\(\int \frac {\log (c (d+e x^n)^p)}{x (f+g x^{2 n})} \, dx\) [370]

Optimal result

Integrand size = 27, antiderivative size = 266 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{2 n}\right )} \, dx=\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x^n\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f n}-\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} \left (d+e x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f n}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {g} \left (d+e x^n\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f n}+\frac {p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{f n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2525, 272, 36, 29, 31, 2463, 2441, 2352, 266, 2440, 2438} \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{2 n}\right )} \, dx=-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x^n\right )}{d \sqrt {g}+e \sqrt {-f}}\right )}{2 f n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f n}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} \left (e x^n+d\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f n}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {g} \left (e x^n+d\right )}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 f n}+\frac {p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{f n} \]

[In]

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

[Out]

(Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/(f*n) - (Log[c*(d + e*x^n)^p]*Log[(e*(Sqrt[-f] - Sqrt[g]*x^n))/(e*Sqr
t[-f] + d*Sqrt[g])])/(2*f*n) - (Log[c*(d + e*x^n)^p]*Log[(e*(Sqrt[-f] + Sqrt[g]*x^n))/(e*Sqrt[-f] - d*Sqrt[g])
])/(2*f*n) - (p*PolyLog[2, -((Sqrt[g]*(d + e*x^n))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*f*n) - (p*PolyLog[2, (Sqrt[g
]*(d + e*x^n))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f*n) + (p*PolyLog[2, 1 + (e*x^n)/d])/(f*n)

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

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 )}{x \left (f+g x^2\right )} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\log \left (c (d+e x)^p\right )}{f x}-\frac {g x \log \left (c (d+e x)^p\right )}{f \left (f+g x^2\right )}\right ) \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{f n}-\frac {g \text {Subst}\left (\int \frac {x \log \left (c (d+e x)^p\right )}{f+g x^2} \, dx,x,x^n\right )}{f n} \\ & = \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac {g \text {Subst}\left (\int \left (-\frac {\log \left (c (d+e x)^p\right )}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\log \left (c (d+e x)^p\right )}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx,x,x^n\right )}{f n}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{f n} \\ & = \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f n}+\frac {\sqrt {g} \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-f}-\sqrt {g} x} \, dx,x,x^n\right )}{2 f n}-\frac {\sqrt {g} \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-f}+\sqrt {g} x} \, dx,x,x^n\right )}{2 f n} \\ & = \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x^n\right )}{e \sqrt {-f}+d \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}+\sqrt {g} x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f n}+\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \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 {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx,x,x^n\right )}{2 f n} \\ & = \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x^n\right )}{e \sqrt {-f}+d \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}+\sqrt {g} x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f n}+\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \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 {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x^n\right )}{2 f n} \\ & = \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x^n\right )}{e \sqrt {-f}+d \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}+\sqrt {g} x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f n}-\frac {p \text {Li}_2\left (-\frac {\sqrt {g} \left (d+e x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f n}-\frac {p \text {Li}_2\left (\frac {\sqrt {g} \left (d+e x^n\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{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 = 3.08 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.80

method	result	size
risch	\(\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (x^{n}\right )}{n f}-\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (f +g \,x^{2 n}\right )}{2 n f}-\frac {p \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n f}-\frac {p \ln \left (x^{n}\right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )}{n f}+\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (f +g \,x^{2 n}\right )}{2 n f}-\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (\frac {e \sqrt {-f g}-g \left (d +e \,x^{n}\right )+d g}{e \sqrt {-f g}+d g}\right )}{2 n f}-\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (\frac {e \sqrt {-f g}+g \left (d +e \,x^{n}\right )-d g}{e \sqrt {-f g}-d g}\right )}{2 n f}-\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-f g}-g \left (d +e \,x^{n}\right )+d g}{e \sqrt {-f g}+d g}\right )}{2 n f}-\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-f g}+g \left (d +e \,x^{n}\right )-d g}{e \sqrt {-f g}-d g}\right )}{2 n f}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\ln \left (x^{n}\right )}{n f}-\frac {\ln \left (f +g \,x^{2 n}\right )}{2 n f}\right )\)	\(478\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

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

[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)/((g*x^(2*n) + f)*x), x)

Giac [F]

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

[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)/((g*x^(2*n) + f)*x), x)

Mupad [F(-1)]

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

[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)