expIntegralEi \left (1, -\ln \left ({\mathrm e}^{5}+x \right )\right )\right )+\ln \relax (5) \left ({\mathrm e}^{5}+x \right )^{2}-9 \left ({\mathrm e}^{5}+x \right ) \ln \relax (5)-16 \,{\mathrm e}^{5} \left ({\mathrm e}^{5}+x \right )-\frac {12 \left ({\mathrm e}^{5}+x \right )}{\ln \left ({\mathrm e}^{5}+x \right )}+3 \,{\mathrm e}^{5} \left ({\mathrm e}^{5}+x \right )^{2}-\frac {12 \left ({\mathrm e}^{5}+x \right )^{2}}{\ln \left ({\mathrm e}^{5}+x \right )}+\frac {4 \left ({\mathrm e}^{5}+x \right )^{2}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}+\frac {12 \,{\mathrm e}^{5}}{\ln \left ({\mathrm e}^{5}+x \right )}-\frac {4 \,{\mathrm e}^{5}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}-2 \,{\mathrm e}^{5} \ln \relax (5) \left ({\mathrm e}^{5}+x \right )+\frac {4 \,{\mathrm e}^{5} \ln \relax (5)}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {12 \,{\mathrm e}^{5} \ln \relax (5)}{\ln \left ({\mathrm e}^{5}+x \right )}+\frac {4 \,{\mathrm e}^{5}+4 x}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}\)	\(378\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((2*x-9)*exp(5)+2*x^2-9*x)*ln(5)+(-3*x^2+16*x+9)*exp(5)-3*x^3+16*x^2+9*x)*ln(exp(5)+x)^3+((12*exp(5)+12*
x)*ln(5)+(-24*x-12)*exp(5)-24*x^2-12*x)*ln(exp(5)+x)^2+((-4*exp(5)-16*x)*ln(5)+(8*x+4)*exp(5)+20*x^2+16*x)*ln(
exp(5)+x)+8*x*ln(5)-8*x^2-8*x)/(exp(5)+x)/ln(exp(5)+x)^3,x,method=_RETURNVERBOSE)

[Out]

x^2*ln(5)-x^3-9*x*ln(5)+8*x^2+9*x+4*x*(3*ln(exp(5)+x)*ln(5)-3*ln(exp(5)+x)*x-ln(5)+x-3*ln(exp(5)+x)+1)/ln(exp(
5)+x)^2

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maxima [B]  time = 0.49, size = 68, normalized size = 2.12 \begin {gather*} -\frac {{\left (x^{3} - x^{2} {\left (\log \relax (5) + 8\right )} + 9 \, x {\left (\log \relax (5) - 1\right )}\right )} \log \left (x + e^{5}\right )^{2} - 4 \, x^{2} + 4 \, x {\left (\log \relax (5) - 1\right )} + 12 \, {\left (x^{2} - x {\left (\log \relax (5) - 1\right )}\right )} \log \left (x + e^{5}\right )}{\log \left (x + e^{5}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x-9)*exp(5)+2*x^2-9*x)*log(5)+(-3*x^2+16*x+9)*exp(5)-3*x^3+16*x^2+9*x)*log(exp(5)+x)^3+((12*ex
p(5)+12*x)*log(5)+(-24*x-12)*exp(5)-24*x^2-12*x)*log(exp(5)+x)^2+((-4*exp(5)-16*x)*log(5)+(8*x+4)*exp(5)+20*x^
2+16*x)*log(exp(5)+x)+8*x*log(5)-8*x^2-8*x)/(exp(5)+x)/log(exp(5)+x)^3,x, algorithm="maxima")

[Out]

-((x^3 - x^2*(log(5) + 8) + 9*x*(log(5) - 1))*log(x + e^5)^2 - 4*x^2 + 4*x*(log(5) - 1) + 12*(x^2 - x*(log(5)
- 1))*log(x + e^5))/log(x + e^5)^2

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mupad [B]  time = 3.66, size = 93, normalized size = 2.91 \begin {gather*} 9\,x-\frac {12\,x^2}{\ln \left (x+{\mathrm {e}}^5\right )}+\frac {4\,x^2}{{\ln \left (x+{\mathrm {e}}^5\right )}^2}-9\,x\,\ln \relax (5)+x^2\,\ln \relax (5)+8\,x^2-x^3-\frac {12\,x}{\ln \left (x+{\mathrm {e}}^5\right )}+\frac {4\,x}{{\ln \left (x+{\mathrm {e}}^5\right )}^2}+\frac {12\,x\,\ln \relax (5)}{\ln \left (x+{\mathrm {e}}^5\right )}-\frac {4\,x\,\ln \relax (5)}{{\ln \left (x+{\mathrm {e}}^5\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x - log(x + exp(5))*(16*x - log(5)*(16*x + 4*exp(5)) + 20*x^2 + exp(5)*(8*x + 4)) - 8*x*log(5) - log(x
+ exp(5))^3*(9*x + exp(5)*(16*x - 3*x^2 + 9) + log(5)*(2*x^2 - 9*x + exp(5)*(2*x - 9)) + 16*x^2 - 3*x^3) + lo
g(x + exp(5))^2*(12*x - log(5)*(12*x + 12*exp(5)) + 24*x^2 + exp(5)*(24*x + 12)) + 8*x^2)/(log(x + exp(5))^3*(
x + exp(5))),x)

[Out]

9*x - (12*x^2)/log(x + exp(5)) + (4*x^2)/log(x + exp(5))^2 - 9*x*log(5) + x^2*log(5) + 8*x^2 - x^3 - (12*x)/lo
g(x + exp(5)) + (4*x)/log(x + exp(5))^2 + (12*x*log(5))/log(x + exp(5)) - (4*x*log(5))/log(x + exp(5))^2

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sympy [B]  time = 0.19, size = 65, normalized size = 2.03 \begin {gather*} - x^{3} + x^{2} \left (\log {\relax (5 )} + 8\right ) + x \left (9 - 9 \log {\relax (5 )}\right ) + \frac {4 x^{2} - 4 x \log {\relax (5 )} + 4 x + \left (- 12 x^{2} - 12 x + 12 x \log {\relax (5 )}\right ) \log {\left (x + e^{5} \right )}}{\log {\left (x + e^{5} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x-9)*exp(5)+2*x**2-9*x)*ln(5)+(-3*x**2+16*x+9)*exp(5)-3*x**3+16*x**2+9*x)*ln(exp(5)+x)**3+((12
*exp(5)+12*x)*ln(5)+(-24*x-12)*exp(5)-24*x**2-12*x)*ln(exp(5)+x)**2+((-4*exp(5)-16*x)*ln(5)+(8*x+4)*exp(5)+20*
x**2+16*x)*ln(exp(5)+x)+8*x*ln(5)-8*x**2-8*x)/(exp(5)+x)/ln(exp(5)+x)**3,x)

[Out]

-x**3 + x**2*(log(5) + 8) + x*(9 - 9*log(5)) + (4*x**2 - 4*x*log(5) + 4*x + (-12*x**2 - 12*x + 12*x*log(5))*lo
g(x + exp(5)))/log(x + exp(5))**2

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