3.20 \(\int (a+b \log (c x^n))^3 \log (1+e x) \, dx\)

Optimal. Leaf size=327 \[ -\frac{6 b^2 n^2 \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{6 b^2 n^2 \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac{3 b n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac{6 b^3 n^3 \text{PolyLog}(2,-e x)}{e}+\frac{6 b^3 n^3 \text{PolyLog}(3,-e x)}{e}+\frac{6 b^3 n^3 \text{PolyLog}(4,-e x)}{e}+\frac{6 b^2 n^2 (e x+1) \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{e}-6 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-12 a b^2 n^2 x-\frac{3 b n (e x+1) \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac{(e x+1) \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{e}+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3-12 b^3 n^2 x \log \left (c x^n\right )-\frac{6 b^3 n^3 (e x+1) \log (e x+1)}{e}+24 b^3 n^3 x \]

[Out]

-12*a*b^2*n^2*x + 24*b^3*n^3*x - 12*b^3*n^2*x*Log[c*x^n] - 6*b^2*n^2*x*(a + b*Log[c*x^n]) + 6*b*n*x*(a + b*Log
[c*x^n])^2 - x*(a + b*Log[c*x^n])^3 - (6*b^3*n^3*(1 + e*x)*Log[1 + e*x])/e + (6*b^2*n^2*(1 + e*x)*(a + b*Log[c
*x^n])*Log[1 + e*x])/e - (3*b*n*(1 + e*x)*(a + b*Log[c*x^n])^2*Log[1 + e*x])/e + ((1 + e*x)*(a + b*Log[c*x^n])
^3*Log[1 + e*x])/e + (6*b^3*n^3*PolyLog[2, -(e*x)])/e - (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(e*x)])/e +
(3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(e*x)])/e + (6*b^3*n^3*PolyLog[3, -(e*x)])/e - (6*b^2*n^2*(a + b*Log[c
*x^n])*PolyLog[3, -(e*x)])/e + (6*b^3*n^3*PolyLog[4, -(e*x)])/e

________________________________________________________________________________________

Rubi [A]  time = 0.762996, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 16, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.842, Rules used = {2389, 2295, 2370, 2296, 2346, 2302, 30, 6742, 2301, 2411, 43, 2351, 2315, 2374, 6589, 2383} \[ -\frac{6 b^2 n^2 \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{6 b^2 n^2 \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac{3 b n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac{6 b^3 n^3 \text{PolyLog}(2,-e x)}{e}+\frac{6 b^3 n^3 \text{PolyLog}(3,-e x)}{e}+\frac{6 b^3 n^3 \text{PolyLog}(4,-e x)}{e}+\frac{6 b^2 n^2 (e x+1) \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{e}-6 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-12 a b^2 n^2 x-\frac{3 b n (e x+1) \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac{(e x+1) \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{e}+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3-12 b^3 n^2 x \log \left (c x^n\right )-\frac{6 b^3 n^3 (e x+1) \log (e x+1)}{e}+24 b^3 n^3 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^3*Log[1 + e*x],x]

[Out]

-12*a*b^2*n^2*x + 24*b^3*n^3*x - 12*b^3*n^2*x*Log[c*x^n] - 6*b^2*n^2*x*(a + b*Log[c*x^n]) + 6*b*n*x*(a + b*Log
[c*x^n])^2 - x*(a + b*Log[c*x^n])^3 - (6*b^3*n^3*(1 + e*x)*Log[1 + e*x])/e + (6*b^2*n^2*(1 + e*x)*(a + b*Log[c
*x^n])*Log[1 + e*x])/e - (3*b*n*(1 + e*x)*(a + b*Log[c*x^n])^2*Log[1 + e*x])/e + ((1 + e*x)*(a + b*Log[c*x^n])
^3*Log[1 + e*x])/e + (6*b^3*n^3*PolyLog[2, -(e*x)])/e - (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(e*x)])/e +
(3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(e*x)])/e + (6*b^3*n^3*PolyLog[3, -(e*x)])/e - (6*b^2*n^2*(a + b*Log[c
*x^n])*PolyLog[3, -(e*x)])/e + (6*b^3*n^3*PolyLog[4, -(e*x)])/e

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 2295

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

Rule 2370

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> With[
{u = IntHide[Log[d*(e + f*x^m)^r], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[Dist[(a + b*Log[c*x
^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[
p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1]))

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 2346

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

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 30

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

Rule 6742

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

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 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 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 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

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

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rubi steps

\begin{align*} \int \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x) \, dx &=-x \left (a+b \log \left (c x^n\right )\right )^3+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}-(3 b n) \int \left (-\left (a+b \log \left (c x^n\right )\right )^2+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e x}\right ) \, dx\\ &=-x \left (a+b \log \left (c x^n\right )\right )^3+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}+(3 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac{(3 b n) \int \frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx}{e}\\ &=3 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}-\frac{(3 b n) \int \left (e \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}\right ) \, dx}{e}-\left (6 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=-6 a b^2 n^2 x+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}-(3 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx-\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx}{e}-\left (6 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx\\ &=-6 a b^2 n^2 x+6 b^3 n^3 x-6 b^3 n^2 x \log \left (c x^n\right )+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b n (1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e}+\left (6 b^2 n^2\right ) \int \left (-a-b \log \left (c x^n\right )+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e x}\right ) \, dx-\frac{\left (6 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{x} \, dx}{e}\\ &=-12 a b^2 n^2 x+6 b^3 n^3 x-6 b^3 n^2 x \log \left (c x^n\right )+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b n (1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e}-\left (6 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx+\frac{\left (6 b^2 n^2\right ) \int \frac{(1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{e}+\frac{\left (6 b^3 n^3\right ) \int \frac{\text{Li}_3(-e x)}{x} \, dx}{e}\\ &=-12 a b^2 n^2 x+12 b^3 n^3 x-12 b^3 n^2 x \log \left (c x^n\right )+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b n (1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e}+\frac{6 b^3 n^3 \text{Li}_4(-e x)}{e}+\frac{\left (6 b^2 n^2\right ) \int \left (e \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}\right ) \, dx}{e}\\ &=-12 a b^2 n^2 x+12 b^3 n^3 x-12 b^3 n^2 x \log \left (c x^n\right )+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b n (1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e}+\frac{6 b^3 n^3 \text{Li}_4(-e x)}{e}+\left (6 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx+\frac{\left (6 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{e}\\ &=-12 a b^2 n^2 x+12 b^3 n^3 x-12 b^3 n^2 x \log \left (c x^n\right )-6 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3+\frac{6 b^2 n^2 (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}-\frac{3 b n (1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{e}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e}+\frac{6 b^3 n^3 \text{Li}_4(-e x)}{e}-\left (6 b^3 n^3\right ) \int \left (-1+\frac{(1+e x) \log (1+e x)}{e x}\right ) \, dx+\frac{\left (6 b^3 n^3\right ) \int \frac{\text{Li}_2(-e x)}{x} \, dx}{e}\\ &=-12 a b^2 n^2 x+18 b^3 n^3 x-12 b^3 n^2 x \log \left (c x^n\right )-6 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3+\frac{6 b^2 n^2 (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}-\frac{3 b n (1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{e}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e}+\frac{6 b^3 n^3 \text{Li}_3(-e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e}+\frac{6 b^3 n^3 \text{Li}_4(-e x)}{e}-\frac{\left (6 b^3 n^3\right ) \int \frac{(1+e x) \log (1+e x)}{x} \, dx}{e}\\ &=-12 a b^2 n^2 x+18 b^3 n^3 x-12 b^3 n^2 x \log \left (c x^n\right )-6 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3+\frac{6 b^2 n^2 (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}-\frac{3 b n (1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{e}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e}+\frac{6 b^3 n^3 \text{Li}_3(-e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e}+\frac{6 b^3 n^3 \text{Li}_4(-e x)}{e}-\frac{\left (6 b^3 n^3\right ) \operatorname{Subst}\left (\int \frac{x \log (x)}{-\frac{1}{e}+\frac{x}{e}} \, dx,x,1+e x\right )}{e^2}\\ &=-12 a b^2 n^2 x+18 b^3 n^3 x-12 b^3 n^2 x \log \left (c x^n\right )-6 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3+\frac{6 b^2 n^2 (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}-\frac{3 b n (1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{e}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e}+\frac{6 b^3 n^3 \text{Li}_3(-e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e}+\frac{6 b^3 n^3 \text{Li}_4(-e x)}{e}-\frac{\left (6 b^3 n^3\right ) \operatorname{Subst}\left (\int \left (e \log (x)+\frac{e \log (x)}{-1+x}\right ) \, dx,x,1+e x\right )}{e^2}\\ &=-12 a b^2 n^2 x+18 b^3 n^3 x-12 b^3 n^2 x \log \left (c x^n\right )-6 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3+\frac{6 b^2 n^2 (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}-\frac{3 b n (1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{e}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e}+\frac{6 b^3 n^3 \text{Li}_3(-e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e}+\frac{6 b^3 n^3 \text{Li}_4(-e x)}{e}-\frac{\left (6 b^3 n^3\right ) \operatorname{Subst}(\int \log (x) \, dx,x,1+e x)}{e}-\frac{\left (6 b^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{-1+x} \, dx,x,1+e x\right )}{e}\\ &=-12 a b^2 n^2 x+24 b^3 n^3 x-12 b^3 n^2 x \log \left (c x^n\right )-6 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3-\frac{6 b^3 n^3 (1+e x) \log (1+e x)}{e}+\frac{6 b^2 n^2 (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}-\frac{3 b n (1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}+\frac{6 b^3 n^3 \text{Li}_2(-e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{e}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e}+\frac{6 b^3 n^3 \text{Li}_3(-e x)}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e}+\frac{6 b^3 n^3 \text{Li}_4(-e x)}{e}\\ \end{align*}

Mathematica [A]  time = 0.177127, size = 584, normalized size = 1.79 \[ \frac{3 b n \text{PolyLog}(2,-e x) \left (a^2+2 b (a-b n) \log \left (c x^n\right )-2 a b n+b^2 \log ^2\left (c x^n\right )+2 b^2 n^2\right )-6 b^2 n^2 \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )-b n\right )+6 b^3 n^3 \text{PolyLog}(4,-e x)-3 a^2 b e x \log \left (c x^n\right )+3 a^2 b \log (e x+1) \log \left (c x^n\right )+3 a^2 b e x \log (e x+1) \log \left (c x^n\right )+6 a^2 b e n x-3 a^2 b n \log (e x+1)-3 a^2 b e n x \log (e x+1)+a^3 (-e) x+a^3 e x \log (e x+1)+a^3 \log (e x+1)-3 a b^2 e x \log ^2\left (c x^n\right )+3 a b^2 \log (e x+1) \log ^2\left (c x^n\right )+3 a b^2 e x \log (e x+1) \log ^2\left (c x^n\right )+12 a b^2 e n x \log \left (c x^n\right )-6 a b^2 n \log (e x+1) \log \left (c x^n\right )-6 a b^2 e n x \log (e x+1) \log \left (c x^n\right )-18 a b^2 e n^2 x+6 a b^2 n^2 \log (e x+1)+6 a b^2 e n^2 x \log (e x+1)-18 b^3 e n^2 x \log \left (c x^n\right )+6 b^3 n^2 \log (e x+1) \log \left (c x^n\right )+6 b^3 e n^2 x \log (e x+1) \log \left (c x^n\right )-b^3 e x \log ^3\left (c x^n\right )+6 b^3 e n x \log ^2\left (c x^n\right )+b^3 \log (e x+1) \log ^3\left (c x^n\right )+b^3 e x \log (e x+1) \log ^3\left (c x^n\right )-3 b^3 n \log (e x+1) \log ^2\left (c x^n\right )-3 b^3 e n x \log (e x+1) \log ^2\left (c x^n\right )+24 b^3 e n^3 x-6 b^3 n^3 \log (e x+1)-6 b^3 e n^3 x \log (e x+1)}{e} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])^3*Log[1 + e*x],x]

[Out]

(-(a^3*e*x) + 6*a^2*b*e*n*x - 18*a*b^2*e*n^2*x + 24*b^3*e*n^3*x - 3*a^2*b*e*x*Log[c*x^n] + 12*a*b^2*e*n*x*Log[
c*x^n] - 18*b^3*e*n^2*x*Log[c*x^n] - 3*a*b^2*e*x*Log[c*x^n]^2 + 6*b^3*e*n*x*Log[c*x^n]^2 - b^3*e*x*Log[c*x^n]^
3 + a^3*Log[1 + e*x] - 3*a^2*b*n*Log[1 + e*x] + 6*a*b^2*n^2*Log[1 + e*x] - 6*b^3*n^3*Log[1 + e*x] + a^3*e*x*Lo
g[1 + e*x] - 3*a^2*b*e*n*x*Log[1 + e*x] + 6*a*b^2*e*n^2*x*Log[1 + e*x] - 6*b^3*e*n^3*x*Log[1 + e*x] + 3*a^2*b*
Log[c*x^n]*Log[1 + e*x] - 6*a*b^2*n*Log[c*x^n]*Log[1 + e*x] + 6*b^3*n^2*Log[c*x^n]*Log[1 + e*x] + 3*a^2*b*e*x*
Log[c*x^n]*Log[1 + e*x] - 6*a*b^2*e*n*x*Log[c*x^n]*Log[1 + e*x] + 6*b^3*e*n^2*x*Log[c*x^n]*Log[1 + e*x] + 3*a*
b^2*Log[c*x^n]^2*Log[1 + e*x] - 3*b^3*n*Log[c*x^n]^2*Log[1 + e*x] + 3*a*b^2*e*x*Log[c*x^n]^2*Log[1 + e*x] - 3*
b^3*e*n*x*Log[c*x^n]^2*Log[1 + e*x] + b^3*Log[c*x^n]^3*Log[1 + e*x] + b^3*e*x*Log[c*x^n]^3*Log[1 + e*x] + 3*b*
n*(a^2 - 2*a*b*n + 2*b^2*n^2 + 2*b*(a - b*n)*Log[c*x^n] + b^2*Log[c*x^n]^2)*PolyLog[2, -(e*x)] - 6*b^2*n^2*(a
- b*n + b*Log[c*x^n])*PolyLog[3, -(e*x)] + 6*b^3*n^3*PolyLog[4, -(e*x)])/e

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Maple [F]  time = 0.246, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( ex+1 \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^3*ln(e*x+1),x)

[Out]

int((a+b*ln(c*x^n))^3*ln(e*x+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (b^{3} e x -{\left (b^{3} e x + b^{3}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{3}}{e} + \frac{-{\left (e x -{\left (e x + 1\right )} \log \left (e x + 1\right ) + 1\right )} b^{3} \log \left (c\right )^{3} - 3 \,{\left (e x -{\left (e x + 1\right )} \log \left (e x + 1\right ) + 1\right )} a b^{2} \log \left (c\right )^{2} - 3 \,{\left (e x -{\left (e x + 1\right )} \log \left (e x + 1\right ) + 1\right )} a^{2} b \log \left (c\right ) -{\left (e x -{\left (e x + 1\right )} \log \left (e x + 1\right ) + 1\right )} a^{3} + \int \frac{3 \,{\left ({\left (b^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} e \log \left (c\right ) + a^{2} b e\right )} x \log \left (e x + 1\right ) \log \left (x^{n}\right ) +{\left (b^{3} e n x -{\left (b^{3} n +{\left ({\left (e n - e \log \left (c\right )\right )} b^{3} - a b^{2} e\right )} x\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{2}\right )}}{x}\,{d x}}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(e*x+1),x, algorithm="maxima")

[Out]

-(b^3*e*x - (b^3*e*x + b^3)*log(e*x + 1))*log(x^n)^3/e + integrate((3*(b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2
*b*e)*x*log(e*x + 1)*log(x^n) + (b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*log(c) + a^3*e)*x*log(e*x + 1
) + 3*(b^3*e*n*x - (b^3*n + ((e*n - e*log(c))*b^3 - a*b^2*e)*x)*log(e*x + 1))*log(x^n)^2)/x, x)/e

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} \log \left (c x^{n}\right )^{3} \log \left (e x + 1\right ) + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} \log \left (e x + 1\right ) + 3 \, a^{2} b \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a^{3} \log \left (e x + 1\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(e*x+1),x, algorithm="fricas")

[Out]

integral(b^3*log(c*x^n)^3*log(e*x + 1) + 3*a*b^2*log(c*x^n)^2*log(e*x + 1) + 3*a^2*b*log(c*x^n)*log(e*x + 1) +
a^3*log(e*x + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**3*ln(e*x+1),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left (e x + 1\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(e*x+1),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^3*log(e*x + 1), x)