Integrand size = 28, antiderivative size = 70 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=-\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f x^2\right )+\frac {1}{2} b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f x^2\right )-\frac {1}{4} b^2 n^2 \operatorname {PolyLog}\left (4,-d f x^2\right ) \]
-1/2*(a+b*ln(c*x^n))^2*polylog(2,-d*f*x^2)+1/2*b*n*(a+b*ln(c*x^n))*polylog (3,-d*f*x^2)-1/4*b^2*n^2*polylog(4,-d*f*x^2)
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 484, normalized size of antiderivative = 6.91 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\frac {1}{3} \left (\log (x) \left (b^2 n^2 \log ^2(x)-3 b n \log (x) \left (a+b \log \left (c x^n\right )\right )+3 \left (a+b \log \left (c x^n\right )\right )^2\right ) \log \left (1+d f x^2\right )-3 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (\log (x) \left (\log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )+\operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )\right )+3 b n \left (-a+b n \log (x)-b \log \left (c x^n\right )\right ) \left (\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )-2 \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )-2 \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )\right )-b^2 n^2 \left (\log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )+6 \operatorname {PolyLog}\left (4,-i \sqrt {d} \sqrt {f} x\right )+6 \operatorname {PolyLog}\left (4,i \sqrt {d} \sqrt {f} x\right )\right )\right ) \]
(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)*Log[1 + d*f*x^2] - 3*(a - b*n*Log[x] + b*Log[c*x^n])^2*(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] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]) + 3*b*n*(-a + b*n*Log[x] - b*Log[c*x^n])*(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]) - b^2*n^2*(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*Sqrt[d]*S qrt[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*PolyLo g[4, I*Sqrt[d]*Sqrt[f]*x]))/3
Time = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.97, 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.
\(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx\) |
\(\Big \downarrow \) 2821 |
\(\displaystyle b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f x^2\right )}{x}dx-\frac {1}{2} \operatorname {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2\) |
\(\Big \downarrow \) 2830 |
\(\displaystyle b n \left (\frac {1}{2} \operatorname {PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \int \frac {\operatorname {PolyLog}\left (3,-d f x^2\right )}{x}dx\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle b n \left (\frac {1}{2} \operatorname {PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \operatorname {PolyLog}\left (4,-d f x^2\right )\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2\) |
-1/2*((a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*x^2)]) + b*n*(((a + b*Log[c*x^ n])*PolyLog[3, -(d*f*x^2)])/2 - (b*n*PolyLog[4, -(d*f*x^2)])/4)
3.1.34.3.1 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 = 14.67 (sec) , antiderivative size = 825, normalized size of antiderivative = 11.79
ln(d*f*x^2+1)*ln(x)^3*b^2*n^2-ln(x)^3*ln(1+x*(-d*f)^(1/2))*b^2*n^2-ln(x)^3 *ln(1-x*(-d*f)^(1/2))*b^2*n^2-2*ln(d*f*x^2+1)*ln(x)^2*ln(x^n)*b^2*n+2*ln(x )^2*ln(x^n)*ln(1+x*(-d*f)^(1/2))*b^2*n+2*ln(x)^2*ln(x^n)*ln(1-x*(-d*f)^(1/ 2))*b^2*n-ln(x)^2*dilog(1+x*(-d*f)^(1/2))*b^2*n^2-ln(x)^2*dilog(1-x*(-d*f) ^(1/2))*b^2*n^2+ln(d*f*x^2+1)*ln(x)*ln(x^n)^2*b^2-ln(x)*ln(x^n)^2*ln(1+x*( -d*f)^(1/2))*b^2-ln(x)*ln(x^n)^2*ln(1-x*(-d*f)^(1/2))*b^2+2*ln(x)*ln(x^n)* dilog(1+x*(-d*f)^(1/2))*b^2*n+2*ln(x)*ln(x^n)*dilog(1-x*(-d*f)^(1/2))*b^2* n-ln(x^n)^2*dilog(1+x*(-d*f)^(1/2))*b^2-ln(x^n)^2*dilog(1-x*(-d*f)^(1/2))* b^2+1/2*ln(x)^2*polylog(2,-d*f*x^2)*b^2*n^2-1/4*b^2*n^2*polylog(4,-d*f*x^2 )-ln(x)*ln(x^n)*polylog(2,-d*f*x^2)*b^2*n+1/2*ln(x^n)*polylog(3,-d*f*x^2)* b^2*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*csgn(I*c*x^n)^3+2*b*ln (c)+2*a)*b*((ln(x^n)-n*ln(x))*(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))+dil og(1-x*(-d*f)^(1/2)))/d/f))+n*(-1/2*ln(x)*polylog(2,-d*f*x^2)+1/4*polylog( 3,-d*f*x^2)))+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*(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))
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x} \,d x } \]
integral((b^2*log(d*f*x^2 + 1)*log(c*x^n)^2 + 2*a*b*log(d*f*x^2 + 1)*log(c *x^n) + a^2*log(d*f*x^2 + 1))/x, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x} \,d x } \]
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^2 + 1) - integrate(2 /3*(b^2*d*f*n^2*x*log(x)^3 + 3*b^2*d*f*x*log(x)*log(x^n)^2 - 3*(b^2*d*f*n* log(c) + a*b*d*f*n)*x*log(x)^2 + 3*(b^2*d*f*log(c)^2 + 2*a*b*d*f*log(c) + a^2*d*f)*x*log(x) - 3*(b^2*d*f*n*x*log(x)^2 - 2*(b^2*d*f*log(c) + a*b*d*f) *x*log(x))*log(x^n))/(d*f*x^2 + 1), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x} \,d x \]