3.175 \(\int \frac{\log ^3(c (d+e x^n)^p)}{x} \, dx\)

Optimal. Leaf size=113 \[ -\frac{6 p^2 \text{PolyLog}\left (3,\frac{e x^n}{d}+1\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac{3 p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{n}+\frac{6 p^3 \text{PolyLog}\left (4,\frac{e x^n}{d}+1\right )}{n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n} \]

[Out]

(Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p]^3)/n + (3*p*Log[c*(d + e*x^n)^p]^2*PolyLog[2, 1 + (e*x^n)/d])/n - (6*p
^2*Log[c*(d + e*x^n)^p]*PolyLog[3, 1 + (e*x^n)/d])/n + (6*p^3*PolyLog[4, 1 + (e*x^n)/d])/n

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Rubi [A]  time = 0.148454, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2454, 2396, 2433, 2374, 2383, 6589} \[ -\frac{6 p^2 \text{PolyLog}\left (3,\frac{e x^n}{d}+1\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac{3 p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{n}+\frac{6 p^3 \text{PolyLog}\left (4,\frac{e x^n}{d}+1\right )}{n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x^n)^p]^3/x,x]

[Out]

(Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p]^3)/n + (3*p*Log[c*(d + e*x^n)^p]^2*PolyLog[2, 1 + (e*x^n)/d])/n - (6*p
^2*Log[c*(d + e*x^n)^p]*PolyLog[3, 1 + (e*x^n)/d])/n + (6*p^3*PolyLog[4, 1 + (e*x^n)/d])/n

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
 q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
 IGtQ[m, 0])

Rule 2396

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*
(f + g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])^p)/g, x] - Dist[(b*e*n*p)/g, Int[(Log[(e*(f + g*x))/(e*f -
d*g)]*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*
f - d*g, 0] && IGtQ[p, 1]

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
 x] && EqQ[e*k - d*l, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log ^3\left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}-\frac{(3 e p) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right ) \log ^2\left (c (d+e x)^p\right )}{d+e x} \, dx,x,x^n\right )}{n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}-\frac{(3 p) \operatorname{Subst}\left (\int \frac{\log ^2\left (c x^p\right ) \log \left (-\frac{e \left (-\frac{d}{e}+\frac{x}{e}\right )}{d}\right )}{x} \, dx,x,d+e x^n\right )}{n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}+\frac{3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \text{Li}_2\left (1+\frac{e x^n}{d}\right )}{n}-\frac{\left (6 p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right ) \text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x^n\right )}{n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}+\frac{3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \text{Li}_2\left (1+\frac{e x^n}{d}\right )}{n}-\frac{6 p^2 \log \left (c \left (d+e x^n\right )^p\right ) \text{Li}_3\left (1+\frac{e x^n}{d}\right )}{n}+\frac{\left (6 p^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x^n\right )}{n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}+\frac{3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \text{Li}_2\left (1+\frac{e x^n}{d}\right )}{n}-\frac{6 p^2 \log \left (c \left (d+e x^n\right )^p\right ) \text{Li}_3\left (1+\frac{e x^n}{d}\right )}{n}+\frac{6 p^3 \text{Li}_4\left (1+\frac{e x^n}{d}\right )}{n}\\ \end{align*}

Mathematica [B]  time = 0.0996806, size = 270, normalized size = 2.39 \[ \frac{-6 p^2 \text{PolyLog}\left (3,\frac{e x^n}{d}+1\right ) \log \left (c \left (d+e x^n\right )^p\right )+3 p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+6 p^3 \text{PolyLog}\left (4,\frac{e x^n}{d}+1\right )+3 n p^2 \log (x) \log ^2\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 p^2 \log \left (-\frac{e x^n}{d}\right ) \log ^2\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 n p \log (x) \log \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+3 p \log \left (-\frac{e x^n}{d}\right ) \log \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+n \log (x) \log ^3\left (c \left (d+e x^n\right )^p\right )-n p^3 \log (x) \log ^3\left (d+e x^n\right )+p^3 \log \left (-\frac{e x^n}{d}\right ) \log ^3\left (d+e x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x^n)^p]^3/x,x]

[Out]

(-(n*p^3*Log[x]*Log[d + e*x^n]^3) + p^3*Log[-((e*x^n)/d)]*Log[d + e*x^n]^3 + 3*n*p^2*Log[x]*Log[d + e*x^n]^2*L
og[c*(d + e*x^n)^p] - 3*p^2*Log[-((e*x^n)/d)]*Log[d + e*x^n]^2*Log[c*(d + e*x^n)^p] - 3*n*p*Log[x]*Log[d + e*x
^n]*Log[c*(d + e*x^n)^p]^2 + 3*p*Log[-((e*x^n)/d)]*Log[d + e*x^n]*Log[c*(d + e*x^n)^p]^2 + n*Log[x]*Log[c*(d +
 e*x^n)^p]^3 + 3*p*Log[c*(d + e*x^n)^p]^2*PolyLog[2, 1 + (e*x^n)/d] - 6*p^2*Log[c*(d + e*x^n)^p]*PolyLog[3, 1
+ (e*x^n)/d] + 6*p^3*PolyLog[4, 1 + (e*x^n)/d])/n

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Maple [C]  time = 4.84, size = 6131, normalized size = 54.3 \begin{align*} \text{output 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)^3/x,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \log \left ({\left (e x^{n} + d\right )}^{p}\right )^{3} \log \left (x\right ) - \int -\frac{e x^{n} \log \left (c\right )^{3} + d \log \left (c\right )^{3} - 3 \,{\left ({\left (e n p \log \left (x\right ) - e \log \left (c\right )\right )} x^{n} - d \log \left (c\right )\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right )^{2} + 3 \,{\left (e x^{n} \log \left (c\right )^{2} + d \log \left (c\right )^{2}\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right )}{e x x^{n} + d x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^n)^p)^3/x,x, algorithm="maxima")

[Out]

log((e*x^n + d)^p)^3*log(x) - integrate(-(e*x^n*log(c)^3 + d*log(c)^3 - 3*((e*n*p*log(x) - e*log(c))*x^n - d*l
og(c))*log((e*x^n + d)^p)^2 + 3*(e*x^n*log(c)^2 + d*log(c)^2)*log((e*x^n + d)^p))/(e*x*x^n + d*x), x)

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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 )^{3}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^n)^p)^3/x,x, algorithm="fricas")

[Out]

integral(log((e*x^n + d)^p*c)^3/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (c \left (d + e x^{n}\right )^{p} \right )}^{3}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(d+e*x**n)**p)**3/x,x)

[Out]

Integral(log(c*(d + e*x**n)**p)**3/x, x)

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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 )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e*x^n)^p)^3/x,x, algorithm="giac")

[Out]

integrate(log((e*x^n + d)^p*c)^3/x, x)