\(\int \frac {(a+b \log (c x^n))^3 \log (1+e x)}{x} \, dx\) [27]

Optimal result

Integrand size = 22, antiderivative size = 81 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x} \, dx=-\left (a+b \log \left (c x^n\right )\right )^3 \operatorname {PolyLog}(2,-e x)+3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}(3,-e x)-6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(4,-e x)+6 b^3 n^3 \operatorname {PolyLog}(5,-e x) \] Output:

-(a+b*ln(c*x^n))^3*polylog(2,-e*x)+3*b*n*(a+b*ln(c*x^n))^2*polylog(3,-e*x) 
-6*b^2*n^2*(a+b*ln(c*x^n))*polylog(4,-e*x)+6*b^3*n^3*polylog(5,-e*x)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x} \, dx=-\left (a+b \log \left (c x^n\right )\right )^3 \operatorname {PolyLog}(2,-e x)+3 b n \left (\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}(3,-e x)+2 b n \left (-\left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(4,-e x)\right )+b n \operatorname {PolyLog}(5,-e x)\right )\right ) \] Input:

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

Output:

-((a + b*Log[c*x^n])^3*PolyLog[2, -(e*x)]) + 3*b*n*((a + b*Log[c*x^n])^2*P 
olyLog[3, -(e*x)] + 2*b*n*(-((a + b*Log[c*x^n])*PolyLog[4, -(e*x)]) + b*n* 
PolyLog[5, -(e*x)]))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2821, 2830, 2830, 7143}

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.

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

\(\displaystyle 3 b n \left (\operatorname {PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \left (\operatorname {PolyLog}(4,-e x) \left (a+b \log \left (c x^n\right )\right )-b n \operatorname {PolyLog}(5,-e x)\right )\right )-\operatorname {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^3\)

Input:

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

Output:

-((a + b*Log[c*x^n])^3*PolyLog[2, -(e*x)]) + 3*b*n*((a + b*Log[c*x^n])^2*P 
olyLog[3, -(e*x)] - 2*b*n*((a + b*Log[c*x^n])*PolyLog[4, -(e*x)] - b*n*Pol 
yLog[5, -(e*x)]))
 

Defintions of rubi rules used

rule 2821

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] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[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 2830

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ 
.)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) 
, x] - Simp[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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

Maple [C] (warning: unable to verify)

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

Time = 22.32 (sec) , antiderivative size = 605, normalized size of antiderivative = 7.47

method	result	size
risch	\(\ln \left (x \right )^{3} \operatorname {dilog}\left (e x +1\right ) b^{3} n^{3}-\ln \left (x \right )^{3} \operatorname {polylog}\left (2, -e x \right ) b^{3} n^{3}-3 \ln \left (x \right )^{2} \ln \left (x^{n}\right ) \operatorname {dilog}\left (e x +1\right ) b^{3} n^{2}+3 \ln \left (x \right )^{2} \ln \left (x^{n}\right ) \operatorname {polylog}\left (2, -e x \right ) b^{3} n^{2}+3 \ln \left (x \right ) \ln \left (x^{n}\right )^{2} \operatorname {dilog}\left (e x +1\right ) b^{3} n -3 \ln \left (x \right ) \ln \left (x^{n}\right )^{2} \operatorname {polylog}\left (2, -e x \right ) b^{3} n -\ln \left (x^{n}\right )^{3} \operatorname {dilog}\left (e x +1\right ) b^{3}+3 \ln \left (x^{n}\right )^{2} \operatorname {polylog}\left (3, -e x \right ) b^{3} n -6 \ln \left (x^{n}\right ) \operatorname {polylog}\left (4, -e x \right ) b^{3} n^{2}+6 b^{3} n^{3} \operatorname {polylog}\left (5, -e x \right )-\frac {{\left (i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 a \right )}^{3} \operatorname {dilog}\left (e x +1\right )}{8}+\frac {3 \left (i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 a \right ) b^{2} \left (-{\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )}^{2} \operatorname {dilog}\left (e x +1\right )+n^{2} \left (-\ln \left (x \right )^{2} \operatorname {polylog}\left (2, -e x \right )+2 \ln \left (x \right ) \operatorname {polylog}\left (3, -e x \right )-2 \operatorname {polylog}\left (4, -e x \right )\right )+2 n \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) \left (-\ln \left (x \right ) \operatorname {polylog}\left (2, -e x \right )+\operatorname {polylog}\left (3, -e x \right )\right )\right )}{2}+\frac {3 {\left (i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 a \right )}^{2} b \left (-\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) \operatorname {dilog}\left (e x +1\right )-\ln \left (x \right ) \operatorname {polylog}\left (2, -e x \right ) n +\operatorname {polylog}\left (3, -e x \right ) n \right )}{4}\)	\(605\)

Input:

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

Output:

ln(x)^3*dilog(e*x+1)*b^3*n^3-ln(x)^3*polylog(2,-e*x)*b^3*n^3-3*ln(x)^2*ln( 
x^n)*dilog(e*x+1)*b^3*n^2+3*ln(x)^2*ln(x^n)*polylog(2,-e*x)*b^3*n^2+3*ln(x 
)*ln(x^n)^2*dilog(e*x+1)*b^3*n-3*ln(x)*ln(x^n)^2*polylog(2,-e*x)*b^3*n-ln( 
x^n)^3*dilog(e*x+1)*b^3+3*ln(x^n)^2*polylog(3,-e*x)*b^3*n-6*ln(x^n)*polylo 
g(4,-e*x)*b^3*n^2+6*b^3*n^3*polylog(5,-e*x)-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*dilog(e*x+1)+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*dilog(e*x+1)+n^2*(-ln(x)^2*polylog(2,-e*x)+2*ln(x)* 
polylog(3,-e*x)-2*polylog(4,-e*x))+2*n*(ln(x^n)-n*ln(x))*(-ln(x)*polylog(2 
,-e*x)+polylog(3,-e*x)))+3/4*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*cs 
gn(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))*dilog(e*x+1)-ln(x)* 
polylog(2,-e*x)*n+polylog(3,-e*x)*n)
 

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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