3.379 \(\int \frac{x^{-1+n} \log (c (d+e x^n))}{-1+c d+c e x^n} \, dx\)

Optimal. Leaf size=25 \[ -\frac{\text{PolyLog}\left (2,1-c \left (d+e x^n\right )\right )}{c e n} \]

[Out]

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

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Rubi [A]  time = 0.0985, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2475, 2393, 2391} \[ -\frac{\text{PolyLog}\left (2,1-c \left (d+e x^n\right )\right )}{c e n} \]

Antiderivative was successfully verified.

[In]

Int[(x^(-1 + n)*Log[c*(d + e*x^n)])/(-1 + c*d + c*e*x^n),x]

[Out]

-(PolyLog[2, 1 - c*(d + e*x^n)]/(c*e*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 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{x^{-1+n} \log \left (c \left (d+e x^n\right )\right )}{-1+c d+c e x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log (c (d+e x))}{-1+c d+c e x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,-1+c d+c e x^n\right )}{c e n}\\ &=-\frac{\text{Li}_2\left (1-c \left (d+e x^n\right )\right )}{c e n}\\ \end{align*}

Mathematica [A]  time = 0.0212684, size = 26, normalized size = 1.04 \[ -\frac{\text{PolyLog}\left (2,-c d-c e x^n+1\right )}{c e n} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(-1 + n)*Log[c*(d + e*x^n)])/(-1 + c*d + c*e*x^n),x]

[Out]

-(PolyLog[2, 1 - c*d - c*e*x^n]/(c*e*n))

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Maple [A]  time = 0.388, size = 23, normalized size = 0.9 \begin{align*} -{\frac{{\it dilog} \left ( ce{x}^{n}+cd \right ) }{nce}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+n)*ln(c*(d+e*x^n))/(-1+c*d+c*e*x^n),x)

[Out]

-1/n/c/e*dilog(c*e*x^n+c*d)

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Maxima [B]  time = 1.0575, size = 147, normalized size = 5.88 \begin{align*} \frac{\log \left (c e x^{n} + c d - 1\right ) \log \left ({\left (e x^{n} + d\right )} c\right )}{c e n} - \frac{\log \left (c e x^{n} + c d - 1\right ) \log \left (e x^{n} + d\right )}{c e n} + \frac{\log \left (-c e x^{n} - c d + 1\right ) \log \left (e x^{n} + d\right ) +{\rm Li}_2\left (c e x^{n} + c d\right )}{c e n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*log(c*(d+e*x^n))/(-1+c*d+c*e*x^n),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.70788, size = 49, normalized size = 1.96 \begin{align*} -\frac{{\rm Li}_2\left (-c e x^{n} - c d + 1\right )}{c e n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*log(c*(d+e*x^n))/(-1+c*d+c*e*x^n),x, algorithm="fricas")

[Out]

-dilog(-c*e*x^n - c*d + 1)/(c*e*n)

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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(x**(-1+n)*ln(c*(d+e*x**n))/(-1+c*d+c*e*x**n),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{n - 1} \log \left ({\left (e x^{n} + d\right )} c\right )}{c e x^{n} + c d - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*log(c*(d+e*x^n))/(-1+c*d+c*e*x^n),x, algorithm="giac")

[Out]

integrate(x^(n - 1)*log((e*x^n + d)*c)/(c*e*x^n + c*d - 1), x)