\(\int \frac {(a+b \log (c x^n))^3 \log (1+e x)}{x^2} \, dx\) [28]

Optimal result

Integrand size = 22, antiderivative size = 342 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=6 b^3 e n^3 \log (x)-6 b^2 e n^2 \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-3 b e n \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-e \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-6 b^3 e n^3 \log (1+e x)-\frac {6 b^3 n^3 \log (1+e x)}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x}+6 b^3 e n^3 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )+6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )+3 b e n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )+6 b^3 e n^3 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right )+6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right )+6 b^3 e n^3 \operatorname {PolyLog}\left (4,-\frac {1}{e x}\right ) \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(770\) vs. \(2(342)=684\).

Time = 0.41 (sec) , antiderivative size = 770, normalized size of antiderivative = 2.25 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=a^3 e \log (x)+3 a^2 b e n \log (x)+6 a b^2 e n^2 \log (x)+6 b^3 e n^3 \log (x)-\frac {3}{2} a^2 b e n \log ^2(x)-3 a b^2 e n^2 \log ^2(x)-3 b^3 e n^3 \log ^2(x)+a b^2 e n^2 \log ^3(x)+b^3 e n^3 \log ^3(x)-\frac {1}{4} b^3 e n^3 \log ^4(x)+3 a^2 b e \log (x) \log \left (c x^n\right )+6 a b^2 e n \log (x) \log \left (c x^n\right )+6 b^3 e n^2 \log (x) \log \left (c x^n\right )-3 a b^2 e n \log ^2(x) \log \left (c x^n\right )-3 b^3 e n^2 \log ^2(x) \log \left (c x^n\right )+b^3 e n^2 \log ^3(x) \log \left (c x^n\right )+3 a b^2 e \log (x) \log ^2\left (c x^n\right )+3 b^3 e n \log (x) \log ^2\left (c x^n\right )-\frac {3}{2} b^3 e n \log ^2(x) \log ^2\left (c x^n\right )+b^3 e \log (x) \log ^3\left (c x^n\right )-a^3 e \log (1+e x)-3 a^2 b e n \log (1+e x)-6 a b^2 e n^2 \log (1+e x)-6 b^3 e n^3 \log (1+e x)-\frac {a^3 \log (1+e x)}{x}-\frac {3 a^2 b n \log (1+e x)}{x}-\frac {6 a b^2 n^2 \log (1+e x)}{x}-\frac {6 b^3 n^3 \log (1+e x)}{x}-3 a^2 b e \log \left (c x^n\right ) \log (1+e x)-6 a b^2 e n \log \left (c x^n\right ) \log (1+e x)-6 b^3 e n^2 \log \left (c x^n\right ) \log (1+e x)-\frac {3 a^2 b \log \left (c x^n\right ) \log (1+e x)}{x}-\frac {6 a b^2 n \log \left (c x^n\right ) \log (1+e x)}{x}-\frac {6 b^3 n^2 \log \left (c x^n\right ) \log (1+e x)}{x}-3 a b^2 e \log ^2\left (c x^n\right ) \log (1+e x)-3 b^3 e n \log ^2\left (c x^n\right ) \log (1+e x)-\frac {3 a b^2 \log ^2\left (c x^n\right ) \log (1+e x)}{x}-\frac {3 b^3 n \log ^2\left (c x^n\right ) \log (1+e x)}{x}-b^3 e \log ^3\left (c x^n\right ) \log (1+e x)-\frac {b^3 \log ^3\left (c x^n\right ) \log (1+e x)}{x}-3 b e n \left (a^2+2 a b n+2 b^2 n^2+2 b (a+b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right ) \operatorname {PolyLog}(2,-e x)+6 b^2 e n^2 \left (a+b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,-e x)-6 b^3 e n^3 \operatorname {PolyLog}(4,-e x) \] Input:

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

Output:

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*Lo 
g[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)]
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2825, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx\)

\(\Big \downarrow \) 2825

\(\displaystyle -e \int \left (-\frac {6 b^3 n^3}{x (e x+1)}-\frac {6 b^2 \left (a+b \log \left (c x^n\right )\right ) n^2}{x (e x+1)}-\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 n}{x (e x+1)}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (e x+1)}\right )dx-\frac {6 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {6 b^3 n^3 \log (e x+1)}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{x}-e \left (-6 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-6 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+6 b^2 n^2 \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-3 b n \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+3 b n \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-6 b^3 n^3 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )-6 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right )-6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {1}{e x}\right )+6 b^3 n^3 \log (e x+1)-6 b^3 n^3 \log (x)\right )-\frac {3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {6 b^3 n^3 \log (e x+1)}{x}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2825

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]}, Simp[Log[d*(e + f*x^m)^r]   u, x] - Simp[f*m*r 
 Int[x^(m - 1)/(e + f*x^m)   u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m 
, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 23.75 (sec) , antiderivative size = 1080, normalized size of antiderivative = 3.16

method	result	size
risch	\(\text {Expression too large to display}\)	\(1080\)

Input:

int((a+b*ln(c*x^n))^3*ln(e*x+1)/x^2,x,method=_RETURNVERBOSE)
 

Output:

-b^3*e*ln(e*x)*ln(x)^3*n^3-3*b^3*n/x*ln(e*x+1)*ln(x^n)^2-3*b^3*n*ln(e*x+1) 
*e*ln(x^n)^2-3*b^3*e*ln(x)^2*ln(x^n)*n^2+3*b^3*n*e*ln(x)*ln(x^n)^2-2*b^3*e 
*ln(x)^3*ln(x^n)*n^2+3/2*b^3*n*e*ln(x)^2*ln(x^n)^2-3*b^3*n*e*polylog(2,-e* 
x)*ln(x^n)^2-6*b^3*n^2/x*ln(e*x+1)*ln(x^n)-6*b^3*n^2*ln(e*x+1)*e*ln(x^n)+6 
*b^3*n^2*e*ln(x)*ln(x^n)-6*b^3*n^2*e*polylog(2,-e*x)*ln(x^n)+6*b^3*n^2*e*p 
olylog(3,-e*x)*ln(x^n)+1/8*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn 
(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n 
)^2*csgn(I*c)+2*b*ln(c)+2*a)^3*e*(ln(e*x)-ln(e*x+1)/x/e*(e*x+1))+3*b^3*e*l 
n(e*x)*ln(x)^2*ln(x^n)*n^2-3*b^3*e*ln(e*x)*ln(x)*ln(x^n)^2*n+6*b^3*n^3*e*p 
olylog(3,-e*x)+3/4*b^3*n^3*e*ln(x)^4-6*b^3*n^3*e*polylog(4,-e*x)-3*b^3*n^3 
*e*ln(x)^2-6*b^3*n^3*e*polylog(2,-e*x)+3/4*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^ 
n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*P 
i*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+2*a)^2*b*((ln(x^n)-n*ln(x))*e*(ln( 
e*x)-ln(e*x+1)/x/e*(e*x+1))+n*((-1-ln(x))/x*ln(e*x+1)-ln(e*x+1)*e+e*ln(x)+ 
1/2*e*ln(x)^2-e*ln(e*x+1)*ln(x)-e*polylog(2,-e*x)))+3/2*(I*Pi*b*csgn(I*x^n 
)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*csgn(I 
*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+2*a)*b^2*((ln(x^n)-n* 
ln(x))^2*e*(ln(e*x)-ln(e*x+1)/x/e*(e*x+1))+n^2*((-ln(x)^2-2*ln(x)-2)/x*ln( 
e*x+1)-2*ln(e*x+1)*e+2*e*ln(x)+e*ln(x)^2-2*e*ln(e*x+1)*ln(x)-2*e*polylog(2 
,-e*x)+1/3*e*ln(x)^3-e*ln(e*x+1)*ln(x)^2-2*e*ln(x)*polylog(2,-e*x)+2*e*...
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left (e x + 1\right )}{x^{2}} \,d x } \] Input:

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

Output:

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^2, x)
 

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{3} \log {\left (e x + 1 \right )}}{x^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left (e x + 1\right )}{x^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left (e x + 1\right )}{x^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=\int \frac {\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx=\frac {-\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right )^{3} b^{3}-\mathrm {log}\left (e x +1\right ) a^{3} e x -3 \,\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right )^{2} a \,b^{2}-3 \,\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right )^{2} b^{3} n -3 \,\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right ) a^{2} b -6 \,\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right ) b^{3} n^{2}-3 \,\mathrm {log}\left (e x +1\right ) a^{2} b n -6 \,\mathrm {log}\left (e x +1\right ) a \,b^{2} n^{2}-3 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e \,x^{3}+x^{2}}d x \right ) a \,b^{2} x -3 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e \,x^{3}+x^{2}}d x \right ) b^{3} n x -3 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{3}+x^{2}}d x \right ) a^{2} b x -6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{3}+x^{2}}d x \right ) b^{3} n^{2} x -6 \,\mathrm {log}\left (e x +1\right ) b^{3} n^{3}+\mathrm {log}\left (x \right ) a^{3} e x -3 \mathrm {log}\left (x^{n} c \right )^{2} a \,b^{2}-6 \mathrm {log}\left (x^{n} c \right )^{2} b^{3} n -3 \,\mathrm {log}\left (x^{n} c \right ) a^{2} b -\mathrm {log}\left (x^{n} c \right )^{3} b^{3}-18 b^{3} n^{3}-12 \,\mathrm {log}\left (x^{n} c \right ) a \,b^{2} n -6 \,\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right ) a \,b^{2} n -6 \,\mathrm {log}\left (e x +1\right ) b^{3} e \,n^{3} x +6 \,\mathrm {log}\left (x \right ) b^{3} e \,n^{3} x -6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{3}+x^{2}}d x \right ) a \,b^{2} n x -\left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{3}}{e \,x^{3}+x^{2}}d x \right ) b^{3} x -18 \,\mathrm {log}\left (x^{n} c \right ) b^{3} n^{2}-3 a^{2} b n -12 a \,b^{2} n^{2}-3 \,\mathrm {log}\left (e x +1\right ) a^{2} b e n x -6 \,\mathrm {log}\left (e x +1\right ) a \,b^{2} e \,n^{2} x +3 \,\mathrm {log}\left (x \right ) a^{2} b e n x +6 \,\mathrm {log}\left (x \right ) a \,b^{2} e \,n^{2} x -\mathrm {log}\left (e x +1\right ) a^{3}}{x} \] Input:

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

Output:

( - int(log(x**n*c)**3/(e*x**3 + x**2),x)*b**3*x - 3*int(log(x**n*c)**2/(e 
*x**3 + x**2),x)*a*b**2*x - 3*int(log(x**n*c)**2/(e*x**3 + x**2),x)*b**3*n 
*x - 3*int(log(x**n*c)/(e*x**3 + x**2),x)*a**2*b*x - 6*int(log(x**n*c)/(e* 
x**3 + x**2),x)*a*b**2*n*x - 6*int(log(x**n*c)/(e*x**3 + x**2),x)*b**3*n** 
2*x - log(e*x + 1)*log(x**n*c)**3*b**3 - 3*log(e*x + 1)*log(x**n*c)**2*a*b 
**2 - 3*log(e*x + 1)*log(x**n*c)**2*b**3*n - 3*log(e*x + 1)*log(x**n*c)*a* 
*2*b - 6*log(e*x + 1)*log(x**n*c)*a*b**2*n - 6*log(e*x + 1)*log(x**n*c)*b* 
*3*n**2 - log(e*x + 1)*a**3*e*x - log(e*x + 1)*a**3 - 3*log(e*x + 1)*a**2* 
b*e*n*x - 3*log(e*x + 1)*a**2*b*n - 6*log(e*x + 1)*a*b**2*e*n**2*x - 6*log 
(e*x + 1)*a*b**2*n**2 - 6*log(e*x + 1)*b**3*e*n**3*x - 6*log(e*x + 1)*b**3 
*n**3 - log(x**n*c)**3*b**3 - 3*log(x**n*c)**2*a*b**2 - 6*log(x**n*c)**2*b 
**3*n - 3*log(x**n*c)*a**2*b - 12*log(x**n*c)*a*b**2*n - 18*log(x**n*c)*b* 
*3*n**2 + log(x)*a**3*e*x + 3*log(x)*a**2*b*e*n*x + 6*log(x)*a*b**2*e*n**2 
*x + 6*log(x)*b**3*e*n**3*x - 3*a**2*b*n - 12*a*b**2*n**2 - 18*b**3*n**3)/ 
x