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

Optimal result

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

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)*polylog(3,1+e*x^n/d)/n+6*p^3*polylog(4,1+e 
*x^n/d)/n
 

Mathematica [B] (verified)

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

Time = 0.18 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.39 \[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {-n p^3 \log (x) \log ^3\left (d+e x^n\right )+p^3 \log \left (-\frac {e x^n}{d}\right ) \log ^3\left (d+e x^n\right )+3 n p^2 \log (x) \log ^2\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 p^2 \log \left (-\frac {e x^n}{d}\right ) \log ^2\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 n p \log (x) \log \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+3 p \log \left (-\frac {e x^n}{d}\right ) \log \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+n \log (x) \log ^3\left (c \left (d+e x^n\right )^p\right )+3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )-6 p^2 \log \left (c \left (d+e x^n\right )^p\right ) \operatorname {PolyLog}\left (3,1+\frac {e x^n}{d}\right )+6 p^3 \operatorname {PolyLog}\left (4,1+\frac {e x^n}{d}\right )}{n} \] Input:

Integrate[Log[c*(d + e*x^n)^p]^3/x,x]
 

Output:

(-(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*Log[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
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2904, 2843, 2881, 2821, 2830, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2904

\(\displaystyle \frac {\int x^{-n} \log ^3\left (c \left (e x^n+d\right )^p\right )dx^n}{n}\)

\(\Big \downarrow \) 2843

\(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )-3 e p \int \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (e x^n+d\right )^p\right )}{e x^n+d}dx^n}{n}\)

\(\Big \downarrow \) 2881

\(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )-3 p \int x^{-n} \log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (e x^n+d\right )^p\right )d\left (e x^n+d\right )}{n}\)

\(\Big \downarrow \) 2821

\(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )-3 p \left (2 p \int x^{-n} \log \left (c \left (e x^n+d\right )^p\right ) \operatorname {PolyLog}\left (2,\frac {e x^n+d}{d}\right )d\left (e x^n+d\right )-\operatorname {PolyLog}\left (2,\frac {e x^n+d}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\)

\(\Big \downarrow \) 2830

\(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )-3 p \left (2 p \left (\operatorname {PolyLog}\left (3,\frac {e x^n+d}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )-p \int x^{-n} \operatorname {PolyLog}\left (3,\frac {e x^n+d}{d}\right )d\left (e x^n+d\right )\right )-\operatorname {PolyLog}\left (2,\frac {e x^n+d}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )-3 p \left (2 p \left (\operatorname {PolyLog}\left (3,\frac {e x^n+d}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )-p \operatorname {PolyLog}\left (4,\frac {e x^n+d}{d}\right )\right )-\operatorname {PolyLog}\left (2,\frac {e x^n+d}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\)

Input:

Int[Log[c*(d + e*x^n)^p]^3/x,x]
 

Output:

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

Defintions of rubi rules used

rule 2821

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] + Simp[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 2830

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ 
.)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) 
, x] - Simp[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 2843

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] - Simp[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 2881

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym 
bol] :> Simp[1/e   Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[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 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

Maple [C] (warning: unable to verify)

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

Time = 5.04 (sec) , antiderivative size = 1409, normalized size of antiderivative = 12.47

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

Input:

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

Output:

3/n*ln(d+e*x^n)^3*ln(-e*x^n/d)*p^3-2/n*ln(d+e*x^n)^3*ln(1-(d+e*x^n)/d)*p^3 
-1/n*ln(d+e*x^n)^3*ln(e*x^n)*p^3-6/n*ln((d+e*x^n)^p)*ln(d+e*x^n)^2*ln(-e*x 
^n/d)*p^2+3/n*ln((d+e*x^n)^p)*ln(d+e*x^n)^2*ln(1-(d+e*x^n)/d)*p^2+3/n*ln(( 
d+e*x^n)^p)*ln(d+e*x^n)^2*ln(e*x^n)*p^2+3/n*ln(d+e*x^n)^2*dilog(-e*x^n/d)* 
p^3-3/n*ln(d+e*x^n)^2*polylog(2,(d+e*x^n)/d)*p^3+3/n*ln((d+e*x^n)^p)^2*ln( 
d+e*x^n)*ln(-e*x^n/d)*p-3/n*ln((d+e*x^n)^p)^2*ln(d+e*x^n)*ln(e*x^n)*p-6/n* 
ln((d+e*x^n)^p)*ln(d+e*x^n)*dilog(-e*x^n/d)*p^2+6/n*ln((d+e*x^n)^p)*ln(d+e 
*x^n)*polylog(2,(d+e*x^n)/d)*p^2+1/n*ln((d+e*x^n)^p)^3*ln(e*x^n)+3/n*ln((d 
+e*x^n)^p)^2*dilog(-e*x^n/d)*p-6/n*ln((d+e*x^n)^p)*polylog(3,(d+e*x^n)/d)* 
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*cs 
gn(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*l 
n(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*...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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*log(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {4 \left (\int \frac {{\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right )}^{3}}{x^{n} e x +d x}d x \right ) d n p +{\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right )}^{4}}{4 n p} \] Input:

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

Output:

(4*int(log((x**n*e + d)**p*c)**3/(x**n*e*x + d*x),x)*d*n*p + log((x**n*e + 
 d)**p*c)**4)/(4*n*p)