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

Optimal. Leaf size=221 \[ \frac{p \text{PolyLog}\left (2,\frac{\sqrt{-f} \left (d+e x^n\right )}{d \sqrt{-f}-e \sqrt{g}}\right )}{2 f n}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt{-f} \left (d+e x^n\right )}{d \sqrt{-f}+e \sqrt{g}}\right )}{2 f n}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (\sqrt{g}-\sqrt{-f} x^n\right )}{d \sqrt{-f}+e \sqrt{g}}\right )}{2 f n}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac{e \left (\sqrt{-f} x^n+\sqrt{g}\right )}{d \sqrt{-f}-e \sqrt{g}}\right )}{2 f n} \]

[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)

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Rubi [A]  time = 0.41202, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2475, 263, 260, 2416, 2394, 2393, 2391} \[ \frac{p \text{PolyLog}\left (2,\frac{\sqrt{-f} \left (d+e x^n\right )}{d \sqrt{-f}-e \sqrt{g}}\right )}{2 f n}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt{-f} \left (d+e x^n\right )}{d \sqrt{-f}+e \sqrt{g}}\right )}{2 f n}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (\sqrt{g}-\sqrt{-f} x^n\right )}{d \sqrt{-f}+e \sqrt{g}}\right )}{2 f n}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac{e \left (\sqrt{-f} x^n+\sqrt{g}\right )}{d \sqrt{-f}-e \sqrt{g}}\right )}{2 f n} \]

Antiderivative was successfully verified.

[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 2475

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])

Rule 263

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 260

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 2416

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 2394

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 2393

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 2391

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]

Rubi steps

\begin{align*} \int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )} \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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) \operatorname{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) \operatorname{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 \operatorname{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 \operatorname{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]  time = 1.42749, size = 0, normalized size = 0. \[ \int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [C]  time = 1.067, size = 461, normalized size = 2.1 \begin{align*}{\frac{\ln \left ( f \left ({x}^{n} \right ) ^{2}+g \right ) \ln \left ( \left ( d+e{x}^{n} \right ) ^{p} \right ) }{2\,nf}}-{\frac{p\ln \left ( d+e{x}^{n} \right ) \ln \left ( f \left ({x}^{n} \right ) ^{2}+g \right ) }{2\,nf}}+{\frac{p\ln \left ( d+e{x}^{n} \right ) }{2\,nf}\ln \left ({ \left ( e\sqrt{-fg}-f \left ( d+e{x}^{n} \right ) +df \right ) \left ( e\sqrt{-fg}+df \right ) ^{-1}} \right ) }+{\frac{p\ln \left ( d+e{x}^{n} \right ) }{2\,nf}\ln \left ({ \left ( e\sqrt{-fg}+f \left ( d+e{x}^{n} \right ) -df \right ) \left ( e\sqrt{-fg}-df \right ) ^{-1}} \right ) }+{\frac{p}{2\,nf}{\it dilog} \left ({ \left ( e\sqrt{-fg}-f \left ( d+e{x}^{n} \right ) +df \right ) \left ( e\sqrt{-fg}+df \right ) ^{-1}} \right ) }+{\frac{p}{2\,nf}{\it dilog} \left ({ \left ( e\sqrt{-fg}+f \left ( d+e{x}^{n} \right ) -df \right ) \left ( e\sqrt{-fg}-df \right ) ^{-1}} \right ) }+{\frac{{\frac{i}{4}}\ln \left ( f \left ({x}^{n} \right ) ^{2}+g \right ) \pi \,{\it csgn} \left ( i \left ( d+e{x}^{n} \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}}{nf}}-{\frac{{\frac{i}{4}}\ln \left ( f \left ({x}^{n} \right ) ^{2}+g \right ) \pi \,{\it csgn} \left ( i \left ( d+e{x}^{n} \right ) ^{p} \right ){\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ){\it csgn} \left ( ic \right ) }{nf}}-{\frac{{\frac{i}{4}}\ln \left ( f \left ({x}^{n} \right ) ^{2}+g \right ) \pi \, \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{3}}{nf}}+{\frac{{\frac{i}{4}}\ln \left ( f \left ({x}^{n} \right ) ^{2}+g \right ) \pi \, \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) }{nf}}+{\frac{\ln \left ( f \left ({x}^{n} \right ) ^{2}+g \right ) \ln \left ( c \right ) }{2\,nf}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/n/f*ln(f*(x^n)^2+g)*ln((d+e*x^n)^p)-1/2/n/f*p*ln(d+e*x^n)*ln(f*(x^n)^2+g)+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))+1/4*I/n/f*ln(f*(x^n)^2+g)*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d
+e*x^n)^p)^2-1/4*I/n/f*ln(f*(x^n)^2+g)*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)*csgn(I*c)-1/4*I/n/f*ln(f*(
x^n)^2+g)*Pi*csgn(I*c*(d+e*x^n)^p)^3+1/4*I/n/f*ln(f*(x^n)^2+g)*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)+1/2/n/f*ln
(f*(x^n)^2+g)*ln(c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac{g}{x^{2 \, n}}\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2 \, n} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{f x x^{2 \, n} + g x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac{g}{x^{2 \, n}}\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)