Integrand size = 21, antiderivative size = 88 \[ \int \frac {\log ^2\left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {2 p \log \left (c \left (d+e x^n\right )^p\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{f n}-\frac {2 p^2 \operatorname {PolyLog}\left (3,1+\frac {e x^n}{d}\right )}{f n} \]
ln(-e*x^n/d)*ln(c*(d+e*x^n)^p)^2/f/n+2*p*ln(c*(d+e*x^n)^p)*polylog(2,1+e*x ^n/d)/f/n-2*p^2*polylog(3,1+e*x^n/d)/f/n
Time = 0.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.91 \[ \int \frac {\log ^2\left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\frac {\log (x) \left (-p \log \left (d+e x^n\right )+\log \left (c \left (d+e x^n\right )^p\right )\right )^2+2 p \left (-p \log \left (d+e x^n\right )+\log \left (c \left (d+e x^n\right )^p\right )\right ) \left (\log (x) \left (\log \left (d+e x^n\right )-\log \left (1+\frac {e x^n}{d}\right )\right )-\frac {\operatorname {PolyLog}\left (2,-\frac {e x^n}{d}\right )}{n}\right )+\frac {p^2 \left (\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (d+e x^n\right )+2 \log \left (d+e x^n\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {e x^n}{d}\right )\right )}{n}}{f} \]
(Log[x]*(-(p*Log[d + e*x^n]) + Log[c*(d + e*x^n)^p])^2 + 2*p*(-(p*Log[d + e*x^n]) + Log[c*(d + e*x^n)^p])*(Log[x]*(Log[d + e*x^n] - Log[1 + (e*x^n)/ d]) - PolyLog[2, -((e*x^n)/d)]/n) + (p^2*(Log[-((e*x^n)/d)]*Log[d + e*x^n] ^2 + 2*Log[d + e*x^n]*PolyLog[2, 1 + (e*x^n)/d] - 2*PolyLog[3, 1 + (e*x^n) /d]))/n)/f
Time = 0.40 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {27, 2904, 2843, 2881, 2821, 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.
| \(\displaystyle \int \frac {\log ^2\left (c \left (d+e x^n\right )^p\right )}{f x} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\log ^2\left (c \left (e x^n+d\right )^p\right )}{x}dx}{f}\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle \frac {\int x^{-n} \log ^2\left (c \left (e x^n+d\right )^p\right )dx^n}{f n}\) |
| \(\Big \downarrow \) 2843 |
| \(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )-2 e p \int \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (e x^n+d\right )^p\right )}{e x^n+d}dx^n}{f n}\) |
| \(\Big \downarrow \) 2881 |
| \(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )-2 p \int x^{-n} \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (e x^n+d\right )^p\right )d\left (e x^n+d\right )}{f n}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )-2 p \left (p \int x^{-n} \operatorname {PolyLog}\left (2,\frac {e x^n+d}{d}\right )d\left (e x^n+d\right )-\operatorname {PolyLog}\left (2,\frac {e x^n+d}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )\right )}{f n}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )-2 p \left (p \operatorname {PolyLog}\left (3,\frac {e x^n+d}{d}\right )-\operatorname {PolyLog}\left (2,\frac {e x^n+d}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )\right )}{f n}\) |
(Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p]^2 - 2*p*(-(Log[c*(d + e*x^n)^p]*Po lyLog[2, (d + e*x^n)/d]) + p*PolyLog[3, (d + e*x^n)/d]))/(f*n)
3.2.66.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_. )*(x_)), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])^p/g), x] - Simp[b*e*n*(p/g) Int[Log[(e*(f + g*x))/(e*f - d*g)] *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[p, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log [(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym bol] :> Simp[1/e Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[h* ((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 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[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 = 2.30 (sec) , antiderivative size = 614, normalized size of antiderivative = 6.98
| method | result | size |
| risch | \(-\frac {2 \ln \left (-\frac {e \,x^{n}}{d}\right ) \ln \left (d +e \,x^{n}\right )^{2} p^{2}}{n f}+\frac {\ln \left (e \,x^{n}\right ) \ln \left (d +e \,x^{n}\right )^{2} p^{2}}{n f}+\frac {\ln \left (1-\frac {d +e \,x^{n}}{d}\right ) \ln \left (d +e \,x^{n}\right )^{2} p^{2}}{n f}-\frac {2 \operatorname {dilog}\left (-\frac {e \,x^{n}}{d}\right ) \ln \left (d +e \,x^{n}\right ) p^{2}}{n f}+\frac {2 \ln \left (-\frac {e \,x^{n}}{d}\right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (d +e \,x^{n}\right ) p}{n f}-\frac {2 \ln \left (e \,x^{n}\right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (d +e \,x^{n}\right ) p}{n f}+\frac {2 \,\operatorname {Li}_{2}\left (\frac {d +e \,x^{n}}{d}\right ) \ln \left (d +e \,x^{n}\right ) p^{2}}{n f}+\frac {2 \operatorname {dilog}\left (-\frac {e \,x^{n}}{d}\right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right ) p}{n f}+\frac {\ln \left (e \,x^{n}\right ) {\ln \left (\left (d +e \,x^{n}\right )^{p}\right )}^{2}}{n f}-\frac {2 \,\operatorname {Li}_{3}\left (\frac {d +e \,x^{n}}{d}\right ) p^{2}}{n f}+\frac {\left (i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right ) \left (\ln \left (x^{n}\right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )-p e \left (\frac {\operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{e}+\frac {\ln \left (x^{n}\right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )}{e}\right )\right )}{f n}+\frac {{\left (i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right )}^{2} \ln \left (x \right )}{4 f}\) | \(614\) |
-2/n/f*ln(-e*x^n/d)*ln(d+e*x^n)^2*p^2+1/n/f*ln(e*x^n)*ln(d+e*x^n)^2*p^2+1/ n/f*ln(1-(d+e*x^n)/d)*ln(d+e*x^n)^2*p^2-2/n/f*dilog(-e*x^n/d)*ln(d+e*x^n)* p^2+2/n/f*ln(-e*x^n/d)*ln((d+e*x^n)^p)*ln(d+e*x^n)*p-2/n/f*ln(e*x^n)*ln((d +e*x^n)^p)*ln(d+e*x^n)*p+2/n/f*polylog(2,(d+e*x^n)/d)*ln(d+e*x^n)*p^2+2/n/ f*dilog(-e*x^n/d)*ln((d+e*x^n)^p)*p+1/n/f*ln(e*x^n)*ln((d+e*x^n)^p)^2-2/n/ f*polylog(3,(d+e*x^n)/d)*p^2+1/f*(I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x ^n)^p)^2-I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)*csgn(I*c)-I*Pi*csg n(I*c*(d+e*x^n)^p)^3+I*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)+2*ln(c))/n*(ln (x^n)*ln((d+e*x^n)^p)-p*e*(dilog((d+e*x^n)/d)/e+ln(x^n)*ln((d+e*x^n)/d)/e) )+1/4/f*(I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2-I*Pi*csgn(I*(d+e *x^n)^p)*csgn(I*c*(d+e*x^n)^p)*csgn(I*c)-I*Pi*csgn(I*c*(d+e*x^n)^p)^3+I*Pi *csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)+2*ln(c))^2*ln(x)
\[ \int \frac {\log ^2\left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2}}{f x} \,d x } \]
\[ \int \frac {\log ^2\left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\frac {\int \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}^{2}}{x}\, dx}{f} \]
\[ \int \frac {\log ^2\left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2}}{f x} \,d x } \]
(log((e*x^n + d)^p)^2*log(x) - integrate(-(e*x^n*log(c)^2 + d*log(c)^2 - 2 *((e*n*p*log(x) - e*log(c))*x^n - d*log(c))*log((e*x^n + d)^p))/(e*x*x^n + d*x), x))/f
\[ \int \frac {\log ^2\left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2}}{f x} \,d x } \]
Timed out. \[ \int \frac {\log ^2\left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^2}{f\,x} \,d x \]