Integrand size = 25, antiderivative size = 126 \[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {e g p x^{-n}}{2 d n}-\frac {e^2 g p \log (x)}{2 d^2}+\frac {e^2 g p \log \left (d+e x^n\right )}{2 d^2 n}-\frac {g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{n} \] Output:
-1/2*e*g*p/d/n/(x^n)-1/2*e^2*g*p*ln(x)/d^2+1/2*e^2*g*p*ln(d+e*x^n)/d^2/n-1 /2*g*ln(c*(d+e*x^n)^p)/n/(x^(2*n))+f*ln(-e*x^n/d)*ln(c*(d+e*x^n)^p)/n+f*p* polylog(2,1+e*x^n/d)/n
Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.83 \[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {\frac {e g p x^{-n} \left (d+e n x^n \log (x)-e x^n \log \left (d+e x^n\right )\right )}{d^2}+g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )-2 f \left (\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )\right )}{2 n} \] Input:
Integrate[((f + g/x^(2*n))*Log[c*(d + e*x^n)^p])/x,x]
Output:
-1/2*((e*g*p*(d + e*n*x^n*Log[x] - e*x^n*Log[d + e*x^n]))/(d^2*x^n) + (g*L og[c*(d + e*x^n)^p])/x^(2*n) - 2*f*(Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p] + p*PolyLog[2, 1 + (e*x^n)/d]))/n
Time = 0.64 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2005, 2925, 2863, 2009}
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.
| \(\displaystyle \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle \int x^{-2 n-1} \left (f x^{2 n}+g\right ) \log \left (c \left (d+e x^n\right )^p\right )dx\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {\int x^{-3 n} \left (f x^{2 n}+g\right ) \log \left (c \left (e x^n+d\right )^p\right )dx^n}{n}\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {\int \left (g \log \left (c \left (e x^n+d\right )^p\right ) x^{-3 n}+f \log \left (c \left (e x^n+d\right )^p\right ) x^{-n}\right )dx^n}{n}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )-\frac {1}{2} g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )-\frac {e^2 g p \log \left (x^n\right )}{2 d^2}+\frac {e^2 g p \log \left (d+e x^n\right )}{2 d^2}+f p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )-\frac {e g p x^{-n}}{2 d}}{n}\) |
Input:
Int[((f + g/x^(2*n))*Log[c*(d + e*x^n)^p])/x,x]
Output:
(-1/2*(e*g*p)/(d*x^n) - (e^2*g*p*Log[x^n])/(2*d^2) + (e^2*g*p*Log[d + e*x^ n])/(2*d^2) - (g*Log[c*(d + e*x^n)^p])/(2*x^(2*n)) + f*Log[-((e*x^n)/d)]*L og[c*(d + e*x^n)^p] + f*p*PolyLog[2, 1 + (e*x^n)/d])/n
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Si mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 ] || IGtQ[q, 0])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.71 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.12
| method | result | size |
| risch | \(\frac {\left (2 f \ln \left (x \right ) n \,x^{2 n}-g \right ) x^{-2 n} \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{2 n}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\ln \left (x^{n}\right ) f}{n}-\frac {g \,x^{-2 n}}{2 n}\right )+\frac {e^{2} g p \ln \left (d +e \,x^{n}\right )}{2 d^{2} n}-\frac {e g p \,x^{-n}}{2 d n}-\frac {e^{2} p g \ln \left (x^{n}\right )}{2 n \,d^{2}}-\frac {p f \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n}-p f \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )\) | \(267\) |
Input:
int((f+g/(x^(2*n)))*ln(c*(d+e*x^n)^p)/x,x,method=_RETURNVERBOSE)
Output:
1/2*(2*f*ln(x)*n*(x^n)^2-g)/n/(x^n)^2*ln((d+e*x^n)^p)+(1/2*I*Pi*csgn(I*(d+ e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2-1/2*I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d +e*x^n)^p)*csgn(I*c)-1/2*I*Pi*csgn(I*c*(d+e*x^n)^p)^3+1/2*I*Pi*csgn(I*c*(d +e*x^n)^p)^2*csgn(I*c)+ln(c))*(1/n*ln(x^n)*f-1/2/n*g/(x^n)^2)+1/2*e^2*g*p* ln(d+e*x^n)/d^2/n-1/2*e*g*p/d/n/(x^n)-1/2*e^2*p/n*g/d^2*ln(x^n)-p/n*f*dilo g((d+e*x^n)/d)-p*f*ln(x)*ln((d+e*x^n)/d)
Time = 0.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.19 \[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {2 \, d^{2} f n p x^{2 \, n} \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) + 2 \, d^{2} f p x^{2 \, n} {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + d e g p x^{n} + d^{2} g \log \left (c\right ) + {\left (e^{2} g n p - 2 \, d^{2} f n \log \left (c\right )\right )} x^{2 \, n} \log \left (x\right ) + {\left (d^{2} g p - {\left (2 \, d^{2} f n p \log \left (x\right ) + e^{2} g p\right )} x^{2 \, n}\right )} \log \left (e x^{n} + d\right )}{2 \, d^{2} n x^{2 \, n}} \] Input:
integrate((f+g/(x^(2*n)))*log(c*(d+e*x^n)^p)/x,x, algorithm="fricas")
Output:
-1/2*(2*d^2*f*n*p*x^(2*n)*log(x)*log((e*x^n + d)/d) + 2*d^2*f*p*x^(2*n)*di log(-(e*x^n + d)/d + 1) + d*e*g*p*x^n + d^2*g*log(c) + (e^2*g*n*p - 2*d^2* f*n*log(c))*x^(2*n)*log(x) + (d^2*g*p - (2*d^2*f*n*p*log(x) + e^2*g*p)*x^( 2*n))*log(e*x^n + d))/(d^2*n*x^(2*n))
\[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {x^{- 2 n} \left (f x^{2 n} + g\right ) \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \] Input:
integrate((f+g/(x**(2*n)))*ln(c*(d+e*x**n)**p)/x,x)
Output:
Integral((f*x**(2*n) + g)*log(c*(d + e*x**n)**p)/(x*x**(2*n)), x)
\[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (f + \frac {g}{x^{2 \, n}}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \] Input:
integrate((f+g/(x^(2*n)))*log(c*(d+e*x^n)^p)/x,x, algorithm="maxima")
Output:
-1/2*(e*g*p*x^n + d*g*log(c) + (d*f*n^2*p*log(x)^2 - 2*d*f*n*log(c)*log(x) )*x^(2*n) - (2*d*f*n*x^(2*n)*log(x) - d*g)*log((e*x^n + d)^p))/(d*n*x^(2*n )) + integrate(1/2*(2*d^2*f*n*p*log(x) - e^2*g*p)/(d*e*x*x^n + d^2*x), x)
\[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (f + \frac {g}{x^{2 \, n}}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \] Input:
integrate((f+g/(x^(2*n)))*log(c*(d+e*x^n)^p)/x,x, algorithm="giac")
Output:
integrate((f + g/x^(2*n))*log((e*x^n + d)^p*c)/x, x)
Timed out. \[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+\frac {g}{x^{2\,n}}\right )}{x} \,d x \] Input:
int((log(c*(d + e*x^n)^p)*(f + g/x^(2*n)))/x,x)
Output:
int((log(c*(d + e*x^n)^p)*(f + g/x^(2*n)))/x, x)
\[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {2 x^{2 n} \left (\int \frac {\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right )}{x^{n} e x +d x}d x \right ) d^{3} f n p -3 x^{2 n} \mathrm {log}\left (x^{n} e +d \right ) d^{2} f \,p^{2}+x^{2 n} \mathrm {log}\left (x^{n} e +d \right ) e^{2} g \,p^{2}+x^{2 n} {\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right )}^{2} d^{2} f +3 x^{2 n} \mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right ) d^{2} f p -x^{2 n} \mathrm {log}\left (x \right ) e^{2} g n \,p^{2}-x^{n} d e g \,p^{2}-\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right ) d^{2} g p}{2 x^{2 n} d^{2} n p} \] Input:
int((f+g/(x^(2*n)))*log(c*(d+e*x^n)^p)/x,x)
Output:
(2*x**(2*n)*int(log((x**n*e + d)**p*c)/(x**n*e*x + d*x),x)*d**3*f*n*p - 3* x**(2*n)*log(x**n*e + d)*d**2*f*p**2 + x**(2*n)*log(x**n*e + d)*e**2*g*p** 2 + x**(2*n)*log((x**n*e + d)**p*c)**2*d**2*f + 3*x**(2*n)*log((x**n*e + d )**p*c)*d**2*f*p - x**(2*n)*log(x)*e**2*g*n*p**2 - x**n*d*e*g*p**2 - log(( x**n*e + d)**p*c)*d**2*g*p)/(2*x**(2*n)*d**2*n*p)