3.14.33 $\int \frac{729+27x-972x^3+270x^6-28x^9+x^{12}+e(81-2187x^2+729x^5-81x^8+3x^{11})+(729-891x^3+171x^6-9x^9+e(-2187x^2+486x^5-27x^8))\log (\frac{1}{3}(3e+x))+(243-270x^3+27x^6+e(-729x^2+81x^5)){\log }^2(\frac{1}{3}(3e+x))+(27-81e{x}^2-27x^3){\log }^3(\frac{1}{3}(3e+x))}{81e+27x}dx$

Optimal. Leaf size=27 $$x+\frac{1}{4}{(-3+\frac{x^3}{3}-\log (e+\frac{x}{3}))}^4$$

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Rubi [B]  time = 2.66, antiderivative size = 1236, normalized size of antiderivative = 45.78, number of steps used = 87, number of rules used = 19, integrand size = 167, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.114, Rules used = {6742, 43, 1850, 2418, 2389, 2295, 2395, 2390, 12, 2301, 2296, 2401, 2305, 2304, 2398, 2411, 2334, 2302, 30}

result too large to display

Antiderivative was successfully verified.

[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 - 18*E^2*(3*E + x)*Log[E + x/3] - 27*E^2*(2 + 3*E^3)*(3*E + x)*Log[E + x/3] + 3*E*(3*E + x
)^2*Log[E + x/3] - (2*(3*E + x)^3*Log[E + x/3])/9 + 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 + (10*Log[E + x/3]*(162*E^2*(3*E + x) - 27*E*(3*E + x)^2 + 2*(3*E + x)^3 - 162*E^3*Log[E + x/3]))/9 + (Log[E
+ x/3]*(87480*E^5*(3*E + x) - 36450*E^4*(3*E + x)^2 + 10800*E^3*(3*E + x)^3 - 2025*E^2*(3*E + x)^4 + 216*E*(3
*E + x)^5 - 10*(3*E + x)^6 - 43740*E^6*Log[E + x/3]))/180 + 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] + 19683*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 43

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 1850

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 2295

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

Rule 2296

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 2301

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 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 2304

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 2305

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 2334

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]}, Simp[u*(a + b*Log[c*x^n]), 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 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 2395

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 2398

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 2401

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 2411

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 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 6742

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (\frac{27}{3e+x}+\frac{x}{3e+x}-\frac{36x^3}{3e+x}+\frac{10x^6}{3e+x}-\frac{28x^9}{27(3e+x)}+\frac{x^{12}}{27(3e+x)}+\frac{e(27-729x^2+243x^5-27x^8+x^{11})}{9(3e+x)}-\frac{{(-9+x^3)}^2(-1+3e{x}^2+x^3)\log (e+\frac{x}{3})}{3(3e+x)}+\frac{(-9+x^3)(-1+3e{x}^2+x^3){\log }^2(e+\frac{x}{3})}{3e+x}-\frac{(-1+3e{x}^2+x^3){\log }^3(e+\frac{x}{3})}{3e+x})dx\\ & =27\log (3e+x)+\frac{1}{27}\int \frac{x^{12}}{3e+x}dx-\frac{1}{3}\int \frac{{(-9+x^3)}^2(-1+3e{x}^2+x^3)\log (e+\frac{x}{3})}{3e+x}dx-\frac{28}{27}\int \frac{x^9}{3e+x}dx+10\int \frac{x^6}{3e+x}dx-36\int \frac{x^3}{3e+x}dx+\frac{1}{9}e\int \frac{27-729x^2+243x^5-27x^8+x^{11}}{3e+x}dx+\int \frac{x}{3e+x}dx+\int \frac{(-9+x^3)(-1+3e{x}^2+x^3){\log }^2(e+\frac{x}{3})}{3e+x}dx-\int \frac{(-1+3e{x}^2+x^3){\log }^3(e+\frac{x}{3})}{3e+x}dx\\ & =27\log (3e+x)+\frac{1}{27}\int (-177147e^{11}+59049e^{10}x-19683e^9{x}^2+6561e^8{x}^3-2187e^7{x}^4+729e^6{x}^5-243e^5{x}^6+81e^4{x}^7-27e^3{x}^8+9e^2{x}^9-3e{x}^{10}+x^{11}+\frac{531441e^{12}}{3e+x})dx-\frac{1}{3}\int (81e^2(2+3e^3)\log (e+\frac{x}{3})-27e(2+3e^3)x\log (e+\frac{x}{3})+9(11+3e^3)x^2\log (e+\frac{x}{3})-9e^2{x}^3\log (e+\frac{x}{3})+3e{x}^4\log (e+\frac{x}{3})-19x^5\log (e+\frac{x}{3})+x^8\log (e+\frac{x}{3})-\frac{81{(1+3e^3)}^2\log (e+\frac{x}{3})}{3e+x})dx-\frac{28}{27}\int (6561e^8-2187e^{7}x+729e^6{x}^2-243e^5{x}^3+81e^4{x}^4-27e^3{x}^5+9e^2{x}^6-3e{x}^7+x^8-\frac{19683e^9}{3e+x})dx+10\int (-243e^5+81e^{4}x-27e^3{x}^2+9e^2{x}^3-3e{x}^4+x^5+\frac{729e^6}{3e+x})dx-36\int (9e^2-3ex+x^2-\frac{27e^3}{3e+x})dx+\frac{1}{9}e\int (2187e{(1+3e^3)}^3-729{(1+3e^3)}^{3}x+2187e^2(1+3e^3+3e^6)x^2-729e(1+3e^3+3e^6)x^3+243(1+3e^3+3e^6)x^4-243e^2(1+e^3)x^5+81e(1+e^3)x^6-27(1+e^3)x^7+9e^2{x}^8-3e{x}^9+x^{10}-\frac{27(-1+243e^2+2187e^5+6561e^8+6561e^{11})}{3e+x})dx+\int (1-\frac{3e}{3e+x})dx+\int (-9e^2{\log }^2(e+\frac{x}{3})+3ex{\log }^2(e+\frac{x}{3})-10x^2{\log }^2(e+\frac{x}{3})+x^5{\log }^2(e+\frac{x}{3})+\frac{9(1+3e^3){\log }^2(e+\frac{x}{3})}{3e+x})dx-\int (\frac{{\log }^3(e+\frac{x}{3})}{-3e-x}+x^2{\log }^3(e+\frac{x}{3}))dx\\ & =x-324e^{2}x-2430e^{5}x-6804e^{8}x-6561e^{11}x+243e^2{(1+3e^3)}^{3}x+54e{x}^2+405e^4{x}^2+1134e^7{x}^2+\frac{2187e^{10}{x}^2}{2}-\frac{81}{2}e{(1+3e^3)}^3{x}^2-12x^3-90e^3{x}^3-252e^6{x}^3-243e^9{x}^3+81e^3(1+3e^3+3e^6)x^3+\frac{45e^2{x}^4}{2}+63e^5{x}^4+\frac{243e^8{x}^4}{4}-\frac{81}{4}{e}^2(1+3e^3+3e^6)x^4-6e{x}^5-\frac{84e^4{x}^5}{5}-\frac{81e^7{x}^5}{5}+\frac{27}{5}e(1+3e^3+3e^6)x^5+\frac{5x^6}{3}+\frac{14e^3{x}^6}{3}+\frac{9e^6{x}^6}{2}-\frac{9}{2}{e}^3(1+e^3)x^6-\frac{4e^2{x}^7}{3}-\frac{9e^5{x}^7}{7}+\frac{9}{7}{e}^2(1+e^3)x^7+\frac{7e{x}^8}{18}+\frac{3e^4{x}^8}{8}-\frac{3}{8}e(1+e^3)x^8-\frac{28x^9}{243}+\frac{x^{12}}{324}+27\log (3e+x)-3e\log (3e+x)+972e^3\log (3e+x)+7290e^6\log (3e+x)+20412e^9\log (3e+x)+19683e^{12}\log (3e+x)+3e(1-243e^2-2187e^5-6561e^8-6561e^{11})\log (3e+x)-\frac{1}{3}\int x^8\log (e+\frac{x}{3})dx+\frac{19}{3}\int x^5\log (e+\frac{x}{3})dx-10\int x^2{\log }^2(e+\frac{x}{3})dx-e\int x^4\log (e+\frac{x}{3})dx+(3e)\int x{\log }^2(e+\frac{x}{3})dx+\left(3e^2\right)\int x^3\log (e+\frac{x}{3})dx-\left(9e^2\right)\int {\log }^2(e+\frac{x}{3})dx+\left(9(1+3e^3)\right)\int \frac{{\log }^2(e+\frac{x}{3})}{3e+x}dx+\left(27{(1+3e^3)}^2\right)\int \frac{\log (e+\frac{x}{3})}{3e+x}dx+\left(9e(2+3e^3)\right)\int x\log (e+\frac{x}{3})dx-\left(27e^2(2+3e^3)\right)\int \log (e+\frac{x}{3})dx-\left(3(11+3e^3)\right)\int x^2\log (e+\frac{x}{3})dx+\int x^5{\log }^2(e+\frac{x}{3})dx-\int \frac{{\log }^3(e+\frac{x}{3})}{-3e-x}dx-\int x^2{\log }^3(e+\frac{x}{3})dx\\ & =\text{Rest of rules removed due to large latex content}\end{array}\end{array}$$

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Mathematica [B]  time = 0.10, size = 115, normalized size = 4.26 $$\begin{array}{c}\frac{1}{27}(27x-243x^3+\frac{81x^6}{2}-3x^9+\frac{x^{12}}{12}-x^3(243-27x^3+x^6)\log (e+\frac{x}{3})+\frac{9}{2}{(-9+x^3)}^2{\log }^2(e+\frac{x}{3})-9(-9+x^3){\log }^3(e+\frac{x}{3})+\frac{27}{4}{\log }^4(e+\frac{x}{3})+729\log (3e+x))\end{array}$$

Antiderivative was successfully verified.

[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

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fricas [B]  time = 0.93, size = 94, normalized size = 3.48 $$\begin{array}{c}\frac{1}{324}{x}^{12}-\frac{1}{9}{x}^9+\frac{3}{2}{x}^6-\frac{1}{3}(x^3-9)\log {(\frac{1}{3}x+e)}^3+\frac{1}{4}\log {(\frac{1}{3}x+e)}^4-9x^3+\frac{1}{6}(x^6-18x^3+81)\log {(\frac{1}{3}x+e)}^2-\frac{1}{27}(x^9-27x^6+243x^3-729)\log (\frac{1}{3}x+e)+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [B]  time = 0.76, size = 141, normalized size = 5.22 $$\begin{array}{c}\frac{1}{324}{x}^{12}-\frac{1}{27}{x}^9\log (\frac{1}{3}x+e)-\frac{1}{9}{x}^9+\frac{1}{6}{x}^6\log {(\frac{1}{3}x+e)}^2+x^6\log (\frac{1}{3}x+e)+\frac{3}{2}{x}^6-\frac{1}{3}{x}^3\log {(\frac{1}{3}x+e)}^3-3x^3\log {(\frac{1}{3}x+e)}^2-9x^3\log (\frac{1}{3}x+e)+\frac{1}{4}\log {(\frac{1}{3}x+e)}^4-9x^3+3\log {(\frac{1}{3}x+e)}^3+\frac{27}{2}\log {(\frac{1}{3}x+e)}^2+x+27\log (x+3e)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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(x + 3*e)

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maple [B]  time = 0.27, size = 104, normalized size = 3.85

method	result	size
risch	$\frac{\ln {(\mathrm{e}+\frac{x}{3})}^4}{4}+(-\frac{x^3}{3}+3)\ln {(\mathrm{e}+\frac{x}{3})}^3+(\frac{1}{6}{x}^6-3x^3+\frac{27}{2})\ln {(\mathrm{e}+\frac{x}{3})}^2+(-\frac{1}{27}{x}^9+x^6-9x^3)\ln (\mathrm{e}+\frac{x}{3})+\frac{x^{12}}{324}-\frac{x^9}{9}+\frac{3x^6}{2}-9x^3+x+27\ln (3\mathrm{e}+x)$	$104$
derivativedivides	$\text{Expression too large to display}$	$1598$
default	$\text{Expression too large to display}$	$1598$

Verification of antiderivative is not currently implemented for this CAS.

[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^6-3*x^3+27/2)*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(3*exp(1)+x)

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maxima [B]  time = 0.94, size = 1976, normalized size = 73.19 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 1.36, size = 107, normalized size = 3.96 $$\begin{array}{c}x+27\ln (x+3\mathrm{e})-{\ln (\frac{x}{3}+\mathrm{e})}^3(\frac{x^3}{3}-3)+\frac{{\ln (\frac{x}{3}+\mathrm{e})}^4}{4}-\ln (\frac{x}{3}+\mathrm{e})(\frac{x^9}{27}-x^6+9x^3)+{\ln (\frac{x}{3}+\mathrm{e})}^2(\frac{x^6}{6}-3x^3+\frac{27}{2})-9x^3+\frac{3x^6}{2}-\frac{x^9}{9}+\frac{x^{12}}{324}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [B]  time = 0.40, size = 107, normalized size = 3.96 $$\begin{array}{c}\frac{x^{12}}{324}-\frac{x^9}{9}+\frac{3x^6}{2}-9x^3+x+(3-\frac{x^3}{3})\log {(\frac{x}{3}+e)}^3+(\frac{x^6}{6}-3x^3+\frac{27}{2})\log {(\frac{x}{3}+e)}^2+(-\frac{x^9}{27}+x^6-9x^3)\log (\frac{x}{3}+e)+\frac{\log {(\frac{x}{3}+e)}^4}{4}+27\log (x+3e)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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