\(\int \frac {(a+b \log (c x^n))^2 \log (1+e x)}{x^2} \, dx\) [21]

Optimal result

Integrand size = 22, antiderivative size = 203 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x^2} \, dx=2 b^2 e n^2 \log (x)-2 b e n \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-e \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b^2 e n^2 \log (1+e x)-\frac {2 b^2 n^2 \log (1+e x)}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}+2 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )+2 b e n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {1}{e x}\right )+2 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {1}{e x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x^2} \, dx=\frac {1}{3} b^2 e n^2 \log ^3(x)-b e n \log ^2(x) \left (a+b n+b \log \left (c x^n\right )\right )+e \log (x) \left (a^2+2 a b n+2 b^2 n^2+2 b (a+b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right )-\frac {(1+e x) \left (a^2+2 a b n+2 b^2 n^2+2 b (a+b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right ) \log (1+e x)}{x}-2 b e n \left (a+b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,-e x)+2 b^2 e n^2 \operatorname {PolyLog}(3,-e x) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 199, 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.

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

\(\Big \downarrow \) 2825

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-2*b^2*n^2*Log[1 + e*x])/x - (2*b*n*(a + b*Log[c*x^n])*Log[1 + e*x])/x - 
((a + b*Log[c*x^n])^2*Log[1 + e*x])/x - e*(-2*b^2*n^2*Log[x] + 2*b*n*Log[1 
 + 1/(e*x)]*(a + b*Log[c*x^n]) + Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^2 + 2 
*b^2*n^2*Log[1 + e*x] - 2*b^2*n^2*PolyLog[2, -(1/(e*x))] - 2*b*n*(a + b*Lo 
g[c*x^n])*PolyLog[2, -(1/(e*x))] - 2*b^2*n^2*PolyLog[3, -(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 = 7.41 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.84

method	result	size
risch	\(b^{2} n e \ln \left (x \right )^{2} \ln \left (x^{n}\right )-2 b^{2} n e \operatorname {polylog}\left (2, -e x \right ) \ln \left (x^{n}\right )+b^{2} e \ln \left (e x \right ) \ln \left (x \right )^{2} n^{2}-\frac {\ln \left (x^{n}\right )^{2} \ln \left (e x +1\right ) b^{2}}{x}-\frac {2 b^{2} n \ln \left (e x +1\right ) \ln \left (x^{n}\right )}{x}-2 b^{2} n \ln \left (e x +1\right ) e \ln \left (x^{n}\right )+2 b^{2} n e \ln \left (x \right ) \ln \left (x^{n}\right )-2 b^{2} e \ln \left (e x \right ) \ln \left (x \right ) \ln \left (x^{n}\right ) n +b^{2} e \ln \left (e x \right ) \ln \left (x^{n}\right )^{2}-b^{2} e \ln \left (e x +1\right ) \ln \left (x^{n}\right )^{2}-\frac {2 b^{2} n^{2} \ln \left (e x +1\right )}{x}-2 b^{2} e \,n^{2} \ln \left (e x +1\right )+2 b^{2} e \,n^{2} \ln \left (x \right )-b^{2} n^{2} e \ln \left (x \right )^{2}-2 b^{2} n^{2} e \operatorname {polylog}\left (2, -e x \right )-\frac {2 b^{2} n^{2} e \ln \left (x \right )^{3}}{3}+2 b^{2} n^{2} e \operatorname {polylog}\left (3, -e x \right )+\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 \left (\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) e \left (\ln \left (e x \right )-\frac {\ln \left (e x +1\right ) \left (e x +1\right )}{x e}\right )+n \left (\frac {\left (-1-\ln \left (x \right )\right ) \ln \left (e x +1\right )}{x}-\ln \left (e x +1\right ) e +e \ln \left (x \right )+\frac {e \ln \left (x \right )^{2}}{2}-e \ln \left (e x +1\right ) \ln \left (x \right )-e \operatorname {polylog}\left (2, -e x \right )\right )\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 )}^{2} e \left (\ln \left (e x \right )-\frac {\ln \left (e x +1\right ) \left (e x +1\right )}{x e}\right )}{4}\)	\(576\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x^2} \, dx=\frac {-\left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e \,x^{3}+x^{2}}d x \right ) b^{2} x -2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{3}+x^{2}}d x \right ) a b x -2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{3}+x^{2}}d x \right ) b^{2} n x -\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right )^{2} b^{2}-2 \,\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right ) a b -2 \,\mathrm {log}\left (e x +1\right ) \mathrm {log}\left (x^{n} c \right ) b^{2} n -\mathrm {log}\left (e x +1\right ) a^{2} e x -\mathrm {log}\left (e x +1\right ) a^{2}-2 \,\mathrm {log}\left (e x +1\right ) a b e n x -2 \,\mathrm {log}\left (e x +1\right ) a b n -2 \,\mathrm {log}\left (e x +1\right ) b^{2} e \,n^{2} x -2 \,\mathrm {log}\left (e x +1\right ) b^{2} n^{2}-\mathrm {log}\left (x^{n} c \right )^{2} b^{2}-2 \,\mathrm {log}\left (x^{n} c \right ) a b -4 \,\mathrm {log}\left (x^{n} c \right ) b^{2} n +\mathrm {log}\left (x \right ) a^{2} e x +2 \,\mathrm {log}\left (x \right ) a b e n x +2 \,\mathrm {log}\left (x \right ) b^{2} e \,n^{2} x -2 a b n -4 b^{2} n^{2}}{x} \] Input:

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

Output:

( - int(log(x**n*c)**2/(e*x**3 + x**2),x)*b**2*x - 2*int(log(x**n*c)/(e*x* 
*3 + x**2),x)*a*b*x - 2*int(log(x**n*c)/(e*x**3 + x**2),x)*b**2*n*x - log( 
e*x + 1)*log(x**n*c)**2*b**2 - 2*log(e*x + 1)*log(x**n*c)*a*b - 2*log(e*x 
+ 1)*log(x**n*c)*b**2*n - log(e*x + 1)*a**2*e*x - log(e*x + 1)*a**2 - 2*lo 
g(e*x + 1)*a*b*e*n*x - 2*log(e*x + 1)*a*b*n - 2*log(e*x + 1)*b**2*e*n**2*x 
 - 2*log(e*x + 1)*b**2*n**2 - log(x**n*c)**2*b**2 - 2*log(x**n*c)*a*b - 4* 
log(x**n*c)*b**2*n + log(x)*a**2*e*x + 2*log(x)*a*b*e*n*x + 2*log(x)*b**2* 
e*n**2*x - 2*a*b*n - 4*b**2*n**2)/x