\(\int \frac {(a+b \log (c x^n))^2 \log (d (\frac {1}{d}+f x^m))}{x} \, dx\) [72]

Optimal result

Integrand size = 28, antiderivative size = 73 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f x^m\right )}{m}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f x^m\right )}{m^2}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (4,-d f x^m\right )}{m^3} \] Output:

-(a+b*ln(c*x^n))^2*polylog(2,-d*f*x^m)/m+2*b*n*(a+b*ln(c*x^n))*polylog(3,- 
d*f*x^m)/m^2-2*b^2*n^2*polylog(4,-d*f*x^m)/m^3
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(526\) vs. \(2(73)=146\).

Time = 0.31 (sec) , antiderivative size = 526, normalized size of antiderivative = 7.21 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=-\frac {1}{3} a b m n \log ^3(x)+\frac {1}{4} b^2 m n^2 \log ^4(x)-\frac {1}{3} b^2 m n \log ^3(x) \log \left (c x^n\right )-a b n \log ^2(x) \log \left (1+\frac {x^{-m}}{d f}\right )+\frac {2}{3} b^2 n^2 \log ^3(x) \log \left (1+\frac {x^{-m}}{d f}\right )-b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {x^{-m}}{d f}\right )+a b n \log ^2(x) \log \left (1+d f x^m\right )-\frac {2}{3} b^2 n^2 \log ^3(x) \log \left (1+d f x^m\right )+\frac {a^2 \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}-\frac {2 a b n \log (x) \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}+\frac {b^2 n^2 \log ^2(x) \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}+b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (1+d f x^m\right )+\frac {2 a b \log \left (-d f x^m\right ) \log \left (c x^n\right ) \log \left (1+d f x^m\right )}{m}-\frac {2 b^2 n \log (x) \log \left (-d f x^m\right ) \log \left (c x^n\right ) \log \left (1+d f x^m\right )}{m}+\frac {b^2 \log \left (-d f x^m\right ) \log ^2\left (c x^n\right ) \log \left (1+d f x^m\right )}{m}+\frac {b n \log (x) \left (-b n \log (x)+2 \left (a+b \log \left (c x^n\right )\right )\right ) \operatorname {PolyLog}\left (2,-\frac {x^{-m}}{d f}\right )}{m}+\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,1+d f x^m\right )}{m}+\frac {2 a b n \operatorname {PolyLog}\left (3,-\frac {x^{-m}}{d f}\right )}{m^2}+\frac {2 b^2 n \log \left (c x^n\right ) \operatorname {PolyLog}\left (3,-\frac {x^{-m}}{d f}\right )}{m^2}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (4,-\frac {x^{-m}}{d f}\right )}{m^3} \] Input:

Integrate[((a + b*Log[c*x^n])^2*Log[d*(d^(-1) + f*x^m)])/x,x]
 

Output:

-1/3*(a*b*m*n*Log[x]^3) + (b^2*m*n^2*Log[x]^4)/4 - (b^2*m*n*Log[x]^3*Log[c 
*x^n])/3 - a*b*n*Log[x]^2*Log[1 + 1/(d*f*x^m)] + (2*b^2*n^2*Log[x]^3*Log[1 
 + 1/(d*f*x^m)])/3 - b^2*n*Log[x]^2*Log[c*x^n]*Log[1 + 1/(d*f*x^m)] + a*b* 
n*Log[x]^2*Log[1 + d*f*x^m] - (2*b^2*n^2*Log[x]^3*Log[1 + d*f*x^m])/3 + (a 
^2*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])/m - (2*a*b*n*Log[x]*Log[-(d*f*x^m)]*L 
og[1 + d*f*x^m])/m + (b^2*n^2*Log[x]^2*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])/m 
 + b^2*n*Log[x]^2*Log[c*x^n]*Log[1 + d*f*x^m] + (2*a*b*Log[-(d*f*x^m)]*Log 
[c*x^n]*Log[1 + d*f*x^m])/m - (2*b^2*n*Log[x]*Log[-(d*f*x^m)]*Log[c*x^n]*L 
og[1 + d*f*x^m])/m + (b^2*Log[-(d*f*x^m)]*Log[c*x^n]^2*Log[1 + d*f*x^m])/m 
 + (b*n*Log[x]*(-(b*n*Log[x]) + 2*(a + b*Log[c*x^n]))*PolyLog[2, -(1/(d*f* 
x^m))])/m + ((a - b*n*Log[x] + b*Log[c*x^n])^2*PolyLog[2, 1 + d*f*x^m])/m 
+ (2*a*b*n*PolyLog[3, -(1/(d*f*x^m))])/m^2 + (2*b^2*n*Log[c*x^n]*PolyLog[3 
, -(1/(d*f*x^m))])/m^2 + (2*b^2*n^2*PolyLog[4, -(1/(d*f*x^m))])/m^3
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2821, 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.

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

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

Input:

Int[((a + b*Log[c*x^n])^2*Log[d*(d^(-1) + f*x^m)])/x,x]
 

Output:

-(((a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*x^m)])/m) + (2*b*n*(((a + b*Log[c 
*x^n])*PolyLog[3, -(d*f*x^m)])/m - (b*n*PolyLog[4, -(d*f*x^m)])/m^2))/m
 

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 = 136.58 (sec) , antiderivative size = 684, normalized size of antiderivative = 9.37

method	result	size
risch	\(\frac {b^{2} n^{2} \ln \left (x \right )^{2} \operatorname {polylog}\left (2, -d f \,x^{m}\right )}{m}-b^{2} n \ln \left (\frac {1}{d}+f \,x^{m}\right ) \ln \left (x \right )^{2} \ln \left (x^{n}\right )+b^{2} \ln \left (\frac {1}{d}+f \,x^{m}\right ) \ln \left (x \right ) \ln \left (x^{n}\right )^{2}-\frac {b^{2} \operatorname {dilog}\left (d f \,x^{m}+1\right ) \ln \left (x \right )^{2} n^{2}}{m}+\frac {2 b^{2} \operatorname {dilog}\left (d f \,x^{m}+1\right ) \ln \left (x \right ) \ln \left (x^{n}\right ) n}{m}-\frac {2 b^{2} n \ln \left (x \right ) \operatorname {polylog}\left (2, -d f \,x^{m}\right ) \ln \left (x^{n}\right )}{m}+\frac {2 b^{2} n \operatorname {polylog}\left (3, -d f \,x^{m}\right ) \ln \left (x^{n}\right )}{m^{2}}+b^{2} \ln \left (x \right )^{2} \ln \left (d f \,x^{m}+1\right ) \ln \left (x^{n}\right ) n +\frac {b^{2} \ln \left (d \left (\frac {1}{d}+f \,x^{m}\right )\right ) \ln \left (x^{n}\right )^{3}}{3 n}+\frac {b^{2} n^{2} \ln \left (\frac {1}{d}+f \,x^{m}\right ) \ln \left (x \right )^{3}}{3}-\frac {b^{2} \ln \left (\frac {1}{d}+f \,x^{m}\right ) \ln \left (x^{n}\right )^{3}}{3 n}-\frac {b^{2} \operatorname {dilog}\left (d f \,x^{m}+1\right ) \ln \left (x^{n}\right )^{2}}{m}-b^{2} \ln \left (x \right ) \ln \left (d f \,x^{m}+1\right ) \ln \left (x^{n}\right )^{2}-\frac {2 b^{2} n^{2} \operatorname {polylog}\left (4, -d f \,x^{m}\right )}{m^{3}}-\frac {b^{2} n^{2} \ln \left (x \right )^{3} \ln \left (d f \,x^{m}+1\right )}{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 \left (\frac {\ln \left (x \right )^{2} n \ln \left (d \left (\frac {1}{d}+f \,x^{m}\right )\right )}{2}+\ln \left (d \left (\frac {1}{d}+f \,x^{m}\right )\right ) \ln \left (x \right ) \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )-\frac {n \ln \left (x \right )^{2} \ln \left (d f \,x^{m}+1\right )}{2}-\frac {n \ln \left (x \right ) \operatorname {polylog}\left (2, -d f \,x^{m}\right )}{m}+\frac {n \operatorname {polylog}\left (3, -d f \,x^{m}\right )}{m^{2}}-\frac {\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) \operatorname {dilog}\left (d f \,x^{m}+1\right )}{m}-\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) \ln \left (x \right ) \ln \left (d f \,x^{m}+1\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} \operatorname {dilog}\left (d f \,x^{m}+1\right )}{4 m}\)	\(684\)

Input:

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

Output:

b^2*n^2/m*ln(x)^2*polylog(2,-d*f*x^m)-b^2*n*ln(1/d+f*x^m)*ln(x)^2*ln(x^n)+ 
b^2*ln(1/d+f*x^m)*ln(x)*ln(x^n)^2-b^2/m*dilog(d*f*x^m+1)*ln(x)^2*n^2+2*b^2 
/m*dilog(d*f*x^m+1)*ln(x)*ln(x^n)*n-2*b^2*n/m*ln(x)*polylog(2,-d*f*x^m)*ln 
(x^n)+2*b^2*n/m^2*polylog(3,-d*f*x^m)*ln(x^n)+b^2*ln(x)^2*ln(d*f*x^m+1)*ln 
(x^n)*n+1/3*b^2*ln(d*(1/d+f*x^m))/n*ln(x^n)^3+1/3*b^2*n^2*ln(1/d+f*x^m)*ln 
(x)^3-1/3*b^2/n*ln(1/d+f*x^m)*ln(x^n)^3-b^2/m*dilog(d*f*x^m+1)*ln(x^n)^2-b 
^2*ln(x)*ln(d*f*x^m+1)*ln(x^n)^2-2*b^2*n^2*polylog(4,-d*f*x^m)/m^3-1/3*b^2 
*n^2*ln(x)^3*ln(d*f*x^m+1)+(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*(1/2*ln(x)^2*n*ln(d*(1/d+f*x^m))+ln(d*(1/d+ 
f*x^m))*ln(x)*(ln(x^n)-n*ln(x))-1/2*n*ln(x)^2*ln(d*f*x^m+1)-n/m*ln(x)*poly 
log(2,-d*f*x^m)+n/m^2*polylog(3,-d*f*x^m)-1/m*(ln(x^n)-n*ln(x))*dilog(d*f* 
x^m+1)-(ln(x^n)-n*ln(x))*ln(x)*ln(d*f*x^m+1))-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/m*dilog(d*f*x^m+1)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=-\frac {2 \, b^{2} n^{2} {\rm polylog}\left (4, -d f x^{m}\right ) + {\left (b^{2} m^{2} n^{2} \log \left (x\right )^{2} + b^{2} m^{2} \log \left (c\right )^{2} + 2 \, a b m^{2} \log \left (c\right ) + a^{2} m^{2} + 2 \, {\left (b^{2} m^{2} n \log \left (c\right ) + a b m^{2} n\right )} \log \left (x\right )\right )} {\rm Li}_2\left (-d f x^{m}\right ) - 2 \, {\left (b^{2} m n^{2} \log \left (x\right ) + b^{2} m n \log \left (c\right ) + a b m n\right )} {\rm polylog}\left (3, -d f x^{m}\right )}{m^{3}} \] Input:

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

Output:

-(2*b^2*n^2*polylog(4, -d*f*x^m) + (b^2*m^2*n^2*log(x)^2 + b^2*m^2*log(c)^ 
2 + 2*a*b*m^2*log(c) + a^2*m^2 + 2*(b^2*m^2*n*log(c) + a*b*m^2*n)*log(x))* 
dilog(-d*f*x^m) - 2*(b^2*m*n^2*log(x) + b^2*m*n*log(c) + a*b*m*n)*polylog( 
3, -d*f*x^m))/m^3
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*ln(c*x**n))**2*ln(d*(1/d+f*x**m))/x,x)
 

Output:

Exception raised: TypeError >> Invalid comparison of non-real zoo
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*int(log(x**m*d*f + 1)/(x**m*d*f*x + x),x)*a**2*m + 2*int((log(x**m*d*f 
+ 1)*log(x**n*c)**2)/x,x)*b**2*m + 4*int((log(x**m*d*f + 1)*log(x**n*c))/x 
,x)*a*b*m + log(x**m*d*f + 1)**2*a**2)/(2*m)