Integrand size = 24, antiderivative size = 124 \[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {2 e p x^{1+n} (f x)^{-1-n} \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d n}-\frac {x (f x)^{-1-n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}+\frac {2 e p^2 x^{1+n} (f x)^{-1-n} \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{d n} \]
2*e*p*x^(1+n)*(f*x)^(-1-n)*ln(-e*x^n/d)*ln(c*(d+e*x^n)^p)/d/n-x*(f*x)^(-1- n)*(d+e*x^n)*ln(c*(d+e*x^n)^p)^2/d/n+2*e*p^2*x^(1+n)*(f*x)^(-1-n)*polylog( 2,1+e*x^n/d)/d/n
Time = 0.07 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.21 \[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {x^{1+n} (f x)^{-1-n} \left (x^{-n} \log ^2\left (c \left (d+e x^n\right )^p\right )-2 e p \left (-\frac {p \log \left (-\frac {d x^{-n}}{e}\right ) \log \left (-e-d x^{-n}\right )}{d}+\frac {p \log ^2\left (-e-d x^{-n}\right )}{2 d}-\frac {\log \left (-e-d x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {e+d x^{-n}}{e}\right )}{d}\right )\right )}{n} \]
-((x^(1 + n)*(f*x)^(-1 - n)*(Log[c*(d + e*x^n)^p]^2/x^n - 2*e*p*(-((p*Log[ -(d/(e*x^n))]*Log[-e - d/x^n])/d) + (p*Log[-e - d/x^n]^2)/(2*d) - (Log[-e - d/x^n]*Log[c*(d + e*x^n)^p])/d - (p*PolyLog[2, (e + d/x^n)/e])/d)))/n)
Time = 0.37 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2906, 2904, 2844, 2841, 2752}
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.
| \(\displaystyle \int (f x)^{-n-1} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx\) |
| \(\Big \downarrow \) 2906 |
| \(\displaystyle x^{n+1} (f x)^{-n-1} \int x^{-n-1} \log ^2\left (c \left (e x^n+d\right )^p\right )dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle \frac {x^{n+1} (f x)^{-n-1} \int x^{-2 n} \log ^2\left (c \left (e x^n+d\right )^p\right )dx^n}{n}\) |
| \(\Big \downarrow \) 2844 |
| \(\displaystyle \frac {x^{n+1} (f x)^{-n-1} \left (\frac {2 e p \int x^{-n} \log \left (c \left (e x^n+d\right )^p\right )dx^n}{d}-\frac {x^{-n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d}\right )}{n}\) |
| \(\Big \downarrow \) 2841 |
| \(\displaystyle \frac {x^{n+1} (f x)^{-n-1} \left (\frac {2 e p \left (\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )-e p \int \frac {\log \left (-\frac {e x^n}{d}\right )}{e x^n+d}dx^n\right )}{d}-\frac {x^{-n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d}\right )}{n}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {x^{n+1} (f x)^{-n-1} \left (\frac {2 e p \left (\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )\right )}{d}-\frac {x^{-n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d}\right )}{n}\) |
(x^(1 + n)*(f*x)^(-1 - n)*(-(((d + e*x^n)*Log[c*(d + e*x^n)^p]^2)/(d*x^n)) + (2*e*p*(Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p] + p*PolyLog[2, 1 + (e*x^ n)/d]))/d))/n
3.2.67.3.1 Defintions of rubi rules used
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_ )), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x )^n])/g), x] - Simp[b*e*(n/g) Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_. )*(x_))^2, x_Symbol] :> Simp[(d + e*x)*((a + b*Log[c*(d + e*x)^n])^p/((e*f - d*g)*(f + g*x))), x] - Simp[b*e*n*(p/(e*f - d*g)) Int[(a + b*Log[c*(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] & & NeQ[e*f - d*g, 0] && GtQ[p, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_)*( x_))^(m_), x_Symbol] :> Simp[(f*x)^m/x^m Int[x^m*(a + b*Log[c*(d + e*x^n) ^p])^q, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && IntegerQ[Simp lify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])
\[\int \left (f x \right )^{-1-n} {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}^{2}d x\]
Time = 0.37 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.59 \[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {2 \, e f^{-n - 1} n p^{2} x^{n} \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) - 2 \, e f^{-n - 1} n p x^{n} \log \left (c\right ) \log \left (x\right ) + 2 \, e f^{-n - 1} p^{2} x^{n} {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + d f^{-n - 1} \log \left (c\right )^{2} + {\left (e f^{-n - 1} p^{2} x^{n} + d f^{-n - 1} p^{2}\right )} \log \left (e x^{n} + d\right )^{2} + 2 \, {\left (d f^{-n - 1} p \log \left (c\right ) - {\left (e n p^{2} \log \left (x\right ) - e p \log \left (c\right )\right )} f^{-n - 1} x^{n}\right )} \log \left (e x^{n} + d\right )}{d n x^{n}} \]
-(2*e*f^(-n - 1)*n*p^2*x^n*log(x)*log((e*x^n + d)/d) - 2*e*f^(-n - 1)*n*p* x^n*log(c)*log(x) + 2*e*f^(-n - 1)*p^2*x^n*dilog(-(e*x^n + d)/d + 1) + d*f ^(-n - 1)*log(c)^2 + (e*f^(-n - 1)*p^2*x^n + d*f^(-n - 1)*p^2)*log(e*x^n + d)^2 + 2*(d*f^(-n - 1)*p*log(c) - (e*n*p^2*log(x) - e*p*log(c))*f^(-n - 1 )*x^n)*log(e*x^n + d))/(d*n*x^n)
\[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int \left (f x\right )^{- n - 1} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}^{2}\, dx \]
\[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{-n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2} \,d x } \]
-(e*n^2*p^2*x^n*log(x)^2 - e*p^2*x^n*log(e*x^n + d)^2 + d*log((e*x^n + d)^ p)^2 + d*log(c)^2 - 2*(e*n*p*x^n*log(x) - e*p*x^n*log(e*x^n + d) - d*log(c ))*log((e*x^n + d)^p))*f^(-n - 1)/(d*n*x^n) + integrate(2*(e*n*p^2*log(x) + e*p*log(c))/(e*f^(n + 1)*x*x^n + d*f^(n + 1)*x), x)
\[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{-n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2} \,d x } \]
Timed out. \[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^2}{{\left (f\,x\right )}^{n+1}} \,d x \]