Integrand size = 28, antiderivative size = 105 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \operatorname {PolyLog}\left (2,-d f x^m\right )}{m}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (3,-d f x^m\right )}{m^2}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (4,-d f x^m\right )}{m^3}+\frac {6 b^3 n^3 \operatorname {PolyLog}\left (5,-d f x^m\right )}{m^4} \]
-(a+b*ln(c*x^n))^3*polylog(2,-d*f*x^m)/m+3*b*n*(a+b*ln(c*x^n))^2*polylog(3 ,-d*f*x^m)/m^2-6*b^2*n^2*(a+b*ln(c*x^n))*polylog(4,-d*f*x^m)/m^3+6*b^3*n^3 *polylog(5,-d*f*x^m)/m^4
Leaf count is larger than twice the leaf count of optimal. \(1035\) vs. \(2(105)=210\).
Time = 0.26 (sec) , antiderivative size = 1035, normalized size of antiderivative = 9.86 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=-\frac {1}{2} a^2 b m n \log ^3(x)+\frac {3}{4} a b^2 m n^2 \log ^4(x)-\frac {3}{10} b^3 m n^3 \log ^5(x)-a b^2 m n \log ^3(x) \log \left (c x^n\right )+\frac {3}{4} b^3 m n^2 \log ^4(x) \log \left (c x^n\right )-\frac {1}{2} b^3 m n \log ^3(x) \log ^2\left (c x^n\right )-\frac {3}{2} a^2 b n \log ^2(x) \log \left (1+\frac {x^{-m}}{d f}\right )+2 a b^2 n^2 \log ^3(x) \log \left (1+\frac {x^{-m}}{d f}\right )-\frac {3}{4} b^3 n^3 \log ^4(x) \log \left (1+\frac {x^{-m}}{d f}\right )-3 a b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {x^{-m}}{d f}\right )+2 b^3 n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (1+\frac {x^{-m}}{d f}\right )-\frac {3}{2} b^3 n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (1+\frac {x^{-m}}{d f}\right )+\frac {3}{2} a^2 b n \log ^2(x) \log \left (1+d f x^m\right )-2 a b^2 n^2 \log ^3(x) \log \left (1+d f x^m\right )+\frac {3}{4} b^3 n^3 \log ^4(x) \log \left (1+d f x^m\right )+\frac {a^3 \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}-\frac {3 a^2 b n \log (x) \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}+\frac {3 a b^2 n^2 \log ^2(x) \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}-\frac {b^3 n^3 \log ^3(x) \log \left (-d f x^m\right ) \log \left (1+d f x^m\right )}{m}+3 a b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (1+d f x^m\right )-2 b^3 n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (1+d f x^m\right )+\frac {3 a^2 b \log \left (-d f x^m\right ) \log \left (c x^n\right ) \log \left (1+d f x^m\right )}{m}-\frac {6 a 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 {3 b^3 n^2 \log ^2(x) \log \left (-d f x^m\right ) \log \left (c x^n\right ) \log \left (1+d f x^m\right )}{m}+\frac {3}{2} b^3 n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (1+d f x^m\right )+\frac {3 a 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 {3 b^3 n \log (x) \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^3 \log \left (-d f x^m\right ) \log ^3\left (c x^n\right ) \log \left (1+d f x^m\right )}{m}+\frac {b n \log (x) \left (b^2 n^2 \log ^2(x)-3 b n \log (x) \left (a+b \log \left (c x^n\right )\right )+3 \left (a+b \log \left (c x^n\right )\right )^2\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 )^3 \operatorname {PolyLog}\left (2,1+d f x^m\right )}{m}+\frac {3 a^2 b n \operatorname {PolyLog}\left (3,-\frac {x^{-m}}{d f}\right )}{m^2}+\frac {6 a b^2 n \log \left (c x^n\right ) \operatorname {PolyLog}\left (3,-\frac {x^{-m}}{d f}\right )}{m^2}+\frac {3 b^3 n \log ^2\left (c x^n\right ) \operatorname {PolyLog}\left (3,-\frac {x^{-m}}{d f}\right )}{m^2}+\frac {6 a b^2 n^2 \operatorname {PolyLog}\left (4,-\frac {x^{-m}}{d f}\right )}{m^3}+\frac {6 b^3 n^2 \log \left (c x^n\right ) \operatorname {PolyLog}\left (4,-\frac {x^{-m}}{d f}\right )}{m^3}+\frac {6 b^3 n^3 \operatorname {PolyLog}\left (5,-\frac {x^{-m}}{d f}\right )}{m^4} \]
-1/2*(a^2*b*m*n*Log[x]^3) + (3*a*b^2*m*n^2*Log[x]^4)/4 - (3*b^3*m*n^3*Log[ x]^5)/10 - a*b^2*m*n*Log[x]^3*Log[c*x^n] + (3*b^3*m*n^2*Log[x]^4*Log[c*x^n ])/4 - (b^3*m*n*Log[x]^3*Log[c*x^n]^2)/2 - (3*a^2*b*n*Log[x]^2*Log[1 + 1/( d*f*x^m)])/2 + 2*a*b^2*n^2*Log[x]^3*Log[1 + 1/(d*f*x^m)] - (3*b^3*n^3*Log[ x]^4*Log[1 + 1/(d*f*x^m)])/4 - 3*a*b^2*n*Log[x]^2*Log[c*x^n]*Log[1 + 1/(d* f*x^m)] + 2*b^3*n^2*Log[x]^3*Log[c*x^n]*Log[1 + 1/(d*f*x^m)] - (3*b^3*n*Lo g[x]^2*Log[c*x^n]^2*Log[1 + 1/(d*f*x^m)])/2 + (3*a^2*b*n*Log[x]^2*Log[1 + d*f*x^m])/2 - 2*a*b^2*n^2*Log[x]^3*Log[1 + d*f*x^m] + (3*b^3*n^3*Log[x]^4* Log[1 + d*f*x^m])/4 + (a^3*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])/m - (3*a^2*b* n*Log[x]*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])/m + (3*a*b^2*n^2*Log[x]^2*Log[- (d*f*x^m)]*Log[1 + d*f*x^m])/m - (b^3*n^3*Log[x]^3*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])/m + 3*a*b^2*n*Log[x]^2*Log[c*x^n]*Log[1 + d*f*x^m] - 2*b^3*n^2* Log[x]^3*Log[c*x^n]*Log[1 + d*f*x^m] + (3*a^2*b*Log[-(d*f*x^m)]*Log[c*x^n] *Log[1 + d*f*x^m])/m - (6*a*b^2*n*Log[x]*Log[-(d*f*x^m)]*Log[c*x^n]*Log[1 + d*f*x^m])/m + (3*b^3*n^2*Log[x]^2*Log[-(d*f*x^m)]*Log[c*x^n]*Log[1 + d*f *x^m])/m + (3*b^3*n*Log[x]^2*Log[c*x^n]^2*Log[1 + d*f*x^m])/2 + (3*a*b^2*L og[-(d*f*x^m)]*Log[c*x^n]^2*Log[1 + d*f*x^m])/m - (3*b^3*n*Log[x]*Log[-(d* f*x^m)]*Log[c*x^n]^2*Log[1 + d*f*x^m])/m + (b^3*Log[-(d*f*x^m)]*Log[c*x^n] ^3*Log[1 + d*f*x^m])/m + (b*n*Log[x]*(b^2*n^2*Log[x]^2 - 3*b*n*Log[x]*(a + b*Log[c*x^n]) + 3*(a + b*Log[c*x^n])^2)*PolyLog[2, -(1/(d*f*x^m))])/m ...
Time = 0.43 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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.
| \(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f x^m\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {3 b n \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f x^m\right )}{x}dx}{m}-\frac {\operatorname {PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {3 b n \left (\frac {\operatorname {PolyLog}\left (3,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac {2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f x^m\right )}{x}dx}{m}\right )}{m}-\frac {\operatorname {PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {3 b n \left (\frac {\operatorname {PolyLog}\left (3,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac {2 b n \left (\frac {\operatorname {PolyLog}\left (4,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m}-\frac {b n \int \frac {\operatorname {PolyLog}\left (4,-d f x^m\right )}{x}dx}{m}\right )}{m}\right )}{m}-\frac {\operatorname {PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3 b n \left (\frac {\operatorname {PolyLog}\left (3,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac {2 b n \left (\frac {\operatorname {PolyLog}\left (4,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m}-\frac {b n \operatorname {PolyLog}\left (5,-d f x^m\right )}{m^2}\right )}{m}\right )}{m}-\frac {\operatorname {PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}\) |
-(((a + b*Log[c*x^n])^3*PolyLog[2, -(d*f*x^m)])/m) + (3*b*n*(((a + b*Log[c *x^n])^2*PolyLog[3, -(d*f*x^m)])/m - (2*b*n*(((a + b*Log[c*x^n])*PolyLog[4 , -(d*f*x^m)])/m - (b*n*PolyLog[5, -(d*f*x^m)])/m^2))/m))/m
3.1.65.3.1 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 = 294.23 (sec) , antiderivative size = 1261, normalized size of antiderivative = 12.01
-3*b^3*n^2/m*dilog(d*f*x^m+1)*ln(x)^2*ln(x^n)+3*b^3*n/m*dilog(d*f*x^m+1)*l n(x)*ln(x^n)^2+3*b^3*n^2/m*ln(x)^2*ln(x^n)*polylog(2,-d*f*x^m)-3*b^3*n/m*l n(x)*ln(x^n)^2*polylog(2,-d*f*x^m)+3/2*b^3*n*ln(x)^2*ln(d*f*x^m+1)*ln(x^n) ^2-b^3*ln(x)*ln(d*f*x^m+1)*ln(x^n)^3+3*b^3*n/m^2*ln(x^n)^2*polylog(3,-d*f* x^m)-6*b^3*n^2/m^3*ln(x^n)*polylog(4,-d*f*x^m)-b^3*n^3/m*ln(x)^3*polylog(2 ,-d*f*x^m)+b^3*n^2*ln(1/d+f*x^m)*ln(x)^3*ln(x^n)-3/2*b^3*n*ln(1/d+f*x^m)*l n(x)^2*ln(x^n)^2+b^3*ln(1/d+f*x^m)*ln(x)*ln(x^n)^3-1/4*b^3/n*ln(1/d+f*x^m) *ln(x^n)^4+6*b^3*n^3*polylog(5,-d*f*x^m)/m^4-1/4*b^3*n^3*ln(1/d+f*x^m)*ln( x)^4+1/4*b^3*n^3*ln(x)^4*ln(d*f*x^m+1)+b^3*n^3/m*dilog(d*f*x^m+1)*ln(x)^3- b^3/m*dilog(d*f*x^m+1)*ln(x^n)^3-b^3*n^2*ln(x)^3*ln(d*f*x^m+1)*ln(x^n)+1/4 *b^3*ln(d*(1/d+f*x^m))/n*ln(x^n)^4-1/8*(-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)^3/m*dilog(d*f*x^m+1)+3/2*(-I*b*P i*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)*b^2* (1/3*ln(x)^3*n^2*ln(d*(1/d+f*x^m))+ln(d*(1/d+f*x^m))*n*ln(x)^2*(ln(x^n)-n* ln(x))+ln(d*(1/d+f*x^m))*ln(x)*(ln(x^n)-n*ln(x))^2+1/3*ln(d*(1/d+f*x^m))/n *(ln(x^n)-n*ln(x))^3-1/3/n*(ln(x^n)-n*ln(x))^3*ln(1/d+f*x^m)-1/3*n^2*ln(x) ^3*ln(d*f*x^m+1)-n^2/m*ln(x)^2*polylog(2,-d*f*x^m)+2*n^2/m^2*ln(x)*polylog (3,-d*f*x^m)-2*n^2/m^3*polylog(4,-d*f*x^m)-1/m*(ln(x^n)-n*ln(x))^2*dilo...
Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (104) = 208\).
Time = 0.27 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.71 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\frac {6 \, b^{3} n^{3} {\rm polylog}\left (5, -d f x^{m}\right ) - {\left (b^{3} m^{3} n^{3} \log \left (x\right )^{3} + b^{3} m^{3} \log \left (c\right )^{3} + 3 \, a b^{2} m^{3} \log \left (c\right )^{2} + 3 \, a^{2} b m^{3} \log \left (c\right ) + a^{3} m^{3} + 3 \, {\left (b^{3} m^{3} n^{2} \log \left (c\right ) + a b^{2} m^{3} n^{2}\right )} \log \left (x\right )^{2} + 3 \, {\left (b^{3} m^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} m^{3} n \log \left (c\right ) + a^{2} b m^{3} n\right )} \log \left (x\right )\right )} {\rm Li}_2\left (-d f x^{m}\right ) - 6 \, {\left (b^{3} m n^{3} \log \left (x\right ) + b^{3} m n^{2} \log \left (c\right ) + a b^{2} m n^{2}\right )} {\rm polylog}\left (4, -d f x^{m}\right ) + 3 \, {\left (b^{3} m^{2} n^{3} \log \left (x\right )^{2} + b^{3} m^{2} n \log \left (c\right )^{2} + 2 \, a b^{2} m^{2} n \log \left (c\right ) + a^{2} b m^{2} n + 2 \, {\left (b^{3} m^{2} n^{2} \log \left (c\right ) + a b^{2} m^{2} n^{2}\right )} \log \left (x\right )\right )} {\rm polylog}\left (3, -d f x^{m}\right )}{m^{4}} \]
(6*b^3*n^3*polylog(5, -d*f*x^m) - (b^3*m^3*n^3*log(x)^3 + b^3*m^3*log(c)^3 + 3*a*b^2*m^3*log(c)^2 + 3*a^2*b*m^3*log(c) + a^3*m^3 + 3*(b^3*m^3*n^2*lo g(c) + a*b^2*m^3*n^2)*log(x)^2 + 3*(b^3*m^3*n*log(c)^2 + 2*a*b^2*m^3*n*log (c) + a^2*b*m^3*n)*log(x))*dilog(-d*f*x^m) - 6*(b^3*m*n^3*log(x) + b^3*m*n ^2*log(c) + a*b^2*m*n^2)*polylog(4, -d*f*x^m) + 3*(b^3*m^2*n^3*log(x)^2 + b^3*m^2*n*log(c)^2 + 2*a*b^2*m^2*n*log(c) + a^2*b*m^2*n + 2*(b^3*m^2*n^2*l og(c) + a*b^2*m^2*n^2)*log(x))*polylog(3, -d*f*x^m))/m^4
Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\text {Exception raised: TypeError} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{m} + \frac {1}{d}\right )} d\right )}{x} \,d x } \]
-1/4*(b^3*n^3*log(x)^4 - 4*b^3*log(x)*log(x^n)^3 - 4*(b^3*n^2*log(c) + a*b ^2*n^2)*log(x)^3 + 6*(b^3*n*log(c)^2 + 2*a*b^2*n*log(c) + a^2*b*n)*log(x)^ 2 + 6*(b^3*n*log(x)^2 - 2*(b^3*log(c) + a*b^2)*log(x))*log(x^n)^2 - 4*(b^3 *n^2*log(x)^3 - 3*(b^3*n*log(c) + a*b^2*n)*log(x)^2 + 3*(b^3*log(c)^2 + 2* a*b^2*log(c) + a^2*b)*log(x))*log(x^n) - 4*(b^3*log(c)^3 + 3*a*b^2*log(c)^ 2 + 3*a^2*b*log(c) + a^3)*log(x))*log(d*f*x^m + 1) - integrate(1/4*(4*b^3* d*f*m*x^m*log(x)*log(x^n)^3 - 6*(b^3*d*f*m*n*log(x)^2 - 2*(b^3*d*f*m*log(c ) + a*b^2*d*f*m)*log(x))*x^m*log(x^n)^2 + 4*(b^3*d*f*m*n^2*log(x)^3 - 3*(b ^3*d*f*m*n*log(c) + a*b^2*d*f*m*n)*log(x)^2 + 3*(b^3*d*f*m*log(c)^2 + 2*a* b^2*d*f*m*log(c) + a^2*b*d*f*m)*log(x))*x^m*log(x^n) - (b^3*d*f*m*n^3*log( x)^4 - 4*(b^3*d*f*m*n^2*log(c) + a*b^2*d*f*m*n^2)*log(x)^3 + 6*(b^3*d*f*m* n*log(c)^2 + 2*a*b^2*d*f*m*n*log(c) + a^2*b*d*f*m*n)*log(x)^2 - 4*(b^3*d*f *m*log(c)^3 + 3*a*b^2*d*f*m*log(c)^2 + 3*a^2*b*d*f*m*log(c) + a^3*d*f*m)*l og(x))*x^m)/(d*f*x*x^m + x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{m} + \frac {1}{d}\right )} d\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\int \frac {\ln \left (d\,\left (f\,x^m+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x} \,d x \]