\(\int \frac {\log ^3(c (d+e x^n)^p)}{x} \, dx\) [175]

Optimal result

Integrand size = 18, antiderivative size = 113 \[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}+\frac {3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{n}-\frac {6 p^2 \log \left (c \left (d+e x^n\right )^p\right ) \operatorname {PolyLog}\left (3,1+\frac {e x^n}{d}\right )}{n}+\frac {6 p^3 \operatorname {PolyLog}\left (4,1+\frac {e x^n}{d}\right )}{n} \]

[Out]

ln(-e*x^n/d)*ln(c*(d+e*x^n)^p)^3/n+3*p*ln(c*(d+e*x^n)^p)^2*polylog(2,1+e*x^n/d)/n-6*p^2*ln(c*(d+e*x^n)^p)*poly
log(3,1+e*x^n/d)/n+6*p^3*polylog(4,1+e*x^n/d)/n

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2443, 2481, 2421, 2430, 6724} \[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {6 p^2 \operatorname {PolyLog}\left (3,\frac {e x^n}{d}+1\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {3 p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{n}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}+\frac {6 p^3 \operatorname {PolyLog}\left (4,\frac {e x^n}{d}+1\right )}{n} \]

[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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[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 2443

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 2481

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 2504

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 6724

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*} \text {integral}& = \frac {\text {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) \text {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) \text {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 ) \text {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 ) \text {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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(270\) vs. \(2(113)=226\).

Time = 0.11 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.39 \[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {-n p^3 \log (x) \log ^3\left (d+e x^n\right )+p^3 \log \left (-\frac {e x^n}{d}\right ) \log ^3\left (d+e x^n\right )+3 n p^2 \log (x) \log ^2\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 p^2 \log \left (-\frac {e x^n}{d}\right ) \log ^2\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 n p \log (x) \log \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+3 p \log \left (-\frac {e x^n}{d}\right ) \log \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+n \log (x) \log ^3\left (c \left (d+e x^n\right )^p\right )+3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )-6 p^2 \log \left (c \left (d+e x^n\right )^p\right ) \operatorname {PolyLog}\left (3,1+\frac {e x^n}{d}\right )+6 p^3 \operatorname {PolyLog}\left (4,1+\frac {e x^n}{d}\right )}{n} \]

[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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.99 (sec) , antiderivative size = 1409, normalized size of antiderivative = 12.47

method	result	size
risch	\(\text {Expression too large to display}\)	\(1409\)

[In]

int(ln(c*(d+e*x^n)^p)^3/x,x,method=_RETURNVERBOSE)

[Out]

-1/n*ln(e*x^n)*ln(d+e*x^n)^3*p^3+3/n*ln(-e*x^n/d)*ln(d+e*x^n)^3*p^3-2/n*ln(1-(d+e*x^n)/d)*ln(d+e*x^n)^3*p^3+3/
n*ln(e*x^n)*ln((d+e*x^n)^p)*ln(d+e*x^n)^2*p^2+3/n*dilog(-e*x^n/d)*ln(d+e*x^n)^2*p^3-6/n*ln(-e*x^n/d)*ln((d+e*x
^n)^p)*ln(d+e*x^n)^2*p^2+3/n*ln(1-(d+e*x^n)/d)*ln((d+e*x^n)^p)*ln(d+e*x^n)^2*p^2-3/n*polylog(2,(d+e*x^n)/d)*ln
(d+e*x^n)^2*p^3-3/n*ln(e*x^n)*ln((d+e*x^n)^p)^2*ln(d+e*x^n)*p-6/n*dilog(-e*x^n/d)*ln((d+e*x^n)^p)*ln(d+e*x^n)*
p^2+3/n*ln(-e*x^n/d)*ln((d+e*x^n)^p)^2*ln(d+e*x^n)*p+6/n*polylog(2,(d+e*x^n)/d)*ln((d+e*x^n)^p)*ln(d+e*x^n)*p^
2+1/n*ln(e*x^n)*ln((d+e*x^n)^p)^3+3/n*dilog(-e*x^n/d)*ln((d+e*x^n)^p)^2*p-6/n*polylog(3,(d+e*x^n)/d)*ln((d+e*x
^n)^p)*p^2+6/n*polylog(4,(d+e*x^n)/d)*p^3+1/8*(I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2-I*Pi*csgn(I*(d
+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)*csgn(I*c)-I*Pi*csgn(I*c*(d+e*x^n)^p)^3+I*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)
+2*ln(c))^3*ln(x)+(3/2*I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2-3/2*I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*
(d+e*x^n)^p)*csgn(I*c)-3/2*I*Pi*csgn(I*c*(d+e*x^n)^p)^3+3/2*I*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)+3*ln(c))/n*
((ln((d+e*x^n)^p)-p*ln(d+e*x^n))^2*ln(e*x^n)+p^2*(ln(d+e*x^n)^2*ln(1-(d+e*x^n)/d)+2*ln(d+e*x^n)*polylog(2,(d+e
*x^n)/d)-2*polylog(3,(d+e*x^n)/d))+2*p*(ln((d+e*x^n)^p)-p*ln(d+e*x^n))*(dilog(-e*x^n/d)+ln(d+e*x^n)*ln(-e*x^n/
d)))+(-3/4*Pi^2*csgn(I*(d+e*x^n)^p)^2*csgn(I*c*(d+e*x^n)^p)^4+3/2*Pi^2*csgn(I*(d+e*x^n)^p)^2*csgn(I*c*(d+e*x^n
)^p)^3*csgn(I*c)-3/4*Pi^2*csgn(I*(d+e*x^n)^p)^2*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)^2+3/2*Pi^2*csgn(I*(d+e*x^n)^
p)*csgn(I*c*(d+e*x^n)^p)^5-3*Pi^2*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^4*csgn(I*c)+3/2*Pi^2*csgn(I*(d+e*x
^n)^p)*csgn(I*c*(d+e*x^n)^p)^3*csgn(I*c)^2-3/4*Pi^2*csgn(I*c*(d+e*x^n)^p)^6+3/2*Pi^2*csgn(I*c*(d+e*x^n)^p)^5*c
sgn(I*c)-3/4*Pi^2*csgn(I*c*(d+e*x^n)^p)^4*csgn(I*c)^2+3*I*Pi*ln(c)*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2
-3*I*Pi*ln(c)*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)*csgn(I*c)-3*I*Pi*ln(c)*csgn(I*c*(d+e*x^n)^p)^3+3*I*Pi*
ln(c)*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)+3*ln(c)^2)/n*(ln(x^n)*ln((d+e*x^n)^p)-p*e*(dilog((d+e*x^n)/d)/e+ln(x^n
)*ln((d+e*x^n)/d)/e))

Fricas [F]

\[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{3}}{x} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}^{3}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{3}}{x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{3}}{x} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^3}{x} \,d x \]

[In]

int(log(c*(d + e*x^n)^p)^3/x,x)

[Out]

int(log(c*(d + e*x^n)^p)^3/x, x)