\(\int \frac {\log (c (d+e x^n)^p)}{x (f+g x^{-n})} \, dx\) [372]

Optimal result

Integrand size = 27, antiderivative size = 70 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx=\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (g+f x^n\right )}{d f-e g}\right )}{f n}+\frac {p \operatorname {PolyLog}\left (2,\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f n} \]

[Out]

ln(c*(d+e*x^n)^p)*ln(-e*(g+f*x^n)/(d*f-e*g))/f/n+p*polylog(2,f*(d+e*x^n)/(d*f-e*g))/f/n

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2525, 2459, 2441, 2440, 2438} \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx=\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (f x^n+g\right )}{d f-e g}\right )}{f n}+\frac {p \operatorname {PolyLog}\left (2,\frac {f \left (e x^n+d\right )}{d f-e g}\right )}{f n} \]

[In]

Int[Log[c*(d + e*x^n)^p]/(x*(f + g/x^n)),x]

[Out]

(Log[c*(d + e*x^n)^p]*Log[-((e*(g + f*x^n))/(d*f - e*g))])/(f*n) + (p*PolyLog[2, (f*(d + e*x^n))/(d*f - e*g)])
/(f*n)

Rule 2438

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 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

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] - Dist[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]

Rule 2459

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)/(x_))^(q_.)*(x_)^(m_.), x_Symbol]
:> Int[(g + f*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x] && EqQ[m,
q] && IntegerQ[q]

Rule 2525

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(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] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\left (f+\frac {g}{x}\right ) x} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{g+f x} \, dx,x,x^n\right )}{n} \\ & = \frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (g+f x^n\right )}{d f-e g}\right )}{f n}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e (g+f x)}{-d f+e g}\right )}{d+e x} \, dx,x,x^n\right )}{f n} \\ & = \frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (g+f x^n\right )}{d f-e g}\right )}{f n}-\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {f x}{-d f+e g}\right )}{x} \, dx,x,d+e x^n\right )}{f n} \\ & = \frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (g+f x^n\right )}{d f-e g}\right )}{f n}+\frac {p \text {Li}_2\left (\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.91 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx=\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (g+f x^n\right )}{-d f+e g}\right )+p \operatorname {PolyLog}\left (2,\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f n} \]

[In]

Integrate[Log[c*(d + e*x^n)^p]/(x*(f + g/x^n)),x]

[Out]

(Log[c*(d + e*x^n)^p]*Log[(e*(g + f*x^n))/(-(d*f) + e*g)] + p*PolyLog[2, (f*(d + e*x^n))/(d*f - e*g)])/(f*n)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 3.04 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.47

method	result	size
risch	\(\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (g +f \,x^{n}\right )}{n f}-\frac {p \operatorname {dilog}\left (\frac {\left (g +f \,x^{n}\right ) e +d f -e g}{d f -e g}\right )}{n f}-\frac {p \ln \left (g +f \,x^{n}\right ) \ln \left (\frac {\left (g +f \,x^{n}\right ) e +d f -e g}{d f -e g}\right )}{n f}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \ln \left (g +f \,x^{n}\right )}{n f}\)	\(243\)

[In]

int(ln(c*(d+e*x^n)^p)/x/(f+g/(x^n)),x,method=_RETURNVERBOSE)

[Out]

1/n*ln((d+e*x^n)^p)*ln(g+f*x^n)/f-1/n/f*p*dilog(((g+f*x^n)*e+d*f-e*g)/(d*f-e*g))-1/n/f*p*ln(g+f*x^n)*ln(((g+f*
x^n)*e+d*f-e*g)/(d*f-e*g))+(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))/n*ln(g+f*x^n)/f

Fricas [F]

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

[In]

integrate(log(c*(d+e*x^n)^p)/x/(f+g/(x^n)),x, algorithm="fricas")

[Out]

integral(x^n*log((e*x^n + d)^p*c)/(f*x*x^n + g*x), x)

Sympy [F(-2)]

Exception generated. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate(ln(c*(d+e*x**n)**p)/x/(f+g/(x**n)),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.60 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx={\left (\frac {\log \left (f + \frac {g}{x^{n}}\right )}{f n} - \frac {\log \left (\frac {1}{x^{n}}\right )}{f n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) - \frac {{\left (\log \left (f x^{n} + g\right ) \log \left (\frac {e f x^{n} + e g}{d f - e g} + 1\right ) + {\rm Li}_2\left (-\frac {e f x^{n} + e g}{d f - e g}\right )\right )} p}{f n} \]

[In]

integrate(log(c*(d+e*x^n)^p)/x/(f+g/(x^n)),x, algorithm="maxima")

[Out]

(log(f + g/x^n)/(f*n) - log(1/(x^n))/(f*n))*log((e*x^n + d)^p*c) - (log(f*x^n + g)*log((e*f*x^n + e*g)/(d*f -
e*g) + 1) + dilog(-(e*f*x^n + e*g)/(d*f - e*g)))*p/(f*n)

Giac [F]

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

[In]

integrate(log(c*(d+e*x^n)^p)/x/(f+g/(x^n)),x, algorithm="giac")

[Out]

integrate(log((e*x^n + d)^p*c)/((f + g/x^n)*x), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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