3.1.20 \(\int (a+b \log (c x^n))^3 \log (1+e x) \, dx\) [20]

Optimal. Leaf size=327 \[ -12 a b^2 n^2 x+24 b^3 n^3 x-12 b^3 n^2 x \log \left (c x^n\right )-6 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b n x \left (a+b \log \left (c x^n\right )\right )^2-x \left (a+b \log \left (c x^n\right )\right )^3-\frac {6 b^3 n^3 (1+e x) \log (1+e x)}{e}+\frac {6 b^2 n^2 (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}-\frac {3 b n (1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{e}+\frac {6 b^3 n^3 \text {Li}_2(-e x)}{e}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{e}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{e}+\frac {6 b^3 n^3 \text {Li}_3(-e x)}{e}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e}+\frac {6 b^3 n^3 \text {Li}_4(-e x)}{e} \]

[Out]

-12*a*b^2*n^2*x+24*b^3*n^3*x-12*b^3*n^2*x*ln(c*x^n)-6*b^2*n^2*x*(a+b*ln(c*x^n))+6*b*n*x*(a+b*ln(c*x^n))^2-x*(a
+b*ln(c*x^n))^3-6*b^3*n^3*(e*x+1)*ln(e*x+1)/e+6*b^2*n^2*(e*x+1)*(a+b*ln(c*x^n))*ln(e*x+1)/e-3*b*n*(e*x+1)*(a+b
*ln(c*x^n))^2*ln(e*x+1)/e+(e*x+1)*(a+b*ln(c*x^n))^3*ln(e*x+1)/e+6*b^3*n^3*polylog(2,-e*x)/e-6*b^2*n^2*(a+b*ln(
c*x^n))*polylog(2,-e*x)/e+3*b*n*(a+b*ln(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
*ln(c*x^n))*polylog(3,-e*x)/e+6*b^3*n^3*polylog(4,-e*x)/e

________________________________________________________________________________________

Rubi [A]
time = 0.51, antiderivative size = 327, normalized size of antiderivative = 1.00, 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 = {2436, 2332, 2417, 2333, 2388, 2339, 30, 6874, 2338, 2458, 45, 2393, 2352, 2421, 6724, 2430} \begin {gather*} -\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 \end {gather*}

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 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 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 2352

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

Rule 2388

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 2393

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 2417

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[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]

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 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 6724

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 6874

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

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 ) \text {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 ) \text {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 ) \text {Subst}(\int \log (x) \, dx,x,1+e x)}{e}-\frac {\left (6 b^3 n^3\right ) \text {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*}

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Mathematica [A]
time = 0.11, size = 584, normalized size = 1.79 \begin {gather*} \frac {-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 \left (c x^n\right )+12 a b^2 e n x \log \left (c x^n\right )-18 b^3 e n^2 x \log \left (c x^n\right )-3 a b^2 e x \log ^2\left (c x^n\right )+6 b^3 e n x \log ^2\left (c x^n\right )-b^3 e x \log ^3\left (c x^n\right )+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 \log (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 \left (c x^n\right ) \log (1+e x)-6 a b^2 n \log \left (c x^n\right ) \log (1+e x)+6 b^3 n^2 \log \left (c x^n\right ) \log (1+e x)+3 a^2 b e x \log \left (c x^n\right ) \log (1+e x)-6 a b^2 e n x \log \left (c x^n\right ) \log (1+e x)+6 b^3 e n^2 x \log \left (c x^n\right ) \log (1+e x)+3 a b^2 \log ^2\left (c x^n\right ) \log (1+e x)-3 b^3 n \log ^2\left (c x^n\right ) \log (1+e x)+3 a b^2 e x \log ^2\left (c x^n\right ) \log (1+e x)-3 b^3 e n x \log ^2\left (c x^n\right ) \log (1+e x)+b^3 \log ^3\left (c x^n\right ) \log (1+e x)+b^3 e x \log ^3\left (c x^n\right ) \log (1+e x)+3 b n \left (a^2-2 a b n+2 b^2 n^2+2 b (a-b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2(-e x)-6 b^2 n^2 \left (a-b n+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)+6 b^3 n^3 \text {Li}_4(-e x)}{e} \end {gather*}

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 = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (e x +1\right )\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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(x*e + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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