3.1.23 \(\int \frac {(a+b \log (c x^n))^3 \log (1+e x)}{x^3} \, dx\) [23]

Optimal. Leaf size=470 \[ -\frac {45 b^3 e n^3}{8 x}-\frac {3}{8} b^3 e^2 n^3 \log (x)-\frac {21 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}+\frac {3}{4} b^2 e^2 n^2 \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}+\frac {3}{4} b e^2 n \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}+\frac {1}{2} e^2 \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {3}{8} b^3 e^2 n^3 \log (1+e x)-\frac {3 b^3 n^3 \log (1+e x)}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}-\frac {3}{4} b^3 e^2 n^3 \text {Li}_2\left (-\frac {1}{e x}\right )-\frac {3}{2} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {1}{e x}\right )-\frac {3}{2} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {1}{e x}\right )-\frac {3}{2} b^3 e^2 n^3 \text {Li}_3\left (-\frac {1}{e x}\right )-3 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {1}{e x}\right )-3 b^3 e^2 n^3 \text {Li}_4\left (-\frac {1}{e x}\right ) \]

[Out]

-45/8*b^3*e*n^3/x-3/8*b^3*e^2*n^3*ln(x)-21/4*b^2*e*n^2*(a+b*ln(c*x^n))/x+3/4*b^2*e^2*n^2*ln(1+1/e/x)*(a+b*ln(c
*x^n))-9/4*b*e*n*(a+b*ln(c*x^n))^2/x+3/4*b*e^2*n*ln(1+1/e/x)*(a+b*ln(c*x^n))^2-1/2*e*(a+b*ln(c*x^n))^3/x+1/2*e
^2*ln(1+1/e/x)*(a+b*ln(c*x^n))^3+3/8*b^3*e^2*n^3*ln(e*x+1)-3/8*b^3*n^3*ln(e*x+1)/x^2-3/4*b^2*n^2*(a+b*ln(c*x^n
))*ln(e*x+1)/x^2-3/4*b*n*(a+b*ln(c*x^n))^2*ln(e*x+1)/x^2-1/2*(a+b*ln(c*x^n))^3*ln(e*x+1)/x^2-3/4*b^3*e^2*n^3*p
olylog(2,-1/e/x)-3/2*b^2*e^2*n^2*(a+b*ln(c*x^n))*polylog(2,-1/e/x)-3/2*b*e^2*n*(a+b*ln(c*x^n))^2*polylog(2,-1/
e/x)-3/2*b^3*e^2*n^3*polylog(3,-1/e/x)-3*b^2*e^2*n^2*(a+b*ln(c*x^n))*polylog(3,-1/e/x)-3*b^3*e^2*n^3*polylog(4
,-1/e/x)

________________________________________________________________________________________

Rubi [A]
time = 0.53, antiderivative size = 470, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {2342, 2341, 2425, 46, 2380, 2379, 2438, 2421, 6724, 2430} \begin {gather*} -\frac {3}{2} b^2 e^2 n^2 \text {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-3 b^2 e^2 n^2 \text {PolyLog}\left (3,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{2} b e^2 n \text {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{4} b^3 e^2 n^3 \text {PolyLog}\left (2,-\frac {1}{e x}\right )-\frac {3}{2} b^3 e^2 n^3 \text {PolyLog}\left (3,-\frac {1}{e x}\right )-3 b^3 e^2 n^3 \text {PolyLog}\left (4,-\frac {1}{e x}\right )+\frac {3}{4} b^2 e^2 n^2 \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {21 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}+\frac {3}{4} b e^2 n \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} e^2 \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {9 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}-\frac {3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3}{8} b^3 e^2 n^3 \log (x)+\frac {3}{8} b^3 e^2 n^3 \log (e x+1)-\frac {3 b^3 n^3 \log (e x+1)}{8 x^2}-\frac {45 b^3 e n^3}{8 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*b^3*e*n^3)/(8*x) - (3*b^3*e^2*n^3*Log[x])/8 - (21*b^2*e*n^2*(a + b*Log[c*x^n]))/(4*x) + (3*b^2*e^2*n^2*Lo
g[1 + 1/(e*x)]*(a + b*Log[c*x^n]))/4 - (9*b*e*n*(a + b*Log[c*x^n])^2)/(4*x) + (3*b*e^2*n*Log[1 + 1/(e*x)]*(a +
b*Log[c*x^n])^2)/4 - (e*(a + b*Log[c*x^n])^3)/(2*x) + (e^2*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^3)/2 + (3*b^3*
e^2*n^3*Log[1 + e*x])/8 - (3*b^3*n^3*Log[1 + e*x])/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/(4*x^
2) - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/(4*x^2) - ((a + b*Log[c*x^n])^3*Log[1 + e*x])/(2*x^2) - (3*b^3*
e^2*n^3*PolyLog[2, -(1/(e*x))])/4 - (3*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(1/(e*x))])/2 - (3*b*e^2*n*(
a + b*Log[c*x^n])^2*PolyLog[2, -(1/(e*x))])/2 - (3*b^3*e^2*n^3*PolyLog[3, -(1/(e*x))])/2 - 3*b^2*e^2*n^2*(a +
b*Log[c*x^n])*PolyLog[3, -(1/(e*x))] - 3*b^3*e^2*n^3*PolyLog[4, -(1/(e*x))]

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

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 2379

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

Rule 2380

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.))/((d_) + (e_.)*(x_)^(r_.)), x_Symbol] :> Dist[1/d,
Int[x^m*(a + b*Log[c*x^n])^p, x], x] - Dist[e/d, Int[(x^(m + r)*(a + b*Log[c*x^n])^p)/(d + e*x^r), x], x] /;
FreeQ[{a, b, c, d, e, m, n, r}, x] && IGtQ[p, 0] && IGtQ[r, 0] && ILtQ[m, -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 2425

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

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 2438

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

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^3} \, dx &=-\frac {3 b^3 n^3 \log (1+e x)}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}-e \int \left (-\frac {3 b^3 n^3}{8 x^2 (1+e x)}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2 (1+e x)}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2 (1+e x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2 (1+e x)}\right ) \, dx\\ &=-\frac {3 b^3 n^3 \log (1+e x)}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}+\frac {1}{2} e \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^2 (1+e x)} \, dx+\frac {1}{4} (3 b e n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (1+e x)} \, dx+\frac {1}{4} \left (3 b^2 e n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2 (1+e x)} \, dx+\frac {1}{8} \left (3 b^3 e n^3\right ) \int \frac {1}{x^2 (1+e x)} \, dx\\ &=-\frac {3 b^3 n^3 \log (1+e x)}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}+\frac {1}{2} e \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^3}{1+e x}\right ) \, dx+\frac {1}{4} (3 b e n) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{1+e x}\right ) \, dx+\frac {1}{4} \left (3 b^2 e n^2\right ) \int \left (\frac {a+b \log \left (c x^n\right )}{x^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{1+e x}\right ) \, dx+\frac {1}{8} \left (3 b^3 e n^3\right ) \int \left (\frac {1}{x^2}-\frac {e}{x}+\frac {e^2}{1+e x}\right ) \, dx\\ &=-\frac {3 b^3 e n^3}{8 x}-\frac {3}{8} b^3 e^2 n^3 \log (x)+\frac {3}{8} b^3 e^2 n^3 \log (1+e x)-\frac {3 b^3 n^3 \log (1+e x)}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}+\frac {1}{2} e \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx-\frac {1}{2} e^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx+\frac {1}{2} e^3 \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{1+e x} \, dx+\frac {1}{4} (3 b e n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx-\frac {1}{4} \left (3 b e^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\frac {1}{4} \left (3 b e^3 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1+e x} \, dx+\frac {1}{4} \left (3 b^2 e n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx-\frac {1}{4} \left (3 b^2 e^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx+\frac {1}{4} \left (3 b^2 e^3 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{1+e x} \, dx\\ &=-\frac {9 b^3 e n^3}{8 x}-\frac {3}{8} b^3 e^2 n^3 \log (x)-\frac {3 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {3}{8} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}+\frac {3}{8} b^3 e^2 n^3 \log (1+e x)-\frac {3 b^3 n^3 \log (1+e x)}{8 x^2}+\frac {3}{4} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}+\frac {3}{4} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}+\frac {1}{2} e^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}-\frac {1}{4} \left (3 e^2\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )-\frac {e^2 \text {Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {1}{2} (3 b e n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx-\frac {1}{2} \left (3 b e^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx+\frac {1}{2} \left (3 b^2 e n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx-\frac {1}{2} \left (3 b^2 e^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx-\frac {1}{4} \left (3 b^3 e^2 n^3\right ) \int \frac {\log (1+e x)}{x} \, dx\\ &=-\frac {21 b^3 e n^3}{8 x}-\frac {3}{8} b^3 e^2 n^3 \log (x)-\frac {9 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {3}{8} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2-\frac {9 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {1}{4} e^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+\frac {3}{8} b^3 e^2 n^3 \log (1+e x)-\frac {3 b^3 n^3 \log (1+e x)}{8 x^2}+\frac {3}{4} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}+\frac {3}{4} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}+\frac {1}{2} e^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}+\frac {3}{4} b^3 e^2 n^3 \text {Li}_2(-e x)+\frac {3}{2} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)+\frac {3}{2} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)+\left (3 b^2 e n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx-\left (3 b^2 e^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{x} \, dx-\frac {1}{2} \left (3 b^3 e^2 n^3\right ) \int \frac {\text {Li}_2(-e x)}{x} \, dx\\ &=-\frac {45 b^3 e n^3}{8 x}-\frac {3}{8} b^3 e^2 n^3 \log (x)-\frac {21 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {3}{8} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2-\frac {9 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {1}{4} e^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+\frac {3}{8} b^3 e^2 n^3 \log (1+e x)-\frac {3 b^3 n^3 \log (1+e x)}{8 x^2}+\frac {3}{4} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}+\frac {3}{4} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}+\frac {1}{2} e^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}+\frac {3}{4} b^3 e^2 n^3 \text {Li}_2(-e x)+\frac {3}{2} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)+\frac {3}{2} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)-\frac {3}{2} b^3 e^2 n^3 \text {Li}_3(-e x)-3 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)+\left (3 b^3 e^2 n^3\right ) \int \frac {\text {Li}_3(-e x)}{x} \, dx\\ &=-\frac {45 b^3 e n^3}{8 x}-\frac {3}{8} b^3 e^2 n^3 \log (x)-\frac {21 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {3}{8} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2-\frac {9 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {1}{4} e^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+\frac {3}{8} b^3 e^2 n^3 \log (1+e x)-\frac {3 b^3 n^3 \log (1+e x)}{8 x^2}+\frac {3}{4} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}+\frac {3}{4} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}+\frac {1}{2} e^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}+\frac {3}{4} b^3 e^2 n^3 \text {Li}_2(-e x)+\frac {3}{2} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)+\frac {3}{2} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)-\frac {3}{2} b^3 e^2 n^3 \text {Li}_3(-e x)-3 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)+3 b^3 e^2 n^3 \text {Li}_4(-e x)\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1047\) vs. \(2(470)=940\).
time = 0.24, size = 1047, normalized size = 2.23 \begin {gather*} -\frac {4 a^3 e x+18 a^2 b e n x+42 a b^2 e n^2 x+45 b^3 e n^3 x+4 a^3 e^2 x^2 \log (x)+6 a^2 b e^2 n x^2 \log (x)+6 a b^2 e^2 n^2 x^2 \log (x)+3 b^3 e^2 n^3 x^2 \log (x)-6 a^2 b e^2 n x^2 \log ^2(x)-6 a b^2 e^2 n^2 x^2 \log ^2(x)-3 b^3 e^2 n^3 x^2 \log ^2(x)+4 a b^2 e^2 n^2 x^2 \log ^3(x)+2 b^3 e^2 n^3 x^2 \log ^3(x)-b^3 e^2 n^3 x^2 \log ^4(x)+12 a^2 b e x \log \left (c x^n\right )+36 a b^2 e n x \log \left (c x^n\right )+42 b^3 e n^2 x \log \left (c x^n\right )+12 a^2 b e^2 x^2 \log (x) \log \left (c x^n\right )+12 a b^2 e^2 n x^2 \log (x) \log \left (c x^n\right )+6 b^3 e^2 n^2 x^2 \log (x) \log \left (c x^n\right )-12 a b^2 e^2 n x^2 \log ^2(x) \log \left (c x^n\right )-6 b^3 e^2 n^2 x^2 \log ^2(x) \log \left (c x^n\right )+4 b^3 e^2 n^2 x^2 \log ^3(x) \log \left (c x^n\right )+12 a b^2 e x \log ^2\left (c x^n\right )+18 b^3 e n x \log ^2\left (c x^n\right )+12 a b^2 e^2 x^2 \log (x) \log ^2\left (c x^n\right )+6 b^3 e^2 n x^2 \log (x) \log ^2\left (c x^n\right )-6 b^3 e^2 n x^2 \log ^2(x) \log ^2\left (c x^n\right )+4 b^3 e x \log ^3\left (c x^n\right )+4 b^3 e^2 x^2 \log (x) \log ^3\left (c x^n\right )+4 a^3 \log (1+e x)+6 a^2 b n \log (1+e x)+6 a b^2 n^2 \log (1+e x)+3 b^3 n^3 \log (1+e x)-4 a^3 e^2 x^2 \log (1+e x)-6 a^2 b e^2 n x^2 \log (1+e x)-6 a b^2 e^2 n^2 x^2 \log (1+e x)-3 b^3 e^2 n^3 x^2 \log (1+e x)+12 a^2 b \log \left (c x^n\right ) \log (1+e x)+12 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)-12 a^2 b e^2 x^2 \log \left (c x^n\right ) \log (1+e x)-12 a b^2 e^2 n x^2 \log \left (c x^n\right ) \log (1+e x)-6 b^3 e^2 n^2 x^2 \log \left (c x^n\right ) \log (1+e x)+12 a b^2 \log ^2\left (c x^n\right ) \log (1+e x)+6 b^3 n \log ^2\left (c x^n\right ) \log (1+e x)-12 a b^2 e^2 x^2 \log ^2\left (c x^n\right ) \log (1+e x)-6 b^3 e^2 n x^2 \log ^2\left (c x^n\right ) \log (1+e x)+4 b^3 \log ^3\left (c x^n\right ) \log (1+e x)-4 b^3 e^2 x^2 \log ^3\left (c x^n\right ) \log (1+e x)-6 b e^2 n x^2 \left (2 a^2+2 a b n+b^2 n^2+2 b (2 a+b n) \log \left (c x^n\right )+2 b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2(-e x)+12 b^2 e^2 n^2 x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)-24 b^3 e^2 n^3 x^2 \text {Li}_4(-e x)}{8 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(4*a^3*e*x + 18*a^2*b*e*n*x + 42*a*b^2*e*n^2*x + 45*b^3*e*n^3*x + 4*a^3*e^2*x^2*Log[x] + 6*a^2*b*e^2*n*x^
2*Log[x] + 6*a*b^2*e^2*n^2*x^2*Log[x] + 3*b^3*e^2*n^3*x^2*Log[x] - 6*a^2*b*e^2*n*x^2*Log[x]^2 - 6*a*b^2*e^2*n^
2*x^2*Log[x]^2 - 3*b^3*e^2*n^3*x^2*Log[x]^2 + 4*a*b^2*e^2*n^2*x^2*Log[x]^3 + 2*b^3*e^2*n^3*x^2*Log[x]^3 - b^3*
e^2*n^3*x^2*Log[x]^4 + 12*a^2*b*e*x*Log[c*x^n] + 36*a*b^2*e*n*x*Log[c*x^n] + 42*b^3*e*n^2*x*Log[c*x^n] + 12*a^
2*b*e^2*x^2*Log[x]*Log[c*x^n] + 12*a*b^2*e^2*n*x^2*Log[x]*Log[c*x^n] + 6*b^3*e^2*n^2*x^2*Log[x]*Log[c*x^n] - 1
2*a*b^2*e^2*n*x^2*Log[x]^2*Log[c*x^n] - 6*b^3*e^2*n^2*x^2*Log[x]^2*Log[c*x^n] + 4*b^3*e^2*n^2*x^2*Log[x]^3*Log
[c*x^n] + 12*a*b^2*e*x*Log[c*x^n]^2 + 18*b^3*e*n*x*Log[c*x^n]^2 + 12*a*b^2*e^2*x^2*Log[x]*Log[c*x^n]^2 + 6*b^3
*e^2*n*x^2*Log[x]*Log[c*x^n]^2 - 6*b^3*e^2*n*x^2*Log[x]^2*Log[c*x^n]^2 + 4*b^3*e*x*Log[c*x^n]^3 + 4*b^3*e^2*x^
2*Log[x]*Log[c*x^n]^3 + 4*a^3*Log[1 + e*x] + 6*a^2*b*n*Log[1 + e*x] + 6*a*b^2*n^2*Log[1 + e*x] + 3*b^3*n^3*Log
[1 + e*x] - 4*a^3*e^2*x^2*Log[1 + e*x] - 6*a^2*b*e^2*n*x^2*Log[1 + e*x] - 6*a*b^2*e^2*n^2*x^2*Log[1 + e*x] - 3
*b^3*e^2*n^3*x^2*Log[1 + e*x] + 12*a^2*b*Log[c*x^n]*Log[1 + e*x] + 12*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] - 12*a^2*b*e^2*x^2*Log[c*x^n]*Log[1 + e*x] - 12*a*b^2*e^2*n*x^2*Log[c*x^n]*Log[1 +
e*x] - 6*b^3*e^2*n^2*x^2*Log[c*x^n]*Log[1 + e*x] + 12*a*b^2*Log[c*x^n]^2*Log[1 + e*x] + 6*b^3*n*Log[c*x^n]^2*
Log[1 + e*x] - 12*a*b^2*e^2*x^2*Log[c*x^n]^2*Log[1 + e*x] - 6*b^3*e^2*n*x^2*Log[c*x^n]^2*Log[1 + e*x] + 4*b^3*
Log[c*x^n]^3*Log[1 + e*x] - 4*b^3*e^2*x^2*Log[c*x^n]^3*Log[1 + e*x] - 6*b*e^2*n*x^2*(2*a^2 + 2*a*b*n + b^2*n^2
+ 2*b*(2*a + b*n)*Log[c*x^n] + 2*b^2*Log[c*x^n]^2)*PolyLog[2, -(e*x)] + 12*b^2*e^2*n^2*x^2*(2*a + b*n + 2*b*L
og[c*x^n])*PolyLog[3, -(e*x)] - 24*b^3*e^2*n^3*x^2*PolyLog[4, -(e*x)])/x^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.48, size = 17975, normalized size = 38.24

method	result	size
risch	\(\text {Expression too large to display}\)	\(17975\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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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^3,x, algorithm="maxima")

[Out]

-1/2*(b^3*x^2*e^2*log(x) + b^3*x*e - (b^3*x^2*e^2 - b^3)*log(x*e + 1))*log(x^n)^3/x^2 - 1/2*integrate(-(6*(b^3
*log(c)^2 + 2*a*b^2*log(c) + a^2*b)*log(x*e + 1)*log(x^n) + 3*(b^3*n*x^2*e^2*log(x) + b^3*n*x*e - (b^3*n*x^2*e
^2 - b^3*(n + 2*log(c)) - 2*a*b^2)*log(x*e + 1))*log(x^n)^2 + 2*(b^3*log(c)^3 + 3*a*b^2*log(c)^2 + 3*a^2*b*log
(c) + a^3)*log(x*e + 1))/x^3, 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^3,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^3, 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**3,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^3,x, algorithm="giac")

[Out]

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

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

[Out]

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

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