\(\int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e (81-2187 x^2+729 x^5-81 x^8+3 x^{11})+(729-891 x^3+171 x^6-9 x^9+e (-2187 x^2+486 x^5-27 x^8)) \log (\frac {1}{3} (3 e+x))+(243-270 x^3+27 x^6+e (-729 x^2+81 x^5)) \log ^2(\frac {1}{3} (3 e+x))+(27-81 e x^2-27 x^3) \log ^3(\frac {1}{3} (3 e+x))}{81 e+27 x} \, dx\) [1333]

Optimal result

Integrand size = 167, antiderivative size = 27 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=x+\frac {1}{4} \left (-3+\frac {x^3}{3}-\log \left (e+\frac {x}{3}\right )\right )^4 \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1209\) vs. \(2(27)=54\).

Time = 1.37 (sec) , antiderivative size = 1209, normalized size of antiderivative = 44.78, number of steps used = 87, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6874, 45, 1864, 2465, 2436, 2332, 2442, 2437, 12, 2338, 2333, 2448, 2342, 2341, 2445, 2458, 2372, 2339, 30} \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx =\text {Too large to display} \]

[In]

Int[(729 + 27*x - 972*x^3 + 270*x^6 - 28*x^9 + x^12 + E*(81 - 2187*x^2 + 729*x^5 - 81*x^8 + 3*x^11) + (729 - 8
91*x^3 + 171*x^6 - 9*x^9 + E*(-2187*x^2 + 486*x^5 - 27*x^8))*Log[(3*E + x)/3] + (243 - 270*x^3 + 27*x^6 + E*(-
729*x^2 + 81*x^5))*Log[(3*E + x)/3]^2 + (27 - 81*E*x^2 - 27*x^3)*Log[(3*E + x)/3]^3)/(81*E + 27*x),x]

[Out]

x - 486*E^2*x - (52461*E^5*x)/20 - 6561*E^8*x - 6561*E^11*x + 243*E^2*(1 + 3*E^3)^3*x + (81*E^2*(2 + 3*E^3)*x)
/2 + 9*E^2*(11 + 3*E^3)*x + 54*E*x^2 + (14247*E^4*x^2)/40 + (2187*E^7*x^2)/2 + (2187*E^10*x^2)/2 - (81*E*(1 +
3*E^3)^3*x^2)/2 - (9*E*(2 + 3*E^3)*x^2)/4 - (3*E*(11 + 3*E^3)*x^2)/2 - 12*x^3 - (1583*E^3*x^3)/20 - 243*E^6*x^
3 - 243*E^9*x^3 + ((11 + 3*E^3)*x^3)/3 + 81*E^3*(1 + 3*E^3 + 3*E^6)*x^3 + (1583*E^2*x^4)/80 + (243*E^5*x^4)/4
+ (243*E^8*x^4)/4 - (81*E^2*(1 + 3*E^3 + 3*E^6)*x^4)/4 - (799*E*x^5)/150 - (81*E^4*x^5)/5 - (81*E^7*x^5)/5 + (
27*E*(1 + 3*E^3 + 3*E^6)*x^5)/5 + (161*x^6)/108 + (9*E^3*x^6)/2 + (9*E^6*x^6)/2 - (9*E^3*(1 + E^3)*x^6)/2 - (9
*E^2*x^7)/7 - (9*E^5*x^7)/7 + (9*E^2*(1 + E^3)*x^7)/7 + (3*E*x^8)/8 + (3*E^4*x^8)/8 - (3*E*(1 + E^3)*x^8)/8 -
x^9/9 + x^12/324 + (27*E*(3*E + x)^2)/2 + (405*E^4*(3*E + x)^2)/4 - (2*(3*E + x)^3)/3 - 20*E^3*(3*E + x)^3 + (
45*E^2*(3*E + x)^4)/16 - (6*E*(3*E + x)^5)/25 + (3*E + x)^6/108 + (9*E*(2 + 3*E^3)*x^2*Log[E + x/3])/2 - (11 +
3*E^3)*x^3*Log[E + x/3] + (3*E^2*x^4*Log[E + x/3])/4 - (E*x^5*Log[E + x/3])/5 + (19*x^6*Log[E + x/3])/18 - (x
^9*Log[E + x/3])/27 + 162*E^2*(3*E + x)*Log[E + x/3] + 486*E^5*(3*E + x)*Log[E + x/3] - 27*E^2*(2 + 3*E^3)*(3*
E + x)*Log[E + x/3] - 27*E*(3*E + x)^2*Log[E + x/3] - (405*E^4*(3*E + x)^2*Log[E + x/3])/2 + 2*(3*E + x)^3*Log
[E + x/3] + 60*E^3*(3*E + x)^3*Log[E + x/3] - (45*E^2*(3*E + x)^4*Log[E + x/3])/4 + (6*E*(3*E + x)^5*Log[E + x
/3])/5 - ((3*E + x)^6*Log[E + x/3])/18 - 90*E^3*Log[E + x/3]^2 - (243*E^6*Log[E + x/3]^2)/2 + (27*(1 + 3*E^3)^
2*Log[E + x/3]^2)/2 - (10*x^3*Log[E + x/3]^2)/3 + (x^6*Log[E + x/3]^2)/6 + 9*E^2*(3*E + x)*Log[E + x/3]^2 - 3*
E*(3*E + x)^2*Log[E + x/3]^2 + ((3*E + x)^3*Log[E + x/3]^2)/3 + 3*(1 + 3*E^3)*Log[E + x/3]^3 - 9*E^2*(3*E + x)
*Log[E + x/3]^3 + 3*E*(3*E + x)^2*Log[E + x/3]^3 - ((3*E + x)^3*Log[E + x/3]^3)/3 + Log[E + x/3]^4/4 + 27*Log[
3*E + x] - 3*E*Log[3*E + x] + 972*E^3*Log[3*E + x] + (128223*E^6*Log[3*E + x])/20 + 19683*E^9*Log[3*E + x] + 1
9683*E^12*Log[3*E + x] - (81*E^3*(2 + 3*E^3)*Log[3*E + x])/2 - 27*E^3*(11 + 3*E^3)*Log[3*E + x] + 3*E*(1 - 243
*E^2 - 2187*E^5 - 6561*E^8 - 6561*E^11)*Log[3*E + x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2339

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 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2436

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 2437

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 2442

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

Rule 2445

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f
+ g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^p/(g*(q + 1))), x] - Dist[b*e*n*(p/(g*(q + 1))), Int[(f + g*x)^(q
+ 1)*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2448

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

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2465

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {27}{3 e+x}+\frac {x}{3 e+x}-\frac {36 x^3}{3 e+x}+\frac {10 x^6}{3 e+x}-\frac {28 x^9}{27 (3 e+x)}+\frac {x^{12}}{27 (3 e+x)}+\frac {e \left (27-729 x^2+243 x^5-27 x^8+x^{11}\right )}{9 (3 e+x)}-\frac {\left (-9+x^3\right )^2 \left (-1+3 e x^2+x^3\right ) \log \left (e+\frac {x}{3}\right )}{3 (3 e+x)}+\frac {\left (-9+x^3\right ) \left (-1+3 e x^2+x^3\right ) \log ^2\left (e+\frac {x}{3}\right )}{3 e+x}-\frac {\left (-1+3 e x^2+x^3\right ) \log ^3\left (e+\frac {x}{3}\right )}{3 e+x}\right ) \, dx \\ & = 27 \log (3 e+x)+\frac {1}{27} \int \frac {x^{12}}{3 e+x} \, dx-\frac {1}{3} \int \frac {\left (-9+x^3\right )^2 \left (-1+3 e x^2+x^3\right ) \log \left (e+\frac {x}{3}\right )}{3 e+x} \, dx-\frac {28}{27} \int \frac {x^9}{3 e+x} \, dx+10 \int \frac {x^6}{3 e+x} \, dx-36 \int \frac {x^3}{3 e+x} \, dx+\frac {1}{9} e \int \frac {27-729 x^2+243 x^5-27 x^8+x^{11}}{3 e+x} \, dx+\int \frac {x}{3 e+x} \, dx+\int \frac {\left (-9+x^3\right ) \left (-1+3 e x^2+x^3\right ) \log ^2\left (e+\frac {x}{3}\right )}{3 e+x} \, dx-\int \frac {\left (-1+3 e x^2+x^3\right ) \log ^3\left (e+\frac {x}{3}\right )}{3 e+x} \, dx \\ & = 27 \log (3 e+x)+\frac {1}{27} \int \left (-177147 e^{11}+59049 e^{10} x-19683 e^9 x^2+6561 e^8 x^3-2187 e^7 x^4+729 e^6 x^5-243 e^5 x^6+81 e^4 x^7-27 e^3 x^8+9 e^2 x^9-3 e x^{10}+x^{11}+\frac {531441 e^{12}}{3 e+x}\right ) \, dx-\frac {1}{3} \int \left (81 e^2 \left (2+3 e^3\right ) \log \left (e+\frac {x}{3}\right )-27 e \left (2+3 e^3\right ) x \log \left (e+\frac {x}{3}\right )+9 \left (11+3 e^3\right ) x^2 \log \left (e+\frac {x}{3}\right )-9 e^2 x^3 \log \left (e+\frac {x}{3}\right )+3 e x^4 \log \left (e+\frac {x}{3}\right )-19 x^5 \log \left (e+\frac {x}{3}\right )+x^8 \log \left (e+\frac {x}{3}\right )-\frac {81 \left (1+3 e^3\right )^2 \log \left (e+\frac {x}{3}\right )}{3 e+x}\right ) \, dx-\frac {28}{27} \int \left (6561 e^8-2187 e^7 x+729 e^6 x^2-243 e^5 x^3+81 e^4 x^4-27 e^3 x^5+9 e^2 x^6-3 e x^7+x^8-\frac {19683 e^9}{3 e+x}\right ) \, dx+10 \int \left (-243 e^5+81 e^4 x-27 e^3 x^2+9 e^2 x^3-3 e x^4+x^5+\frac {729 e^6}{3 e+x}\right ) \, dx-36 \int \left (9 e^2-3 e x+x^2-\frac {27 e^3}{3 e+x}\right ) \, dx+\frac {1}{9} e \int \left (2187 e \left (1+3 e^3\right )^3-729 \left (1+3 e^3\right )^3 x+2187 e^2 \left (1+3 e^3+3 e^6\right ) x^2-729 e \left (1+3 e^3+3 e^6\right ) x^3+243 \left (1+3 e^3+3 e^6\right ) x^4-243 e^2 \left (1+e^3\right ) x^5+81 e \left (1+e^3\right ) x^6-27 \left (1+e^3\right ) x^7+9 e^2 x^8-3 e x^9+x^{10}-\frac {27 \left (-1+243 e^2+2187 e^5+6561 e^8+6561 e^{11}\right )}{3 e+x}\right ) \, dx+\int \left (1-\frac {3 e}{3 e+x}\right ) \, dx+\int \left (-9 e^2 \log ^2\left (e+\frac {x}{3}\right )+3 e x \log ^2\left (e+\frac {x}{3}\right )-10 x^2 \log ^2\left (e+\frac {x}{3}\right )+x^5 \log ^2\left (e+\frac {x}{3}\right )+\frac {9 \left (1+3 e^3\right ) \log ^2\left (e+\frac {x}{3}\right )}{3 e+x}\right ) \, dx-\int \left (\frac {\log ^3\left (e+\frac {x}{3}\right )}{-3 e-x}+x^2 \log ^3\left (e+\frac {x}{3}\right )\right ) \, dx \\ & = x-324 e^2 x-2430 e^5 x-6804 e^8 x-6561 e^{11} x+243 e^2 \left (1+3 e^3\right )^3 x+54 e x^2+405 e^4 x^2+1134 e^7 x^2+\frac {2187 e^{10} x^2}{2}-\frac {81}{2} e \left (1+3 e^3\right )^3 x^2-12 x^3-90 e^3 x^3-252 e^6 x^3-243 e^9 x^3+81 e^3 \left (1+3 e^3+3 e^6\right ) x^3+\frac {45 e^2 x^4}{2}+63 e^5 x^4+\frac {243 e^8 x^4}{4}-\frac {81}{4} e^2 \left (1+3 e^3+3 e^6\right ) x^4-6 e x^5-\frac {84 e^4 x^5}{5}-\frac {81 e^7 x^5}{5}+\frac {27}{5} e \left (1+3 e^3+3 e^6\right ) x^5+\frac {5 x^6}{3}+\frac {14 e^3 x^6}{3}+\frac {9 e^6 x^6}{2}-\frac {9}{2} e^3 \left (1+e^3\right ) x^6-\frac {4 e^2 x^7}{3}-\frac {9 e^5 x^7}{7}+\frac {9}{7} e^2 \left (1+e^3\right ) x^7+\frac {7 e x^8}{18}+\frac {3 e^4 x^8}{8}-\frac {3}{8} e \left (1+e^3\right ) x^8-\frac {28 x^9}{243}+\frac {x^{12}}{324}+27 \log (3 e+x)-3 e \log (3 e+x)+972 e^3 \log (3 e+x)+7290 e^6 \log (3 e+x)+20412 e^9 \log (3 e+x)+19683 e^{12} \log (3 e+x)+3 e \left (1-243 e^2-2187 e^5-6561 e^8-6561 e^{11}\right ) \log (3 e+x)-\frac {1}{3} \int x^8 \log \left (e+\frac {x}{3}\right ) \, dx+\frac {19}{3} \int x^5 \log \left (e+\frac {x}{3}\right ) \, dx-10 \int x^2 \log ^2\left (e+\frac {x}{3}\right ) \, dx-e \int x^4 \log \left (e+\frac {x}{3}\right ) \, dx+(3 e) \int x \log ^2\left (e+\frac {x}{3}\right ) \, dx+\left (3 e^2\right ) \int x^3 \log \left (e+\frac {x}{3}\right ) \, dx-\left (9 e^2\right ) \int \log ^2\left (e+\frac {x}{3}\right ) \, dx+\left (9 \left (1+3 e^3\right )\right ) \int \frac {\log ^2\left (e+\frac {x}{3}\right )}{3 e+x} \, dx+\left (27 \left (1+3 e^3\right )^2\right ) \int \frac {\log \left (e+\frac {x}{3}\right )}{3 e+x} \, dx+\left (9 e \left (2+3 e^3\right )\right ) \int x \log \left (e+\frac {x}{3}\right ) \, dx-\left (27 e^2 \left (2+3 e^3\right )\right ) \int \log \left (e+\frac {x}{3}\right ) \, dx-\left (3 \left (11+3 e^3\right )\right ) \int x^2 \log \left (e+\frac {x}{3}\right ) \, dx+\int x^5 \log ^2\left (e+\frac {x}{3}\right ) \, dx-\int \frac {\log ^3\left (e+\frac {x}{3}\right )}{-3 e-x} \, dx-\int x^2 \log ^3\left (e+\frac {x}{3}\right ) \, dx \\ & = x-324 e^2 x-2430 e^5 x-6804 e^8 x-6561 e^{11} x+243 e^2 \left (1+3 e^3\right )^3 x+54 e x^2+405 e^4 x^2+1134 e^7 x^2+\frac {2187 e^{10} x^2}{2}-\frac {81}{2} e \left (1+3 e^3\right )^3 x^2-12 x^3-90 e^3 x^3-252 e^6 x^3-243 e^9 x^3+81 e^3 \left (1+3 e^3+3 e^6\right ) x^3+\frac {45 e^2 x^4}{2}+63 e^5 x^4+\frac {243 e^8 x^4}{4}-\frac {81}{4} e^2 \left (1+3 e^3+3 e^6\right ) x^4-6 e x^5-\frac {84 e^4 x^5}{5}-\frac {81 e^7 x^5}{5}+\frac {27}{5} e \left (1+3 e^3+3 e^6\right ) x^5+\frac {5 x^6}{3}+\frac {14 e^3 x^6}{3}+\frac {9 e^6 x^6}{2}-\frac {9}{2} e^3 \left (1+e^3\right ) x^6-\frac {4 e^2 x^7}{3}-\frac {9 e^5 x^7}{7}+\frac {9}{7} e^2 \left (1+e^3\right ) x^7+\frac {7 e x^8}{18}+\frac {3 e^4 x^8}{8}-\frac {3}{8} e \left (1+e^3\right ) x^8-\frac {28 x^9}{243}+\frac {x^{12}}{324}+\frac {9}{2} e \left (2+3 e^3\right ) x^2 \log \left (e+\frac {x}{3}\right )-\left (11+3 e^3\right ) x^3 \log \left (e+\frac {x}{3}\right )+\frac {3}{4} e^2 x^4 \log \left (e+\frac {x}{3}\right )-\frac {1}{5} e x^5 \log \left (e+\frac {x}{3}\right )+\frac {19}{18} x^6 \log \left (e+\frac {x}{3}\right )-\frac {1}{27} x^9 \log \left (e+\frac {x}{3}\right )-\frac {10}{3} x^3 \log ^2\left (e+\frac {x}{3}\right )+\frac {1}{6} x^6 \log ^2\left (e+\frac {x}{3}\right )+27 \log (3 e+x)-3 e \log (3 e+x)+972 e^3 \log (3 e+x)+7290 e^6 \log (3 e+x)+20412 e^9 \log (3 e+x)+19683 e^{12} \log (3 e+x)+3 e \left (1-243 e^2-2187 e^5-6561 e^8-6561 e^{11}\right ) \log (3 e+x)+\frac {1}{81} \int \frac {x^9}{e+\frac {x}{3}} \, dx-\frac {1}{9} \int \frac {x^6 \log \left (e+\frac {x}{3}\right )}{e+\frac {x}{3}} \, dx-\frac {19}{54} \int \frac {x^6}{e+\frac {x}{3}} \, dx+\frac {20}{9} \int \frac {x^3 \log \left (e+\frac {x}{3}\right )}{e+\frac {x}{3}} \, dx-3 \text {Subst}\left (\int -\frac {\log ^3(x)}{3 x} \, dx,x,e+\frac {x}{3}\right )+\frac {1}{15} e \int \frac {x^5}{e+\frac {x}{3}} \, dx+(3 e) \int \left (-3 e \log ^2\left (e+\frac {x}{3}\right )+3 \left (e+\frac {x}{3}\right ) \log ^2\left (e+\frac {x}{3}\right )\right ) \, dx-\frac {1}{4} e^2 \int \frac {x^4}{e+\frac {x}{3}} \, dx-\left (27 e^2\right ) \text {Subst}\left (\int \log ^2(x) \, dx,x,e+\frac {x}{3}\right )-\frac {1}{3} \left (-11-3 e^3\right ) \int \frac {x^3}{e+\frac {x}{3}} \, dx+\left (27 \left (1+3 e^3\right )\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{3 x} \, dx,x,e+\frac {x}{3}\right )+\left (81 \left (1+3 e^3\right )^2\right ) \text {Subst}\left (\int \frac {\log (x)}{3 x} \, dx,x,e+\frac {x}{3}\right )-\frac {1}{2} \left (3 e \left (2+3 e^3\right )\right ) \int \frac {x^2}{e+\frac {x}{3}} \, dx-\left (81 e^2 \left (2+3 e^3\right )\right ) \text {Subst}\left (\int \log (x) \, dx,x,e+\frac {x}{3}\right )-\int \left (9 e^2 \log ^3\left (e+\frac {x}{3}\right )-18 e \left (e+\frac {x}{3}\right ) \log ^3\left (e+\frac {x}{3}\right )+9 \left (e+\frac {x}{3}\right )^2 \log ^3\left (e+\frac {x}{3}\right )\right ) \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(115\) vs. \(2(27)=54\).

Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 4.26 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=\frac {1}{27} \left (27 x-243 x^3+\frac {81 x^6}{2}-3 x^9+\frac {x^{12}}{12}-x^3 \left (243-27 x^3+x^6\right ) \log \left (e+\frac {x}{3}\right )+\frac {9}{2} \left (-9+x^3\right )^2 \log ^2\left (e+\frac {x}{3}\right )-9 \left (-9+x^3\right ) \log ^3\left (e+\frac {x}{3}\right )+\frac {27}{4} \log ^4\left (e+\frac {x}{3}\right )+729 \log (3 e+x)\right ) \]

[In]

Integrate[(729 + 27*x - 972*x^3 + 270*x^6 - 28*x^9 + x^12 + E*(81 - 2187*x^2 + 729*x^5 - 81*x^8 + 3*x^11) + (7
29 - 891*x^3 + 171*x^6 - 9*x^9 + E*(-2187*x^2 + 486*x^5 - 27*x^8))*Log[(3*E + x)/3] + (243 - 270*x^3 + 27*x^6
+ E*(-729*x^2 + 81*x^5))*Log[(3*E + x)/3]^2 + (27 - 81*E*x^2 - 27*x^3)*Log[(3*E + x)/3]^3)/(81*E + 27*x),x]

[Out]

(27*x - 243*x^3 + (81*x^6)/2 - 3*x^9 + x^12/12 - x^3*(243 - 27*x^3 + x^6)*Log[E + x/3] + (9*(-9 + x^3)^2*Log[E
+ x/3]^2)/2 - 9*(-9 + x^3)*Log[E + x/3]^3 + (27*Log[E + x/3]^4)/4 + 729*Log[3*E + x])/27

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs. \(2(22)=44\).

Time = 0.35 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.70

method	result	size
risch	\(\frac {\ln \left ({\mathrm e}+\frac {x}{3}\right )^{4}}{4}+\left (-\frac {x^{3}}{3}+3\right ) \ln \left ({\mathrm e}+\frac {x}{3}\right )^{3}+\frac {\left (x^{3}-9\right )^{2} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2}}{6}+\left (-\frac {1}{27} x^{9}+x^{6}-9 x^{3}\right ) \ln \left ({\mathrm e}+\frac {x}{3}\right )+\frac {x^{12}}{324}-\frac {x^{9}}{9}+\frac {3 x^{6}}{2}-9 x^{3}+x +27 \ln \left (x +3 \,{\mathrm e}\right )\)	\(100\)
parallelrisch	\(\frac {x^{12}}{324}-\frac {\ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{9}}{27}-\frac {x^{9}}{9}+\frac {\ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x^{6}}{6}+\ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{6}+\frac {3 x^{6}}{2}-\frac {\ln \left ({\mathrm e}+\frac {x}{3}\right )^{3} x^{3}}{3}-3 \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x^{3}-9 \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{3}+\frac {\ln \left ({\mathrm e}+\frac {x}{3}\right )^{4}}{4}-9 x^{3}+3 \ln \left ({\mathrm e}+\frac {x}{3}\right )^{3}+\frac {27 \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2}}{2}-6 \,{\mathrm e}+x +27 \ln \left ({\mathrm e}+\frac {x}{3}\right )\)	\(146\)
parts	\(x +\frac {577449 \,{\mathrm e}^{9}}{280}+\frac {35721 \,{\mathrm e}^{6}}{20}+\frac {x^{12}}{324}-\frac {x^{9}}{9}-9 x^{3}+\frac {3 x^{6}}{2}+\frac {891 \,{\mathrm e}^{3}}{2}+\frac {\left (19683 \,{\mathrm e}^{9}+19683 \,{\mathrm e}^{6}+6561 \,{\mathrm e}^{3}+729\right ) \ln \left (x +3 \,{\mathrm e}\right )}{27}-3 \,{\mathrm e} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x^{5}-18 \,{\mathrm e} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{5}+3 \,{\mathrm e} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{3} x^{2}+27 \,{\mathrm e} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x^{2}+81 \,{\mathrm e} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{2}+\frac {\ln \left ({\mathrm e}+\frac {x}{3}\right )^{4}}{4}+\frac {27 \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2}}{2}+{\mathrm e} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{8}-9 \ln \left ({\mathrm e}+\frac {x}{3}\right )^{3} \left ({\mathrm e}+\frac {x}{3}\right )^{3}-81 \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} \left ({\mathrm e}+\frac {x}{3}\right )^{3}-729 \ln \left ({\mathrm e}+\frac {x}{3}\right ) \left ({\mathrm e}+\frac {x}{3}\right )^{9}+81 \,{\mathrm e}^{3} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2}-\frac {243 \,{\mathrm e}^{6} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2}}{2}+9 \,{\mathrm e}^{3} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{3}-1458 \,{\mathrm e}^{6} \ln \left ({\mathrm e}+\frac {x}{3}\right )-243 \ln \left ({\mathrm e}+\frac {x}{3}\right ) \left ({\mathrm e}+\frac {x}{3}\right )^{3}-\frac {45 \,{\mathrm e}^{2} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x^{4}}{2}-135 \,{\mathrm e}^{2} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{4}+9 \,{\mathrm e}^{2} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{3} x +81 \,{\mathrm e}^{2} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x +243 \,{\mathrm e}^{2} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x +12 \,{\mathrm e}^{2} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{7}+378 \,{\mathrm e}^{4} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{5}+84 \,{\mathrm e}^{3} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{6}+1134 \,{\mathrm e}^{5} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{4}+2187 \,{\mathrm e}^{8} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x +2916 \,{\mathrm e}^{7} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{2}+2268 \,{\mathrm e}^{6} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{3}-90 \,{\mathrm e}^{3} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x^{3}-540 \,{\mathrm e}^{3} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{3}-243 \,{\mathrm e}^{5} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x -1458 \,{\mathrm e}^{5} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x -\frac {405 \,{\mathrm e}^{4} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x^{2}}{2}-1215 \,{\mathrm e}^{4} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{2}+\frac {243 \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} \left ({\mathrm e}+\frac {x}{3}\right )^{6}}{2}+729 \ln \left ({\mathrm e}+\frac {x}{3}\right ) \left ({\mathrm e}+\frac {x}{3}\right )^{6}+3 \ln \left ({\mathrm e}+\frac {x}{3}\right )^{3}\)	\(640\)
derivativedivides	\(\text {Expression too large to display}\)	\(1598\)
default	\(\text {Expression too large to display}\)	\(1598\)

[In]

int(((-81*x^2*exp(1)-27*x^3+27)*ln(exp(1)+1/3*x)^3+((81*x^5-729*x^2)*exp(1)+27*x^6-270*x^3+243)*ln(exp(1)+1/3*
x)^2+((-27*x^8+486*x^5-2187*x^2)*exp(1)-9*x^9+171*x^6-891*x^3+729)*ln(exp(1)+1/3*x)+(3*x^11-81*x^8+729*x^5-218
7*x^2+81)*exp(1)+x^12-28*x^9+270*x^6-972*x^3+27*x+729)/(81*exp(1)+27*x),x,method=_RETURNVERBOSE)

[Out]

1/4*ln(exp(1)+1/3*x)^4+(-1/3*x^3+3)*ln(exp(1)+1/3*x)^3+1/6*(x^3-9)^2*ln(exp(1)+1/3*x)^2+(-1/27*x^9+x^6-9*x^3)*
ln(exp(1)+1/3*x)+1/324*x^12-1/9*x^9+3/2*x^6-9*x^3+x+27*ln(x+3*exp(1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (20) = 40\).

Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.48 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=\frac {1}{324} \, x^{12} - \frac {1}{9} \, x^{9} + \frac {3}{2} \, x^{6} - \frac {1}{3} \, {\left (x^{3} - 9\right )} \log \left (\frac {1}{3} \, x + e\right )^{3} + \frac {1}{4} \, \log \left (\frac {1}{3} \, x + e\right )^{4} - 9 \, x^{3} + \frac {1}{6} \, {\left (x^{6} - 18 \, x^{3} + 81\right )} \log \left (\frac {1}{3} \, x + e\right )^{2} - \frac {1}{27} \, {\left (x^{9} - 27 \, x^{6} + 243 \, x^{3} - 729\right )} \log \left (\frac {1}{3} \, x + e\right ) + x \]

[In]

integrate(((-81*x^2*exp(1)-27*x^3+27)*log(exp(1)+1/3*x)^3+((81*x^5-729*x^2)*exp(1)+27*x^6-270*x^3+243)*log(exp
(1)+1/3*x)^2+((-27*x^8+486*x^5-2187*x^2)*exp(1)-9*x^9+171*x^6-891*x^3+729)*log(exp(1)+1/3*x)+(3*x^11-81*x^8+72
9*x^5-2187*x^2+81)*exp(1)+x^12-28*x^9+270*x^6-972*x^3+27*x+729)/(81*exp(1)+27*x),x, algorithm="fricas")

[Out]

1/324*x^12 - 1/9*x^9 + 3/2*x^6 - 1/3*(x^3 - 9)*log(1/3*x + e)^3 + 1/4*log(1/3*x + e)^4 - 9*x^3 + 1/6*(x^6 - 18
*x^3 + 81)*log(1/3*x + e)^2 - 1/27*(x^9 - 27*x^6 + 243*x^3 - 729)*log(1/3*x + e) + x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (19) = 38\).

Time = 0.22 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.96 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=\frac {x^{12}}{324} - \frac {x^{9}}{9} + \frac {3 x^{6}}{2} - 9 x^{3} + x + \left (3 - \frac {x^{3}}{3}\right ) \log {\left (\frac {x}{3} + e \right )}^{3} + \left (\frac {x^{6}}{6} - 3 x^{3} + \frac {27}{2}\right ) \log {\left (\frac {x}{3} + e \right )}^{2} + \left (- \frac {x^{9}}{27} + x^{6} - 9 x^{3}\right ) \log {\left (\frac {x}{3} + e \right )} + \frac {\log {\left (\frac {x}{3} + e \right )}^{4}}{4} + 27 \log {\left (x + 3 e \right )} \]

[In]

integrate(((-81*x**2*exp(1)-27*x**3+27)*ln(exp(1)+1/3*x)**3+((81*x**5-729*x**2)*exp(1)+27*x**6-270*x**3+243)*l
n(exp(1)+1/3*x)**2+((-27*x**8+486*x**5-2187*x**2)*exp(1)-9*x**9+171*x**6-891*x**3+729)*ln(exp(1)+1/3*x)+(3*x**
11-81*x**8+729*x**5-2187*x**2+81)*exp(1)+x**12-28*x**9+270*x**6-972*x**3+27*x+729)/(81*exp(1)+27*x),x)

[Out]

x**12/324 - x**9/9 + 3*x**6/2 - 9*x**3 + x + (3 - x**3/3)*log(x/3 + E)**3 + (x**6/6 - 3*x**3 + 27/2)*log(x/3 +
E)**2 + (-x**9/27 + x**6 - 9*x**3)*log(x/3 + E) + log(x/3 + E)**4/4 + 27*log(x + 3*E)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1976 vs. \(2 (20) = 40\).

Time = 0.31 (sec) , antiderivative size = 1976, normalized size of antiderivative = 73.19 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=\text {Too large to display} \]

[In]

integrate(((-81*x^2*exp(1)-27*x^3+27)*log(exp(1)+1/3*x)^3+((81*x^5-729*x^2)*exp(1)+27*x^6-270*x^3+243)*log(exp
(1)+1/3*x)^2+((-27*x^8+486*x^5-2187*x^2)*exp(1)-9*x^9+171*x^6-891*x^3+729)*log(exp(1)+1/3*x)+(3*x^11-81*x^8+72
9*x^5-2187*x^2+81)*exp(1)+x^12-28*x^9+270*x^6-972*x^3+27*x+729)/(81*exp(1)+27*x),x, algorithm="maxima")

[Out]

1/324*x^12 - 1/99*x^11*e + 1/30*x^10*e^2 - 1/9*x^9*e^3 - 1/9*x^9 + 3/8*x^8*e^4 + 23/64*x^8*e - 9/7*x^7*e^5 - 4
59/392*x^7*e^2 + 1/108*(18*log(1/3*x + e)^2 - 6*log(1/3*x + e) + 1)*(x + 3*e)^6 + 9/2*x^6*e^6 + 431/112*x^6*e^
3 - 18/125*(25*e*log(1/3*x + e)^2 - 10*e*log(1/3*x + e) + 2*e)*(x + 3*e)^5 + 161/108*x^6 - 81/5*x^5*e^7 - 1788
3/1400*x^5*e^4 - 691/150*x^5*e + 135/32*(8*e^2*log(1/3*x + e)^2 - 4*e^2*log(1/3*x + e) + e^2)*(x + 3*e)^4 + 24
3/4*x^4*e^8 + 47979/1120*x^4*e^5 + 1097/80*x^4*e^2 + 27/4*e^3*log(1/3*x + e)^4 - 20*(9*e^3*log(1/3*x + e)^2 -
6*e^3*log(1/3*x + e) + 2*e^3)*(x + 3*e)^3 - 1/27*(9*log(1/3*x + e)^3 - 9*log(1/3*x + e)^2 + 6*log(1/3*x + e) -
2)*(x + 3*e)^3 - 10/27*(9*log(1/3*x + e)^2 - 6*log(1/3*x + e) + 2)*(x + 3*e)^3 - 243*x^3*e^9 - 40419/280*x^3*
e^6 - 717/20*x^3*e^3 + 243*e^6*log(1/3*x + e)^3 + 90*e^3*log(1/3*x + e)^3 + 1/4*log(1/3*x + e)^4 + 9/8*(4*e*lo
g(1/3*x + e)^3 - 6*e*log(1/3*x + e)^2 + 6*e*log(1/3*x + e) - 3*e)*(x + 3*e)^2 + 1215/4*(2*e^4*log(1/3*x + e)^2
- 2*e^4*log(1/3*x + e) + e^4)*(x + 3*e)^2 + 45/2*(2*e*log(1/3*x + e)^2 - 2*e*log(1/3*x + e) + e)*(x + 3*e)^2
- 25/3*x^3 + 2187/2*x^2*e^10 + 261711/560*x^2*e^7 + 1323/40*x^2*e^4 + 51/4*x^2*e - 6561/2*e^9*log(x + 3*e)^2 -
4617/2*e^6*log(x + 3*e)^2 - 891/2*e^3*log(x + 3*e)^2 - 1/280*(35*x^8 - 120*x^7*e + 420*x^6*e^2 - 1512*x^5*e^3
+ 5670*x^4*e^4 - 22680*x^3*e^5 + 102060*x^2*e^6 - 612360*x*e^7 + 1837080*e^8*log(x + 3*e))*e*log(1/3*x + e) +
9/10*(4*x^5 - 15*x^4*e + 60*x^3*e^2 - 270*x^2*e^3 + 1620*x*e^4 - 4860*e^5*log(x + 3*e))*e*log(1/3*x + e) - 81
/2*(x^2 - 6*x*e + 18*e^2*log(x + 3*e))*e*log(1/3*x + e) + 3*log(1/3*x + e)^3 - 27*(e^2*log(1/3*x + e)^3 - 3*e^
2*log(1/3*x + e)^2 + 6*e^2*log(1/3*x + e) - 6*e^2)*(x + 3*e) - 1458*(e^5*log(1/3*x + e)^2 - 2*e^5*log(1/3*x +
e) + 2*e^5)*(x + 3*e) - 270*(e^2*log(1/3*x + e)^2 - 2*e^2*log(1/3*x + e) + 2*e^2)*(x + 3*e) - 6561*x*e^11 - 17
2773/280*x*e^8 + 26811/20*x*e^5 + 441/2*x*e^2 + 1/27720*(280*x^11 - 924*x^10*e + 3080*x^9*e^2 - 10395*x^8*e^3
+ 35640*x^7*e^4 - 124740*x^6*e^5 + 449064*x^5*e^6 - 1683990*x^4*e^7 + 6735960*x^3*e^8 - 30311820*x^2*e^9 + 181
870920*x*e^10 - 545612760*e^11*log(x + 3*e))*e + 1/78400*(1225*x^8 - 9000*x^7*e + 51100*x^6*e^2 - 268632*x^5*e
^3 + 1404270*x^4*e^4 - 7733880*x^3*e^5 + 49090860*x^2*e^6 + 257191200*e^8*log(x + 3*e)^2 - 466005960*x*e^7 + 1
398017880*e^8*log(x + 3*e))*e - 3/280*(35*x^8 - 120*x^7*e + 420*x^6*e^2 - 1512*x^5*e^3 + 5670*x^4*e^4 - 22680*
x^3*e^5 + 102060*x^2*e^6 - 612360*x*e^7 + 1837080*e^8*log(x + 3*e))*e + 1/4000*(96*(25*log(1/3*x + e)^2 - 10*l
og(1/3*x + e) + 2)*(x + 3*e)^5 - 5625*(8*e*log(1/3*x + e)^2 - 4*e*log(1/3*x + e) + e)*(x + 3*e)^4 + 40000*(9*e
^2*log(1/3*x + e)^2 - 6*e^2*log(1/3*x + e) + 2*e^2)*(x + 3*e)^3 - 972000*e^5*log(1/3*x + e)^3 - 810000*(2*e^3*
log(1/3*x + e)^2 - 2*e^3*log(1/3*x + e) + e^3)*(x + 3*e)^2 + 4860000*(e^4*log(1/3*x + e)^2 - 2*e^4*log(1/3*x +
e) + 2*e^4)*(x + 3*e))*e - 9/200*(16*x^5 - 135*x^4*e + 940*x^3*e^2 - 6930*x^2*e^3 - 48600*e^5*log(x + 3*e)^2
+ 73980*x*e^4 - 221940*e^5*log(x + 3*e))*e + 27/20*(4*x^5 - 15*x^4*e + 60*x^3*e^2 - 270*x^2*e^3 + 1620*x*e^4 -
4860*e^5*log(x + 3*e))*e - 3/8*(18*e^2*log(1/3*x + e)^4 + (4*log(1/3*x + e)^3 - 6*log(1/3*x + e)^2 + 6*log(1/
3*x + e) - 3)*(x + 3*e)^2 - 48*(e*log(1/3*x + e)^3 - 3*e*log(1/3*x + e)^2 + 6*e*log(1/3*x + e) - 6*e)*(x + 3*e
))*e - 27/4*(12*e^2*log(1/3*x + e)^3 + (2*log(1/3*x + e)^2 - 2*log(1/3*x + e) + 1)*(x + 3*e)^2 - 24*(e*log(1/3
*x + e)^2 - 2*e*log(1/3*x + e) + 2*e)*(x + 3*e))*e + 81/4*(18*e^2*log(x + 3*e)^2 + x^2 - 18*x*e + 54*e^2*log(x
+ 3*e))*e - 81/2*(x^2 - 6*x*e + 18*e^2*log(x + 3*e))*e + 19683*e^12*log(x + 3*e) + 518319/280*e^9*log(x + 3*e
) - 80433/20*e^6*log(x + 3*e) - 1323/2*e^3*log(x + 3*e) - 27*log(3)*log(x + 3*e) + 27/2*log(x + 3*e)^2 - 1/756
0*(280*x^9 - 945*x^8*e + 3240*x^7*e^2 - 11340*x^6*e^3 + 40824*x^5*e^4 - 153090*x^4*e^5 + 612360*x^3*e^6 - 2755
620*x^2*e^7 + 16533720*x*e^8 - 49601160*e^9*log(x + 3*e))*log(1/3*x + e) + 19/180*(10*x^6 - 36*x^5*e + 135*x^4
*e^2 - 540*x^3*e^3 + 2430*x^2*e^4 - 14580*x*e^5 + 43740*e^6*log(x + 3*e))*log(1/3*x + e) - 11/2*(2*x^3 - 9*x^2
*e + 54*x*e^2 - 162*e^3*log(x + 3*e))*log(1/3*x + e) + x + 27*log(x + 3*e)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (20) = 40\).

Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 5.22 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=\frac {1}{324} \, x^{12} - \frac {1}{27} \, x^{9} \log \left (\frac {1}{3} \, x + e\right ) - \frac {1}{9} \, x^{9} + \frac {1}{6} \, x^{6} \log \left (\frac {1}{3} \, x + e\right )^{2} + x^{6} \log \left (\frac {1}{3} \, x + e\right ) + \frac {3}{2} \, x^{6} - \frac {1}{3} \, x^{3} \log \left (\frac {1}{3} \, x + e\right )^{3} - 3 \, x^{3} \log \left (\frac {1}{3} \, x + e\right )^{2} - 9 \, x^{3} \log \left (\frac {1}{3} \, x + e\right ) + \frac {1}{4} \, \log \left (\frac {1}{3} \, x + e\right )^{4} - 9 \, x^{3} + 3 \, \log \left (\frac {1}{3} \, x + e\right )^{3} + \frac {27}{2} \, \log \left (\frac {1}{3} \, x + e\right )^{2} + x + 27 \, \log \left (\frac {1}{3} \, x + e\right ) \]

[In]

integrate(((-81*x^2*exp(1)-27*x^3+27)*log(exp(1)+1/3*x)^3+((81*x^5-729*x^2)*exp(1)+27*x^6-270*x^3+243)*log(exp
(1)+1/3*x)^2+((-27*x^8+486*x^5-2187*x^2)*exp(1)-9*x^9+171*x^6-891*x^3+729)*log(exp(1)+1/3*x)+(3*x^11-81*x^8+72
9*x^5-2187*x^2+81)*exp(1)+x^12-28*x^9+270*x^6-972*x^3+27*x+729)/(81*exp(1)+27*x),x, algorithm="giac")

[Out]

1/324*x^12 - 1/27*x^9*log(1/3*x + e) - 1/9*x^9 + 1/6*x^6*log(1/3*x + e)^2 + x^6*log(1/3*x + e) + 3/2*x^6 - 1/3
*x^3*log(1/3*x + e)^3 - 3*x^3*log(1/3*x + e)^2 - 9*x^3*log(1/3*x + e) + 1/4*log(1/3*x + e)^4 - 9*x^3 + 3*log(1
/3*x + e)^3 + 27/2*log(1/3*x + e)^2 + x + 27*log(1/3*x + e)

Mupad [B] (verification not implemented)

Time = 8.96 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.96 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=x+27\,\ln \left (x+3\,\mathrm {e}\right )-{\ln \left (\frac {x}{3}+\mathrm {e}\right )}^3\,\left (\frac {x^3}{3}-3\right )+\frac {{\ln \left (\frac {x}{3}+\mathrm {e}\right )}^4}{4}-\ln \left (\frac {x}{3}+\mathrm {e}\right )\,\left (\frac {x^9}{27}-x^6+9\,x^3\right )+{\ln \left (\frac {x}{3}+\mathrm {e}\right )}^2\,\left (\frac {x^6}{6}-3\,x^3+\frac {27}{2}\right )-9\,x^3+\frac {3\,x^6}{2}-\frac {x^9}{9}+\frac {x^{12}}{324} \]

[In]

int((27*x - log(x/3 + exp(1))^2*(exp(1)*(729*x^2 - 81*x^5) + 270*x^3 - 27*x^6 - 243) - log(x/3 + exp(1))*(exp(
1)*(2187*x^2 - 486*x^5 + 27*x^8) + 891*x^3 - 171*x^6 + 9*x^9 - 729) - 972*x^3 + 270*x^6 - 28*x^9 + x^12 - log(
x/3 + exp(1))^3*(81*x^2*exp(1) + 27*x^3 - 27) + exp(1)*(729*x^5 - 2187*x^2 - 81*x^8 + 3*x^11 + 81) + 729)/(27*
x + 81*exp(1)),x)

[Out]

x + 27*log(x + 3*exp(1)) - log(x/3 + exp(1))^3*(x^3/3 - 3) + log(x/3 + exp(1))^4/4 - log(x/3 + exp(1))*(9*x^3
- x^6 + x^9/27) + log(x/3 + exp(1))^2*(x^6/6 - 3*x^3 + 27/2) - 9*x^3 + (3*x^6)/2 - x^9/9 + x^12/324