Integrand size = 24, antiderivative size = 200 \[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {e^2 p^2 x^{1+2 n} (f x)^{-1-2 n} \log (x)}{d^2}-\frac {e p x^{1+n} (f x)^{-1-2 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d^2 n}-\frac {x (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 n}-\frac {e^2 p x^{1+2 n} (f x)^{-1-2 n} \log \left (c \left (d+e x^n\right )^p\right ) \log \left (1-\frac {d}{d+e x^n}\right )}{d^2 n}+\frac {e^2 p^2 x^{1+2 n} (f x)^{-1-2 n} \operatorname {PolyLog}\left (2,\frac {d}{d+e x^n}\right )}{d^2 n} \]
e^2*p^2*x^(1+2*n)*(f*x)^(-1-2*n)*ln(x)/d^2-e*p*x^(1+n)*(f*x)^(-1-2*n)*(d+e *x^n)*ln(c*(d+e*x^n)^p)/d^2/n-1/2*x*(f*x)^(-1-2*n)*ln(c*(d+e*x^n)^p)^2/n-e ^2*p*x^(1+2*n)*(f*x)^(-1-2*n)*ln(c*(d+e*x^n)^p)*ln(1-d/(d+e*x^n))/d^2/n+e^ 2*p^2*x^(1+2*n)*(f*x)^(-1-2*n)*polylog(2,d/(d+e*x^n))/d^2/n
Time = 0.25 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.44 \[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {(f x)^{-2 n} \left (e^2 n^2 p^2 x^{2 n} \log ^2(x)+e^2 p^2 x^{2 n} \log ^2\left (e+d x^{-n}\right )-2 e^2 p^2 x^{2 n} \log \left (e-e x^{-n}\right )-2 e^2 p^2 x^{2 n} \log \left (e+d x^{-n}\right ) \log \left (e-e x^{-n}\right )-2 d e p x^n \log \left (c \left (d+e x^n\right )^p\right )+2 e^2 p x^{2 n} \log \left (e-e x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )-d^2 \log ^2\left (c \left (d+e x^n\right )^p\right )+2 e^2 n p x^{2 n} \log (x) \left (p+p \log \left (e+d x^{-n}\right )-p \log \left (e-e x^{-n}\right )-\log \left (c \left (d+e x^n\right )^p\right )+p \log \left (1+\frac {e x^n}{d}\right )\right )+2 e^2 p^2 x^{2 n} \operatorname {PolyLog}\left (2,-\frac {e x^n}{d}\right )\right )}{2 d^2 f n} \]
(e^2*n^2*p^2*x^(2*n)*Log[x]^2 + e^2*p^2*x^(2*n)*Log[e + d/x^n]^2 - 2*e^2*p ^2*x^(2*n)*Log[e - e/x^n] - 2*e^2*p^2*x^(2*n)*Log[e + d/x^n]*Log[e - e/x^n ] - 2*d*e*p*x^n*Log[c*(d + e*x^n)^p] + 2*e^2*p*x^(2*n)*Log[e - e/x^n]*Log[ c*(d + e*x^n)^p] - d^2*Log[c*(d + e*x^n)^p]^2 + 2*e^2*n*p*x^(2*n)*Log[x]*( p + p*Log[e + d/x^n] - p*Log[e - e/x^n] - Log[c*(d + e*x^n)^p] + p*Log[1 + (e*x^n)/d]) + 2*e^2*p^2*x^(2*n)*PolyLog[2, -((e*x^n)/d)])/(2*d^2*f*n*(f*x )^(2*n))
Time = 0.63 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.73, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2906, 2904, 2845, 2858, 27, 2789, 2751, 16, 2779, 2838}
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.
\(\displaystyle \int (f x)^{-2 n-1} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx\) |
\(\Big \downarrow \) 2906 |
\(\displaystyle x^{2 n+1} (f x)^{-2 n-1} \int x^{-2 n-1} \log ^2\left (c \left (e x^n+d\right )^p\right )dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle \frac {x^{2 n+1} (f x)^{-2 n-1} \int x^{-3 n} \log ^2\left (c \left (e x^n+d\right )^p\right )dx^n}{n}\) |
\(\Big \downarrow \) 2845 |
\(\displaystyle \frac {x^{2 n+1} (f x)^{-2 n-1} \left (e p \int \frac {x^{-2 n} \log \left (c \left (e x^n+d\right )^p\right )}{e x^n+d}dx^n-\frac {1}{2} x^{-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
\(\Big \downarrow \) 2858 |
\(\displaystyle \frac {x^{2 n+1} (f x)^{-2 n-1} \left (p \int x^{-3 n} \log \left (c \left (e x^n+d\right )^p\right )d\left (e x^n+d\right )-\frac {1}{2} x^{-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^{2 n+1} (f x)^{-2 n-1} \left (e^2 p \int \frac {x^{-3 n} \log \left (c \left (e x^n+d\right )^p\right )}{e^2}d\left (e x^n+d\right )-\frac {1}{2} x^{-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
\(\Big \downarrow \) 2789 |
\(\displaystyle \frac {x^{2 n+1} (f x)^{-2 n-1} \left (e^2 p \left (\frac {\int \frac {x^{-2 n} \log \left (c \left (e x^n+d\right )^p\right )}{e^2}d\left (e x^n+d\right )}{d}+\frac {\int -\frac {x^{-2 n} \log \left (c \left (e x^n+d\right )^p\right )}{e}d\left (e x^n+d\right )}{d}\right )-\frac {1}{2} x^{-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
\(\Big \downarrow \) 2751 |
\(\displaystyle \frac {x^{2 n+1} (f x)^{-2 n-1} \left (e^2 p \left (\frac {-\frac {p \int -\frac {x^{-n}}{e}d\left (e x^n+d\right )}{d}-\frac {x^{-n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d e}}{d}+\frac {\int -\frac {x^{-2 n} \log \left (c \left (e x^n+d\right )^p\right )}{e}d\left (e x^n+d\right )}{d}\right )-\frac {1}{2} x^{-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {x^{2 n+1} (f x)^{-2 n-1} \left (e^2 p \left (\frac {\int -\frac {x^{-2 n} \log \left (c \left (e x^n+d\right )^p\right )}{e}d\left (e x^n+d\right )}{d}+\frac {\frac {p \log \left (-e x^n\right )}{d}-\frac {x^{-n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d e}}{d}\right )-\frac {1}{2} x^{-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
\(\Big \downarrow \) 2779 |
\(\displaystyle \frac {x^{2 n+1} (f x)^{-2 n-1} \left (e^2 p \left (\frac {\frac {p \int x^{-n} \log \left (1-d x^{-n}\right )d\left (e x^n+d\right )}{d}-\frac {\log \left (1-d x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d}}{d}+\frac {\frac {p \log \left (-e x^n\right )}{d}-\frac {x^{-n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d e}}{d}\right )-\frac {1}{2} x^{-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {x^{2 n+1} (f x)^{-2 n-1} \left (e^2 p \left (\frac {\frac {p \operatorname {PolyLog}\left (2,d x^{-n}\right )}{d}-\frac {\log \left (1-d x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d}}{d}+\frac {\frac {p \log \left (-e x^n\right )}{d}-\frac {x^{-n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d e}}{d}\right )-\frac {1}{2} x^{-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
(x^(1 + 2*n)*(f*x)^(-1 - 2*n)*(-1/2*Log[c*(d + e*x^n)^p]^2/x^(2*n) + e^2*p *(((p*Log[-(e*x^n)])/d - ((d + e*x^n)*Log[c*(d + e*x^n)^p])/(d*e*x^n))/d + (-((Log[1 - d/x^n]*Log[c*(d + e*x^n)^p])/d) + (p*PolyLog[2, d/x^n])/d)/d) ))/n
3.2.68.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1))) Int[(f + g*x)^(q + 1) *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
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])
\[\int \left (f x \right )^{-1-2 n} {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}^{2}d x\]
Time = 0.33 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.40 \[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {2 \, e^{2} f^{-2 \, n - 1} n p^{2} x^{2 \, n} \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) + 2 \, e^{2} f^{-2 \, n - 1} p^{2} x^{2 \, n} {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) - 2 \, d e f^{-2 \, n - 1} p x^{n} \log \left (c\right ) - d^{2} f^{-2 \, n - 1} \log \left (c\right )^{2} + 2 \, {\left (e^{2} n p^{2} - e^{2} n p \log \left (c\right )\right )} f^{-2 \, n - 1} x^{2 \, n} \log \left (x\right ) + {\left (e^{2} f^{-2 \, n - 1} p^{2} x^{2 \, n} - d^{2} f^{-2 \, n - 1} p^{2}\right )} \log \left (e x^{n} + d\right )^{2} - 2 \, {\left (d e f^{-2 \, n - 1} p^{2} x^{n} + d^{2} f^{-2 \, n - 1} p \log \left (c\right ) + {\left (e^{2} n p^{2} \log \left (x\right ) + e^{2} p^{2} - e^{2} p \log \left (c\right )\right )} f^{-2 \, n - 1} x^{2 \, n}\right )} \log \left (e x^{n} + d\right )}{2 \, d^{2} n x^{2 \, n}} \]
1/2*(2*e^2*f^(-2*n - 1)*n*p^2*x^(2*n)*log(x)*log((e*x^n + d)/d) + 2*e^2*f^ (-2*n - 1)*p^2*x^(2*n)*dilog(-(e*x^n + d)/d + 1) - 2*d*e*f^(-2*n - 1)*p*x^ n*log(c) - d^2*f^(-2*n - 1)*log(c)^2 + 2*(e^2*n*p^2 - e^2*n*p*log(c))*f^(- 2*n - 1)*x^(2*n)*log(x) + (e^2*f^(-2*n - 1)*p^2*x^(2*n) - d^2*f^(-2*n - 1) *p^2)*log(e*x^n + d)^2 - 2*(d*e*f^(-2*n - 1)*p^2*x^n + d^2*f^(-2*n - 1)*p* log(c) + (e^2*n*p^2*log(x) + e^2*p^2 - e^2*p*log(c))*f^(-2*n - 1)*x^(2*n)) *log(e*x^n + d))/(d^2*n*x^(2*n))
\[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int \left (f x\right )^{- 2 n - 1} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}^{2}\, dx \]
\[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{-2 \, n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2} \,d x } \]
1/2*(e^2*n^2*p^2*x^(2*n)*log(x)^2 - e^2*p^2*x^(2*n)*log(e*x^n + d)^2 - 2*d *e*p*x^n*log(c) - d^2*log((e*x^n + d)^p)^2 - d^2*log(c)^2 - 2*(e^2*n*p*x^( 2*n)*log(x) - e^2*p*x^(2*n)*log(e*x^n + d) + d*e*p*x^n + d^2*log(c))*log(( e*x^n + d)^p))*f^(-2*n - 1)/(d^2*n*x^(2*n)) - integrate((e^2*n*p^2*log(x) - e^2*p^2 + e^2*p*log(c))/(d*e*f^(2*n + 1)*x*x^n + d^2*f^(2*n + 1)*x), x)
\[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{-2 \, n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2} \,d x } \]
Timed out. \[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^2}{{\left (f\,x\right )}^{2\,n+1}} \,d x \]