Integrand size = 28, antiderivative size = 101 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=-\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )^3 \operatorname {PolyLog}\left (2,-d f x^2\right )+\frac {3}{4} b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (3,-d f x^2\right )-\frac {3}{4} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (4,-d f x^2\right )+\frac {3}{8} b^3 n^3 \operatorname {PolyLog}\left (5,-d f x^2\right ) \]
-1/2*(a+b*ln(c*x^n))^3*polylog(2,-d*f*x^2)+3/4*b*n*(a+b*ln(c*x^n))^2*polyl og(3,-d*f*x^2)-3/4*b^2*n^2*(a+b*ln(c*x^n))*polylog(4,-d*f*x^2)+3/8*b^3*n^3 *polylog(5,-d*f*x^2)
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 754, normalized size of antiderivative = 7.47 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\frac {1}{4} \left (-\log (x) \left (b^3 n^3 \log ^3(x)-4 b^2 n^2 \log ^2(x) \left (a+b \log \left (c x^n\right )\right )+6 b n \log (x) \left (a+b \log \left (c x^n\right )\right )^2-4 \left (a+b \log \left (c x^n\right )\right )^3\right ) \log \left (1+d f x^2\right )-4 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3 \left (\log (x) \left (\log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )+\operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )\right )-6 b n \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )-2 \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )-2 \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )\right )+4 b^2 n^2 \left (-a+b n \log (x)-b \log \left (c x^n\right )\right ) \left (\log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )+6 \operatorname {PolyLog}\left (4,-i \sqrt {d} \sqrt {f} x\right )+6 \operatorname {PolyLog}\left (4,i \sqrt {d} \sqrt {f} x\right )\right )-b^3 n^3 \left (\log ^4(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log ^4(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+4 \log ^3(x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+4 \log ^3(x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )-12 \log ^2(x) \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )-12 \log ^2(x) \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )+24 \log (x) \operatorname {PolyLog}\left (4,-i \sqrt {d} \sqrt {f} x\right )+24 \log (x) \operatorname {PolyLog}\left (4,i \sqrt {d} \sqrt {f} x\right )-24 \operatorname {PolyLog}\left (5,-i \sqrt {d} \sqrt {f} x\right )-24 \operatorname {PolyLog}\left (5,i \sqrt {d} \sqrt {f} x\right )\right )\right ) \]
(-(Log[x]*(b^3*n^3*Log[x]^3 - 4*b^2*n^2*Log[x]^2*(a + b*Log[c*x^n]) + 6*b* n*Log[x]*(a + b*Log[c*x^n])^2 - 4*(a + b*Log[c*x^n])^3)*Log[1 + d*f*x^2]) - 4*(a - b*n*Log[x] + b*Log[c*x^n])^3*(Log[x]*(Log[1 - I*Sqrt[d]*Sqrt[f]*x ] + Log[1 + I*Sqrt[d]*Sqrt[f]*x]) + PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] + P olyLog[2, I*Sqrt[d]*Sqrt[f]*x]) - 6*b*n*(a - b*n*Log[x] + b*Log[c*x^n])^2* (Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + Log[x]^2*Log[1 + I*Sqrt[d]*Sqrt[f ]*x] + 2*Log[x]*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] + 2*Log[x]*PolyLog[2, I *Sqrt[d]*Sqrt[f]*x] - 2*PolyLog[3, (-I)*Sqrt[d]*Sqrt[f]*x] - 2*PolyLog[3, I*Sqrt[d]*Sqrt[f]*x]) + 4*b^2*n^2*(-a + b*n*Log[x] - b*Log[c*x^n])*(Log[x] ^3*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + Log[x]^3*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + 3*Log[x]^2*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] + 3*Log[x]^2*PolyLog[2, I*Sq rt[d]*Sqrt[f]*x] - 6*Log[x]*PolyLog[3, (-I)*Sqrt[d]*Sqrt[f]*x] - 6*Log[x]* PolyLog[3, I*Sqrt[d]*Sqrt[f]*x] + 6*PolyLog[4, (-I)*Sqrt[d]*Sqrt[f]*x] + 6 *PolyLog[4, I*Sqrt[d]*Sqrt[f]*x]) - b^3*n^3*(Log[x]^4*Log[1 - I*Sqrt[d]*Sq rt[f]*x] + Log[x]^4*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + 4*Log[x]^3*PolyLog[2, ( -I)*Sqrt[d]*Sqrt[f]*x] + 4*Log[x]^3*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] - 12*L og[x]^2*PolyLog[3, (-I)*Sqrt[d]*Sqrt[f]*x] - 12*Log[x]^2*PolyLog[3, I*Sqrt [d]*Sqrt[f]*x] + 24*Log[x]*PolyLog[4, (-I)*Sqrt[d]*Sqrt[f]*x] + 24*Log[x]* PolyLog[4, I*Sqrt[d]*Sqrt[f]*x] - 24*PolyLog[5, (-I)*Sqrt[d]*Sqrt[f]*x] - 24*PolyLog[5, I*Sqrt[d]*Sqrt[f]*x]))/4
Time = 0.40 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, 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.
| \(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {3}{2} b n \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f x^2\right )}{x}dx-\frac {1}{2} \operatorname {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {3}{2} b n \left (\frac {1}{2} \operatorname {PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2-b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f x^2\right )}{x}dx\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {3}{2} b n \left (\frac {1}{2} \operatorname {PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2-b n \left (\frac {1}{2} \operatorname {PolyLog}\left (4,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \int \frac {\operatorname {PolyLog}\left (4,-d f x^2\right )}{x}dx\right )\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3}{2} b n \left (\frac {1}{2} \operatorname {PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2-b n \left (\frac {1}{2} \operatorname {PolyLog}\left (4,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \operatorname {PolyLog}\left (5,-d f x^2\right )\right )\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^3\) |
-1/2*((a + b*Log[c*x^n])^3*PolyLog[2, -(d*f*x^2)]) + (3*b*n*(((a + b*Log[c *x^n])^2*PolyLog[3, -(d*f*x^2)])/2 - b*n*(((a + b*Log[c*x^n])*PolyLog[4, - (d*f*x^2)])/2 - (b*n*PolyLog[5, -(d*f*x^2)])/4)))/2
3.1.42.3.1 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 = 68.08 (sec) , antiderivative size = 1296, normalized size of antiderivative = 12.83
3*ln(d*f*x^2+1)*ln(x)^3*ln(x^n)*b^3*n^2+ln(x)^4*ln(1+x*(-d*f)^(1/2))*b^3*n ^3+ln(x)^4*ln(1-x*(-d*f)^(1/2))*b^3*n^3+ln(x)^3*dilog(1+x*(-d*f)^(1/2))*b^ 3*n^3+ln(x)^3*dilog(1-x*(-d*f)^(1/2))*b^3*n^3-1/2*ln(x)^3*polylog(2,-d*f*x ^2)*b^3*n^3+ln(d*f*x^2+1)*ln(x)*ln(x^n)^3*b^3-ln(x)*ln(x^n)^3*ln(1+x*(-d*f )^(1/2))*b^3-ln(x)*ln(x^n)^3*ln(1-x*(-d*f)^(1/2))*b^3+3/4*ln(x^n)^2*polylo g(3,-d*f*x^2)*b^3*n-3/4*ln(x^n)*polylog(4,-d*f*x^2)*b^3*n^2-ln(d*f*x^2+1)* ln(x)^4*b^3*n^3+3/2*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*cs gn(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*((ln(x^n)-n*ln(x))^2*(ln(x)*ln(d*f*x^2+1)-2*d*f *(1/2*ln(x)*(ln(1+x*(-d*f)^(1/2))+ln(1-x*(-d*f)^(1/2)))/d/f+1/2*(dilog(1+x *(-d*f)^(1/2))+dilog(1-x*(-d*f)^(1/2)))/d/f))+n^2*(-1/2*ln(x)^2*polylog(2, -d*f*x^2)+1/2*ln(x)*polylog(3,-d*f*x^2)-1/4*polylog(4,-d*f*x^2))+2*n*(ln(x ^n)-n*ln(x))*(-1/2*ln(x)*polylog(2,-d*f*x^2)+1/4*polylog(3,-d*f*x^2)))-3*l n(x)^3*ln(x^n)*ln(1+x*(-d*f)^(1/2))*b^3*n^2-3*ln(x)^3*ln(x^n)*ln(1-x*(-d*f )^(1/2))*b^3*n^2-3*ln(d*f*x^2+1)*ln(x)^2*ln(x^n)^2*b^3*n+3*ln(x)^2*ln(x^n) ^2*ln(1+x*(-d*f)^(1/2))*b^3*n+3*ln(x)^2*ln(x^n)^2*ln(1-x*(-d*f)^(1/2))*b^3 *n-3*ln(x)^2*ln(x^n)*dilog(1+x*(-d*f)^(1/2))*b^3*n^2-3*ln(x)^2*ln(x^n)*dil og(1-x*(-d*f)^(1/2))*b^3*n^2+3/2*ln(x)^2*ln(x^n)*polylog(2,-d*f*x^2)*b^3*n ^2+3*ln(x)*ln(x^n)^2*dilog(1+x*(-d*f)^(1/2))*b^3*n+3*ln(x)*ln(x^n)^2*dilog (1-x*(-d*f)^(1/2))*b^3*n-3/2*ln(x)*ln(x^n)^2*polylog(2,-d*f*x^2)*b^3*n+...
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x} \,d x } \]
integral((b^3*log(d*f*x^2 + 1)*log(c*x^n)^3 + 3*a*b^2*log(d*f*x^2 + 1)*log (c*x^n)^2 + 3*a^2*b*log(d*f*x^2 + 1)*log(c*x^n) + a^3*log(d*f*x^2 + 1))/x, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x} \,d x } \]
-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^2 + 1) - integrate(-1/2*(b^3*d *f*n^3*x*log(x)^4 - 4*b^3*d*f*x*log(x)*log(x^n)^3 - 4*(b^3*d*f*n^2*log(c) + a*b^2*d*f*n^2)*x*log(x)^3 + 6*(b^3*d*f*n*log(c)^2 + 2*a*b^2*d*f*n*log(c) + a^2*b*d*f*n)*x*log(x)^2 - 4*(b^3*d*f*log(c)^3 + 3*a*b^2*d*f*log(c)^2 + 3*a^2*b*d*f*log(c) + a^3*d*f)*x*log(x) + 6*(b^3*d*f*n*x*log(x)^2 - 2*(b^3* d*f*log(c) + a*b^2*d*f)*x*log(x))*log(x^n)^2 - 4*(b^3*d*f*n^2*x*log(x)^3 - 3*(b^3*d*f*n*log(c) + a*b^2*d*f*n)*x*log(x)^2 + 3*(b^3*d*f*log(c)^2 + 2*a *b^2*d*f*log(c) + a^2*b*d*f)*x*log(x))*log(x^n))/(d*f*x^2 + 1), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x} \,d x \]