3.1.22 \(\int \frac {(a+b \log (c x^n))^3 \log (1+e x)}{x^2} \, dx\) [22]

Optimal. Leaf size=342 \[ 6 b^3 e n^3 \log (x)-6 b^2 e n^2 \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-3 b e n \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-e \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-6 b^3 e n^3 \log (1+e x)-\frac {6 b^3 n^3 \log (1+e x)}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x}+6 b^3 e n^3 \text {Li}_2\left (-\frac {1}{e x}\right )+6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {1}{e x}\right )+3 b e n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {1}{e x}\right )+6 b^3 e n^3 \text {Li}_3\left (-\frac {1}{e x}\right )+6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {1}{e x}\right )+6 b^3 e n^3 \text {Li}_4\left (-\frac {1}{e x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.30, antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2342, 2341, 2425, 36, 29, 31, 2379, 2438, 2421, 6724, 2430} \begin {gather*} 6 b^2 e n^2 \text {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+6 b^2 e n^2 \text {PolyLog}\left (3,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+3 b e n \text {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+6 b^3 e n^3 \text {PolyLog}\left (2,-\frac {1}{e x}\right )+6 b^3 e n^3 \text {PolyLog}\left (3,-\frac {1}{e x}\right )+6 b^3 e n^3 \text {PolyLog}\left (4,-\frac {1}{e x}\right )-6 b^2 e n^2 \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b e n \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{x}-e \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{x}+6 b^3 e n^3 \log (x)-6 b^3 e n^3 \log (e x+1)-\frac {6 b^3 n^3 \log (e x+1)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])^3*Log[1 + e*x])/x^2,x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 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 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^2} \, dx &=-\frac {6 b^3 n^3 \log (1+e x)}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x}-e \int \left (-\frac {6 b^3 n^3}{x (1+e x)}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{x (1+e x)}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{x (1+e x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (1+e x)}\right ) \, dx\\ &=-\frac {6 b^3 n^3 \log (1+e x)}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x}+e \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (1+e x)} \, dx+(3 b e n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (1+e x)} \, dx+\left (6 b^2 e n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x (1+e x)} \, dx+\left (6 b^3 e n^3\right ) \int \frac {1}{x (1+e x)} \, dx\\ &=-\frac {6 b^3 n^3 \log (1+e x)}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x}+e \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx-e^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{1+e x} \, dx+(3 b e n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx-\left (3 b e^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{1+e x} \, dx+\left (6 b^2 e n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx-\left (6 b^2 e^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{1+e x} \, dx+\left (6 b^3 e n^3\right ) \int \frac {1}{x} \, dx-\left (6 b^3 e^2 n^3\right ) \int \frac {1}{1+e x} \, dx\\ &=6 b^3 e n^3 \log (x)+3 b e n \left (a+b \log \left (c x^n\right )\right )^2-6 b^3 e n^3 \log (1+e x)-\frac {6 b^3 n^3 \log (1+e x)}{x}-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-3 b e 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)}{x}-e \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)}{x}+(3 e) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )+\frac {e \text {Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{b n}+(3 b e n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx+\left (6 b^2 e n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx+\left (6 b^3 e n^3\right ) \int \frac {\log (1+e x)}{x} \, dx\\ &=6 b^3 e n^3 \log (x)+3 b e n \left (a+b \log \left (c x^n\right )\right )^2+e \left (a+b \log \left (c x^n\right )\right )^3+\frac {e \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-6 b^3 e n^3 \log (1+e x)-\frac {6 b^3 n^3 \log (1+e x)}{x}-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-3 b e 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)}{x}-e \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)}{x}-6 b^3 e n^3 \text {Li}_2(-e x)-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)-3 b e n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)+\left (6 b^2 e n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{x} \, dx+\left (6 b^3 e n^3\right ) \int \frac {\text {Li}_2(-e x)}{x} \, dx\\ &=6 b^3 e n^3 \log (x)+3 b e n \left (a+b \log \left (c x^n\right )\right )^2+e \left (a+b \log \left (c x^n\right )\right )^3+\frac {e \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-6 b^3 e n^3 \log (1+e x)-\frac {6 b^3 n^3 \log (1+e x)}{x}-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-3 b e 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)}{x}-e \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)}{x}-6 b^3 e n^3 \text {Li}_2(-e x)-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)-3 b e n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)+6 b^3 e n^3 \text {Li}_3(-e x)+6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)-\left (6 b^3 e n^3\right ) \int \frac {\text {Li}_3(-e x)}{x} \, dx\\ &=6 b^3 e n^3 \log (x)+3 b e n \left (a+b \log \left (c x^n\right )\right )^2+e \left (a+b \log \left (c x^n\right )\right )^3+\frac {e \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-6 b^3 e n^3 \log (1+e x)-\frac {6 b^3 n^3 \log (1+e x)}{x}-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-3 b e 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)}{x}-e \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)}{x}-6 b^3 e n^3 \text {Li}_2(-e x)-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)-3 b e n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)+6 b^3 e n^3 \text {Li}_3(-e x)+6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)-6 b^3 e 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. \(770\) vs. \(2(342)=684\).
time = 0.19, size = 770, normalized size = 2.25 \begin {gather*} a^3 e \log (x)+3 a^2 b e n \log (x)+6 a b^2 e n^2 \log (x)+6 b^3 e n^3 \log (x)-\frac {3}{2} a^2 b e n \log ^2(x)-3 a b^2 e n^2 \log ^2(x)-3 b^3 e n^3 \log ^2(x)+a b^2 e n^2 \log ^3(x)+b^3 e n^3 \log ^3(x)-\frac {1}{4} b^3 e n^3 \log ^4(x)+3 a^2 b e \log (x) \log \left (c x^n\right )+6 a b^2 e n \log (x) \log \left (c x^n\right )+6 b^3 e n^2 \log (x) \log \left (c x^n\right )-3 a b^2 e n \log ^2(x) \log \left (c x^n\right )-3 b^3 e n^2 \log ^2(x) \log \left (c x^n\right )+b^3 e n^2 \log ^3(x) \log \left (c x^n\right )+3 a b^2 e \log (x) \log ^2\left (c x^n\right )+3 b^3 e n \log (x) \log ^2\left (c x^n\right )-\frac {3}{2} b^3 e n \log ^2(x) \log ^2\left (c x^n\right )+b^3 e \log (x) \log ^3\left (c x^n\right )-a^3 e \log (1+e x)-3 a^2 b e n \log (1+e x)-6 a b^2 e n^2 \log (1+e x)-6 b^3 e n^3 \log (1+e x)-\frac {a^3 \log (1+e x)}{x}-\frac {3 a^2 b n \log (1+e x)}{x}-\frac {6 a b^2 n^2 \log (1+e x)}{x}-\frac {6 b^3 n^3 \log (1+e x)}{x}-3 a^2 b e \log \left (c x^n\right ) \log (1+e x)-6 a b^2 e n \log \left (c x^n\right ) \log (1+e x)-6 b^3 e n^2 \log \left (c x^n\right ) \log (1+e x)-\frac {3 a^2 b \log \left (c x^n\right ) \log (1+e x)}{x}-\frac {6 a b^2 n \log \left (c x^n\right ) \log (1+e x)}{x}-\frac {6 b^3 n^2 \log \left (c x^n\right ) \log (1+e x)}{x}-3 a b^2 e \log ^2\left (c x^n\right ) \log (1+e x)-3 b^3 e n \log ^2\left (c x^n\right ) \log (1+e x)-\frac {3 a b^2 \log ^2\left (c x^n\right ) \log (1+e x)}{x}-\frac {3 b^3 n \log ^2\left (c x^n\right ) \log (1+e x)}{x}-b^3 e \log ^3\left (c x^n\right ) \log (1+e x)-\frac {b^3 \log ^3\left (c x^n\right ) \log (1+e x)}{x}-3 b e 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 e n^2 \left (a+b n+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)-6 b^3 e n^3 \text {Li}_4(-e x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])^3*Log[1 + e*x])/x^2,x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.38, size = 14041, normalized size = 41.06

method	result	size
risch	\(\text {Expression too large to display}\)	\(14041\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^3*ln(e*x+1)/x^2,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^2,x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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

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