\(\int (f x)^{-1-2 n} \log ^2(c (d+e x^n)^p) \, dx\) [168]

Optimal result

Integrand size = 24, antiderivative size = 200 \[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {e^2 p^2 x^{1+2 n} (f x)^{-1-2 n} \log (x)}{d^2}-\frac {e p x^{1+n} (f x)^{-1-2 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d^2 n}-\frac {x (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 n}-\frac {e^2 p x^{1+2 n} (f x)^{-1-2 n} \log \left (c \left (d+e x^n\right )^p\right ) \log \left (1-\frac {d}{d+e x^n}\right )}{d^2 n}+\frac {e^2 p^2 x^{1+2 n} (f x)^{-1-2 n} \operatorname {PolyLog}\left (2,\frac {d}{d+e x^n}\right )}{d^2 n} \] Output:

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
 

Mathematica [A] (warning: unable to verify)

Time = 0.40 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.44 \[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {(f x)^{-2 n} \left (e^2 n^2 p^2 x^{2 n} \log ^2(x)+e^2 p^2 x^{2 n} \log ^2\left (e+d x^{-n}\right )-2 e^2 p^2 x^{2 n} \log \left (e-e x^{-n}\right )-2 e^2 p^2 x^{2 n} \log \left (e+d x^{-n}\right ) \log \left (e-e x^{-n}\right )-2 d e p x^n \log \left (c \left (d+e x^n\right )^p\right )+2 e^2 p x^{2 n} \log \left (e-e x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )-d^2 \log ^2\left (c \left (d+e x^n\right )^p\right )+2 e^2 n p x^{2 n} \log (x) \left (p+p \log \left (e+d x^{-n}\right )-p \log \left (e-e x^{-n}\right )-\log \left (c \left (d+e x^n\right )^p\right )+p \log \left (1+\frac {e x^n}{d}\right )\right )+2 e^2 p^2 x^{2 n} \operatorname {PolyLog}\left (2,-\frac {e x^n}{d}\right )\right )}{2 d^2 f n} \] Input:

Integrate[(f*x)^(-1 - 2*n)*Log[c*(d + e*x^n)^p]^2,x]
 

Output:

(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))
 

Rubi [A] (warning: unable to verify)

Time = 1.10 (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

\(\displaystyle \frac {x^{2 n+1} (f x)^{-2 n-1} \left (e^2 p \left (\frac {\frac {p \operatorname {PolyLog}\left (2,d x^{-n}\right )}{d}-\frac {\log \left (1-d x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d}}{d}+\frac {\frac {p \log \left (-e x^n\right )}{d}-\frac {x^{-n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d e}}{d}\right )-\frac {1}{2} x^{-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\)

Input:

Int[(f*x)^(-1 - 2*n)*Log[c*(d + e*x^n)^p]^2,x]
 

Output:

(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
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2751

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]
 

rule 2779

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]
 

rule 2789

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]
 

rule 2838

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]
 

rule 2845

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]))
 

rule 2858

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]
 

rule 2904

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])
 

rule 2906

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])
 

Maple [F]

\[\int \left (f x \right )^{-1-2 n} {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}^{2}d x\]

Input:

int((f*x)^(-1-2*n)*ln(c*(d+e*x^n)^p)^2,x)
 

Output:

int((f*x)^(-1-2*n)*ln(c*(d+e*x^n)^p)^2,x)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.40 \[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {2 \, e^{2} f^{-2 \, n - 1} n p^{2} x^{2 \, n} \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) + 2 \, e^{2} f^{-2 \, n - 1} p^{2} x^{2 \, n} {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) - 2 \, d e f^{-2 \, n - 1} p x^{n} \log \left (c\right ) - d^{2} f^{-2 \, n - 1} \log \left (c\right )^{2} + 2 \, {\left (e^{2} n p^{2} - e^{2} n p \log \left (c\right )\right )} f^{-2 \, n - 1} x^{2 \, n} \log \left (x\right ) + {\left (e^{2} f^{-2 \, n - 1} p^{2} x^{2 \, n} - d^{2} f^{-2 \, n - 1} p^{2}\right )} \log \left (e x^{n} + d\right )^{2} - 2 \, {\left (d e f^{-2 \, n - 1} p^{2} x^{n} + d^{2} f^{-2 \, n - 1} p \log \left (c\right ) + {\left (e^{2} n p^{2} \log \left (x\right ) + e^{2} p^{2} - e^{2} p \log \left (c\right )\right )} f^{-2 \, n - 1} x^{2 \, n}\right )} \log \left (e x^{n} + d\right )}{2 \, d^{2} n x^{2 \, n}} \] Input:

integrate((f*x)^(-1-2*n)*log(c*(d+e*x^n)^p)^2,x, algorithm="fricas")
 

Output:

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))
 

Sympy [F]

\[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int \left (f x\right )^{- 2 n - 1} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}^{2}\, dx \] Input:

integrate((f*x)**(-1-2*n)*ln(c*(d+e*x**n)**p)**2,x)
 

Output:

Integral((f*x)**(-2*n - 1)*log(c*(d + e*x**n)**p)**2, x)
 

Maxima [F]

\[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{-2 \, n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2} \,d x } \] Input:

integrate((f*x)^(-1-2*n)*log(c*(d+e*x^n)^p)^2,x, algorithm="maxima")
 

Output:

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)
 

Giac [F]

\[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{-2 \, n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2} \,d x } \] Input:

integrate((f*x)^(-1-2*n)*log(c*(d+e*x^n)^p)^2,x, algorithm="giac")
 

Output:

integrate((f*x)^(-2*n - 1)*log((e*x^n + d)^p*c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^2}{{\left (f\,x\right )}^{2\,n+1}} \,d x \] Input:

int(log(c*(d + e*x^n)^p)^2/(f*x)^(2*n + 1),x)
                                                                                    
                                                                                    
 

Output:

int(log(c*(d + e*x^n)^p)^2/(f*x)^(2*n + 1), x)
 

Reduce [F]

\[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {-2 x^{2 n} \left (\int \frac {\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right )}{x^{n} e x +d x}d x \right ) d \,e^{2} n p +x^{2 n} \mathrm {log}\left (x^{n} e +d \right ) e^{2} p^{2}-3 x^{2 n} \mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right ) e^{2} p +2 x^{2 n} \mathrm {log}\left (x \right ) e^{2} n \,p^{2}-2 x^{n} \mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right ) d e p -{\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right )}^{2} d^{2}}{2 x^{2 n} f^{2 n} d^{2} f n} \] Input:

int((f*x)^(-1-2*n)*log(c*(d+e*x^n)^p)^2,x)
 

Output:

( - 2*x**(2*n)*int(log((x**n*e + d)**p*c)/(x**n*e*x + d*x),x)*d*e**2*n*p + 
 x**(2*n)*log(x**n*e + d)*e**2*p**2 - 3*x**(2*n)*log((x**n*e + d)**p*c)*e* 
*2*p + 2*x**(2*n)*log(x)*e**2*n*p**2 - 2*x**n*log((x**n*e + d)**p*c)*d*e*p 
 - log((x**n*e + d)**p*c)**2*d**2)/(2*x**(2*n)*f**(2*n)*d**2*f*n)