\(\int \frac {(a+b \log (c x^n))^3 \log (1+e x)}{x^3} \, dx\) [29]

Optimal result

Integrand size = 22, antiderivative size = 470 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^3} \, dx=-\frac {45 b^3 e n^3}{8 x}-\frac {3}{8} b^3 e^2 n^3 \log (x)-\frac {21 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}+\frac {3}{4} b^2 e^2 n^2 \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}+\frac {3}{4} b e^2 n \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}+\frac {1}{2} e^2 \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {3}{8} b^3 e^2 n^3 \log (1+e x)-\frac {3 b^3 n^3 \log (1+e x)}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}-\frac {3}{4} b^3 e^2 n^3 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )-\frac {3}{2} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )-\frac {3}{2} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )-\frac {3}{2} b^3 e^2 n^3 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right )-3 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right )-3 b^3 e^2 n^3 \operatorname {PolyLog}\left (4,-\frac {1}{e x}\right ) \] Output:

-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*polyl 
og(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)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1047\) vs. \(2(470)=940\).

Time = 0.44 (sec) , antiderivative size = 1047, normalized size of antiderivative = 2.23 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-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] - 12*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*L 
og[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*Lo...
 

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 448, normalized size of antiderivative = 0.95, 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.

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

\(\Big \downarrow \) 2825

\(\displaystyle -e \int \left (-\frac {3 b^3 n^3}{8 x^2 (e x+1)}-\frac {3 b^2 \left (a+b \log \left (c x^n\right )\right ) n^2}{4 x^2 (e x+1)}-\frac {3 b \left (a+b \log \left (c x^n\right )\right )^2 n}{4 x^2 (e x+1)}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2 (e x+1)}\right )dx-\frac {3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b^3 n^3 \log (e x+1)}{8 x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-e \left (\frac {3}{2} b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+3 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{4} b^2 e n^2 \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {21 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}+\frac {3}{2} b e n \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{4} b e n \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} e \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {9 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x}+\frac {3}{4} b^3 e n^3 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )+\frac {3}{2} b^3 e n^3 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right )+3 b^3 e n^3 \operatorname {PolyLog}\left (4,-\frac {1}{e x}\right )+\frac {3}{8} b^3 e n^3 \log (x)-\frac {3}{8} b^3 e n^3 \log (e x+1)+\frac {45 b^3 n^3}{8 x}\right )-\frac {3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b^3 n^3 \log (e x+1)}{8 x^2}\)

Input:

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

Output:

(-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) - e*((45*b^3*n^3)/(8*x) + (3*b^3*e*n 
^3*Log[x])/8 + (21*b^2*n^2*(a + b*Log[c*x^n]))/(4*x) - (3*b^2*e*n^2*Log[1 
+ 1/(e*x)]*(a + b*Log[c*x^n]))/4 + (9*b*n*(a + b*Log[c*x^n])^2)/(4*x) - (3 
*b*e*n*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^2)/4 + (a + b*Log[c*x^n])^3/(2* 
x) - (e*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^3)/2 - (3*b^3*e*n^3*Log[1 + e* 
x])/8 + (3*b^3*e*n^3*PolyLog[2, -(1/(e*x))])/4 + (3*b^2*e*n^2*(a + b*Log[c 
*x^n])*PolyLog[2, -(1/(e*x))])/2 + (3*b*e*n*(a + b*Log[c*x^n])^2*PolyLog[2 
, -(1/(e*x))])/2 + (3*b^3*e*n^3*PolyLog[3, -(1/(e*x))])/2 + 3*b^2*e*n^2*(a 
 + b*Log[c*x^n])*PolyLog[3, -(1/(e*x))] + 3*b^3*e*n^3*PolyLog[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 = 23.85 (sec) , antiderivative size = 1329, normalized size of antiderivative = 2.83

method	result	size
risch	\(\text {Expression too large to display}\)	\(1329\)

Input:

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

Output:

3/4*b^3*n^3*e^2*polylog(2,-e*x)-3/2*b^3*n^3*polylog(3,-e*x)*e^2+3*b^3*n^3* 
polylog(4,-e*x)*e^2-1/2*b^3*e^2*ln(e*x)*ln(x^n)^3+1/2*b^3*e^2*ln(e*x+1)*ln 
(x^n)^3+3/8*b^3*n^3*e^2*ln(x)^2-1/4*b^3*n^3*e^2*ln(x)^3-3/8*b^3*n^3*e^2*ln 
(x)^4+3/4*b^3*ln(x)^2*e^2*ln(x^n)*n^2+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*Pi 
*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^2*(-1 
/2*ln(e*x)-1/2/e/x+1/2*ln(e*x+1)*(e*x+1)*(e*x-1)/x^2/e^2)+n*((-1/4-1/2*ln( 
x))/x^2*ln(e*x+1)+1/4*e^2*ln(e*x+1)-3/4*e/x-1/4*e^2*ln(x)+1/2*e^2*ln(e*x+1 
)*ln(x)-1/4*e^2*ln(x)^2+1/2*e^2*polylog(2,-e*x)-1/2*e*ln(x)/x))-3/2*b^3*e^ 
2*ln(e*x)*ln(x)^2*ln(x^n)*n^2+3/2*b^3*e^2*ln(e*x)*ln(x)*ln(x^n)^2*n+b^3*e^ 
2*ln(x)^3*ln(x^n)*n^2-3/4*b^3*n/x^2*ln(e*x+1)*ln(x^n)^2+3/4*b^3*n^2*e^2*ln 
(e*x+1)*ln(x^n)-3/4*b^3*n*ln(x)*e^2*ln(x^n)^2+1/2*b^3*e^2*ln(e*x)*ln(x)^3* 
n^3-3/4*b^3*n*e^2*ln(x)^2*ln(x^n)^2+3/2*b^3*n*e^2*polylog(2,-e*x)*ln(x^n)^ 
2-3/4*b^3*n^2/x^2*ln(e*x+1)*ln(x^n)-3/4*b^3*n^2*ln(x)*e^2*ln(x^n)+3/2*b^3* 
n^2*e^2*polylog(2,-e*x)*ln(x^n)-3*b^3*n^2*polylog(3,-e*x)*e^2*ln(x^n)+3/4* 
b^3*n*e^2*ln(e*x+1)*ln(x^n)^2-1/2*b^3*e/x*ln(x^n)^3+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^2*(-1/2*ln(e*x)-1/2/e/x+1/2*ln(e*x+1)*(e*x+1)*(e*x-1)/x^2/e^2)+ 
n^2*((-1/4-1/2*ln(x)-1/2*ln(x)^2)/x^2*ln(e*x+1)+1/4*e^2*ln(e*x+1)-7/4*e...
 

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 8*int(log(x**n*c)**3/(e*x**4 + x**3),x)*b**3*x**2 - 24*int(log(x**n*c) 
**2/(e*x**4 + x**3),x)*a*b**2*x**2 - 12*int(log(x**n*c)**2/(e*x**4 + x**3) 
,x)*b**3*n*x**2 - 24*int(log(x**n*c)/(e*x**4 + x**3),x)*a**2*b*x**2 - 24*i 
nt(log(x**n*c)/(e*x**4 + x**3),x)*a*b**2*n*x**2 - 12*int(log(x**n*c)/(e*x* 
*4 + x**3),x)*b**3*n**2*x**2 - 8*log(e*x + 1)*log(x**n*c)**3*b**3 - 24*log 
(e*x + 1)*log(x**n*c)**2*a*b**2 - 12*log(e*x + 1)*log(x**n*c)**2*b**3*n - 
24*log(e*x + 1)*log(x**n*c)*a**2*b - 24*log(e*x + 1)*log(x**n*c)*a*b**2*n 
- 12*log(e*x + 1)*log(x**n*c)*b**3*n**2 + 8*log(e*x + 1)*a**3*e**2*x**2 - 
8*log(e*x + 1)*a**3 + 12*log(e*x + 1)*a**2*b*e**2*n*x**2 - 12*log(e*x + 1) 
*a**2*b*n + 12*log(e*x + 1)*a*b**2*e**2*n**2*x**2 - 12*log(e*x + 1)*a*b**2 
*n**2 + 6*log(e*x + 1)*b**3*e**2*n**3*x**2 - 6*log(e*x + 1)*b**3*n**3 - 4* 
log(x**n*c)**3*b**3 - 12*log(x**n*c)**2*a*b**2 - 12*log(x**n*c)**2*b**3*n 
- 12*log(x**n*c)*a**2*b - 24*log(x**n*c)*a*b**2*n - 18*log(x**n*c)*b**3*n* 
*2 - 8*log(x)*a**3*e**2*x**2 - 12*log(x)*a**2*b*e**2*n*x**2 - 12*log(x)*a* 
b**2*e**2*n**2*x**2 - 6*log(x)*b**3*e**2*n**3*x**2 - 8*a**3*e*x - 12*a**2* 
b*e*n*x - 6*a**2*b*n - 12*a*b**2*e*n**2*x - 12*a*b**2*n**2 - 6*b**3*e*n**3 
*x - 9*b**3*n**3)/(16*x**2)