\(\int \frac {\log (c (d+e x^n)^p)}{x (f+g x^{-2 n})} \, dx\) [373]

Optimal result

Integrand size = 27, antiderivative size = 221 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )} \, dx=\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {g}+\sqrt {-f} x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2525, 269, 266, 2463, 2441, 2440, 2438} \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )} \, dx=\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {-f} x^n+\sqrt {g}\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-f} \left (e x^n+d\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-f} \left (e x^n+d\right )}{\sqrt {-f} d+e \sqrt {g}}\right )}{2 f n} \]

[In]

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

[Out]

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[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 2463

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]

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^2}\right ) x} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {\sqrt {-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt {g}-\sqrt {-f} x\right )}+\frac {\sqrt {-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt {g}+\sqrt {-f} x\right )}\right ) \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {g}-\sqrt {-f} x} \, dx,x,x^n\right )}{2 \sqrt {-f} n}-\frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {g}+\sqrt {-f} x} \, dx,x,x^n\right )}{2 \sqrt {-f} n} \\ & = \frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {g}+\sqrt {-f} x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{d+e x} \, dx,x,x^n\right )}{2 f n}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt {g}+\sqrt {-f} x\right )}{-d \sqrt {-f}+e \sqrt {g}}\right )}{d+e x} \, dx,x,x^n\right )}{2 f n} \\ & = \frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {g}+\sqrt {-f} x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}-\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-f} x}{-d \sqrt {-f}+e \sqrt {g}}\right )}{x} \, dx,x,d+e x^n\right )}{2 f n}-\frac {p \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-f} x}{d \sqrt {-f}+e \sqrt {g}}\right )}{x} \, dx,x,d+e x^n\right )}{2 f n} \\ & = \frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {g}+\sqrt {-f} x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}+\frac {p \text {Li}_2\left (\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}+\frac {p \text {Li}_2\left (\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 8.89 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.80

method	result	size
risch	\(\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (g +f \,x^{2 n}\right )}{2 n f}-\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (g +f \,x^{2 n}\right )}{2 n f}+\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (\frac {e \sqrt {-f g}-f \left (d +e \,x^{n}\right )+d f}{e \sqrt {-f g}+d f}\right )}{2 n f}+\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (\frac {e \sqrt {-f g}+f \left (d +e \,x^{n}\right )-d f}{e \sqrt {-f g}-d f}\right )}{2 n f}+\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-f g}-f \left (d +e \,x^{n}\right )+d f}{e \sqrt {-f g}+d f}\right )}{2 n f}+\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-f g}+f \left (d +e \,x^{n}\right )-d f}{e \sqrt {-f g}-d f}\right )}{2 n f}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \ln \left (g +f \,x^{2 n}\right )}{2 n f}\)	\(398\)

[In]

int(ln(c*(d+e*x^n)^p)/x/(f+g/(x^(2*n))),x,method=_RETURNVERBOSE)

[Out]

1/2/n*ln((d+e*x^n)^p)/f*ln(g+f*(x^n)^2)-1/2/n/f*p*ln(d+e*x^n)*ln(g+f*(x^n)^2)+1/2/n/f*p*ln(d+e*x^n)*ln((e*(-f*
g)^(1/2)-f*(d+e*x^n)+d*f)/(e*(-f*g)^(1/2)+d*f))+1/2/n/f*p*ln(d+e*x^n)*ln((e*(-f*g)^(1/2)+f*(d+e*x^n)-d*f)/(e*(
-f*g)^(1/2)-d*f))+1/2/n/f*p*dilog((e*(-f*g)^(1/2)-f*(d+e*x^n)+d*f)/(e*(-f*g)^(1/2)+d*f))+1/2/n/f*p*dilog((e*(-
f*g)^(1/2)+f*(d+e*x^n)-d*f)/(e*(-f*g)^(1/2)-d*f))+1/2*(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/f*ln(g+f*(x^n)^2)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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