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

Optimal. Leaf size=121 \[ -\frac{p \text{PolyLog}\left (2,-\frac{g \left (d+e x^n\right )}{e f-d g}\right )}{f n}+\frac{p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{f n}-\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (f+g x^n\right )}{e f-d g}\right )}{f n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n} \]

[Out]

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

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Rubi [A]  time = 0.195171, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {2475, 36, 29, 31, 2416, 2394, 2315, 2393, 2391} \[ -\frac{p \text{PolyLog}\left (2,-\frac{g \left (d+e x^n\right )}{e f-d g}\right )}{f n}+\frac{p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{f n}-\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (f+g x^n\right )}{e f-d g}\right )}{f n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x^n)^p]/(x*(f + g*x^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*(f + g*x^n))/(e*f - d*g)])/(f*n)
- (p*PolyLog[2, -((g*(d + e*x^n))/(e*f - d*g))])/(f*n) + (p*PolyLog[2, 1 + (e*x^n)/d])/(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 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 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 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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 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^n\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{x (f+g x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\log \left (c (d+e x)^p\right )}{f x}-\frac{g \log \left (c (d+e x)^p\right )}{f (f+g x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{f n}-\frac{g \operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{f+g 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{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (f+g x^n\right )}{e f-d g}\right )}{f n}-\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{f n}+\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\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{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (f+g x^n\right )}{e f-d g}\right )}{f n}+\frac{p \text{Li}_2\left (1+\frac{e x^n}{d}\right )}{f n}+\frac{p \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e 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{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (f+g x^n\right )}{e f-d g}\right )}{f n}-\frac{p \text{Li}_2\left (-\frac{g \left (d+e x^n\right )}{e f-d g}\right )}{f n}+\frac{p \text{Li}_2\left (1+\frac{e x^n}{d}\right )}{f n}\\ \end{align*}

Mathematica [A]  time = 0.0749918, size = 92, normalized size = 0.76 \[ \frac{-p \text{PolyLog}\left (2,\frac{g \left (d+e x^n\right )}{d g-e f}\right )+p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )+\log \left (c \left (d+e x^n\right )^p\right ) \left (\log \left (-\frac{e x^n}{d}\right )-\log \left (\frac{e \left (f+g x^n\right )}{e f-d g}\right )\right )}{f n} \]

Antiderivative was successfully verified.

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

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Maple [C]  time = 1.197, size = 532, normalized size = 4.4 \begin{align*} \text{result too large to display} \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^n),x)

[Out]

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

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Maxima [A]  time = 1.43952, size = 208, normalized size = 1.72 \begin{align*} -e n p{\left (\frac{\log \left (x^{n}\right ) \log \left (\frac{e x^{n}}{d} + 1\right ) +{\rm Li}_2\left (-\frac{e x^{n}}{d}\right )}{e f n^{2}} - \frac{\log \left (g x^{n} + f\right ) \log \left (-\frac{e g x^{n} + e f}{e f - d g} + 1\right ) +{\rm Li}_2\left (\frac{e g x^{n} + e f}{e f - d g}\right )}{e f n^{2}}\right )} -{\left (\frac{\log \left (g x^{n} + f\right )}{f n} - \frac{\log \left (x^{n}\right )}{f n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\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^n),x, algorithm="maxima")

[Out]

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

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

[Out]

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

[Out]

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