Integrand size = 28, antiderivative size = 73 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f x^m\right )}{m}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f x^m\right )}{m^2}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (4,-d f x^m\right )}{m^3} \]
-(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
Leaf count is larger than twice the leaf count of optimal. \(526\) vs. \(2(73)=146\).
Time = 0.16 (sec) , antiderivative size = 526, normalized size of antiderivative = 7.21 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=-\frac {1}{3} a b m n \log ^3(x)+\frac {1}{4} b^2 m n^2 \log ^4(x)-\frac {1}{3} b^2 m n \log ^3(x) \log \left (c x^n\right )-a b n \log ^2(x) \log \left (1+\frac {x^{-m}}{d f}\right )+\frac {2}{3} b^2 n^2 \log ^3(x) \log \left (1+\frac {x^{-m}}{d f}\right )-b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {x^{-m}}{d f}\right )+a b n \log ^2(x) \log \left (1+d f x^m\right )-\frac {2}{3} b^2 n^2 \log ^3(x) \log \left (1+d f x^m\right )+\frac {a^2 \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}-\frac {2 a b n \log (x) \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}+\frac {b^2 n^2 \log ^2(x) \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}+b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (1+d f x^m\right )+\frac {2 a b \log \left (-d f x^m\right ) \log \left (c x^n\right ) \log \left (1+d f x^m\right )}{m}-\frac {2 b^2 n \log (x) \log \left (-d f x^m\right ) \log \left (c x^n\right ) \log \left (1+d f x^m\right )}{m}+\frac {b^2 \log \left (-d f x^m\right ) \log ^2\left (c x^n\right ) \log \left (1+d f x^m\right )}{m}+\frac {b n \log (x) \left (-b n \log (x)+2 \left (a+b \log \left (c x^n\right )\right )\right ) \operatorname {PolyLog}\left (2,-\frac {x^{-m}}{d f}\right )}{m}+\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,1+d f x^m\right )}{m}+\frac {2 a b n \operatorname {PolyLog}\left (3,-\frac {x^{-m}}{d f}\right )}{m^2}+\frac {2 b^2 n \log \left (c x^n\right ) \operatorname {PolyLog}\left (3,-\frac {x^{-m}}{d f}\right )}{m^2}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (4,-\frac {x^{-m}}{d f}\right )}{m^3} \]
-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
Time = 0.34 (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 |
| \(\displaystyle \frac {2 b n \left (\frac {\operatorname {PolyLog}\left (3,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m}-\frac {b n \operatorname {PolyLog}\left (4,-d f x^m\right )}{m^2}\right )}{m}-\frac {\operatorname {PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}\) |
-(((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
3.1.66.3.1 Defintions of rubi rules used
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]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 67.26 (sec) , antiderivative size = 684, normalized size of antiderivative = 9.37
| method | result | size |
| risch | \(b^{2} \ln \left (x \right )^{2} \ln \left (x^{n}\right ) \ln \left (d f \,x^{m}+1\right ) n +\frac {b^{2} n^{2} \ln \left (x \right )^{2} \operatorname {Li}_{2}\left (-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} n \,\operatorname {Li}_{3}\left (-d f \,x^{m}\right ) \ln \left (x^{n}\right )}{m^{2}}-\frac {2 b^{2} n \ln \left (x \right ) \operatorname {Li}_{2}\left (-d f \,x^{m}\right ) \ln \left (x^{n}\right )}{m}-\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 (x^{n}\right )^{2} \ln \left (d f \,x^{m}+1\right )-\frac {b^{2} n^{2} \ln \left (x \right )^{3} \ln \left (d f \,x^{m}+1\right )}{3}+\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 {2 b^{2} n^{2} \operatorname {Li}_{4}\left (-d f \,x^{m}\right )}{m^{3}}+\frac {2 b^{2} \operatorname {dilog}\left (d f \,x^{m}+1\right ) \ln \left (x \right ) \ln \left (x^{n}\right ) n}{m}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \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 )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (\frac {\ln \left (d \left (\frac {1}{d}+f \,x^{m}\right )\right ) n \ln \left (x \right )^{2}}{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 {Li}_{2}\left (-d f \,x^{m}\right )}{m}+\frac {n \,\operatorname {Li}_{3}\left (-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 c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \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 )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2} \operatorname {dilog}\left (d f \,x^{m}+1\right )}{4 m}\) | \(684\) |
b^2*ln(x)^2*ln(x^n)*ln(d*f*x^m+1)*n+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*n/m^2*polylog(3,-d*f*x^m)*ln(x^n)-2*b ^2*n/m*ln(x)*polylog(2,-d*f*x^m)*ln(x^n)-b^2/m*dilog(d*f*x^m+1)*ln(x^n)^2- b^2*ln(x)*ln(x^n)^2*ln(d*f*x^m+1)-1/3*b^2*n^2*ln(x)^3*ln(d*f*x^m+1)+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)^3-2*b^2*n^2*polylog(4,-d*f*x^m)/m^3+2*b^2/m*dilog( d*f*x^m+1)*ln(x)*ln(x^n)*n+(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I* b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*c sgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b*(1/2*ln(d*(1/d+f*x^m))*n*ln(x)^2+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)*pol ylog(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*b*Pi*csgn(I*c)*csgn (I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)* csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)^2/m*dilog(d*f*x^m+1)
Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=-\frac {2 \, b^{2} n^{2} {\rm polylog}\left (4, -d f x^{m}\right ) + {\left (b^{2} m^{2} n^{2} \log \left (x\right )^{2} + b^{2} m^{2} \log \left (c\right )^{2} + 2 \, a b m^{2} \log \left (c\right ) + a^{2} m^{2} + 2 \, {\left (b^{2} m^{2} n \log \left (c\right ) + a b m^{2} n\right )} \log \left (x\right )\right )} {\rm Li}_2\left (-d f x^{m}\right ) - 2 \, {\left (b^{2} m n^{2} \log \left (x\right ) + b^{2} m n \log \left (c\right ) + a b m n\right )} {\rm polylog}\left (3, -d f x^{m}\right )}{m^{3}} \]
-(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
Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\text {Exception raised: TypeError} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{m} + \frac {1}{d}\right )} d\right )}{x} \,d x } \]
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)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{m} + \frac {1}{d}\right )} d\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\int \frac {\ln \left (d\,\left (f\,x^m+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x} \,d x \]