\(\int \frac {\log ^3(f x^p) (a+b \log (c (d+e x^m)^n))}{x} \, dx\) [621]

Optimal result

Integrand size = 28, antiderivative size = 161 \[ \int \frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {b n \log ^4\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{4 p}-\frac {b n \log ^3\left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}+\frac {3 b n p \log ^2\left (f x^p\right ) \operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2}-\frac {6 b n p^2 \log \left (f x^p\right ) \operatorname {PolyLog}\left (4,-\frac {e x^m}{d}\right )}{m^3}+\frac {6 b n p^3 \operatorname {PolyLog}\left (5,-\frac {e x^m}{d}\right )}{m^4} \] Output:

1/4*ln(f*x^p)^4*(a+b*ln(c*(d+e*x^m)^n))/p-1/4*b*n*ln(f*x^p)^4*ln(1+e*x^m/d 
)/p-b*n*ln(f*x^p)^3*polylog(2,-e*x^m/d)/m+3*b*n*p*ln(f*x^p)^2*polylog(3,-e 
*x^m/d)/m^2-6*b*n*p^2*ln(f*x^p)*polylog(4,-e*x^m/d)/m^3+6*b*n*p^3*polylog( 
5,-e*x^m/d)/m^4
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(659\) vs. \(2(161)=322\).

Time = 0.34 (sec) , antiderivative size = 659, normalized size of antiderivative = 4.09 \[ \int \frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx =\text {Too large to display} \] Input:

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

Output:

(-3*b*m*n*p^3*Log[x]^5)/10 + (3*b*m*n*p^2*Log[x]^4*Log[f*x^p])/4 - (b*m*n* 
p*Log[x]^3*Log[f*x^p]^2)/2 + (a*Log[f*x^p]^4)/(4*p) - (3*b*n*p^3*Log[x]^4* 
Log[1 + d/(e*x^m)])/4 + 2*b*n*p^2*Log[x]^3*Log[f*x^p]*Log[1 + d/(e*x^m)] - 
 (3*b*n*p*Log[x]^2*Log[f*x^p]^2*Log[1 + d/(e*x^m)])/2 + b*n*p^3*Log[x]^4*L 
og[d + e*x^m] - (b*n*p^3*Log[x]^3*Log[-((e*x^m)/d)]*Log[d + e*x^m])/m - 3* 
b*n*p^2*Log[x]^3*Log[f*x^p]*Log[d + e*x^m] + (3*b*n*p^2*Log[x]^2*Log[-((e* 
x^m)/d)]*Log[f*x^p]*Log[d + e*x^m])/m + 3*b*n*p*Log[x]^2*Log[f*x^p]^2*Log[ 
d + e*x^m] - (3*b*n*p*Log[x]*Log[-((e*x^m)/d)]*Log[f*x^p]^2*Log[d + e*x^m] 
)/m - b*n*Log[x]*Log[f*x^p]^3*Log[d + e*x^m] + (b*n*Log[-((e*x^m)/d)]*Log[ 
f*x^p]^3*Log[d + e*x^m])/m - (b*p^3*Log[x]^4*Log[c*(d + e*x^m)^n])/4 + b*p 
^2*Log[x]^3*Log[f*x^p]*Log[c*(d + e*x^m)^n] - (3*b*p*Log[x]^2*Log[f*x^p]^2 
*Log[c*(d + e*x^m)^n])/2 + b*Log[x]*Log[f*x^p]^3*Log[c*(d + e*x^m)^n] + (b 
*n*p*Log[x]*(p^2*Log[x]^2 - 3*p*Log[x]*Log[f*x^p] + 3*Log[f*x^p]^2)*PolyLo 
g[2, -(d/(e*x^m))])/m - (b*n*(p*Log[x] - Log[f*x^p])^3*PolyLog[2, 1 + (e*x 
^m)/d])/m + (3*b*n*p*Log[f*x^p]^2*PolyLog[3, -(d/(e*x^m))])/m^2 + (6*b*n*p 
^2*Log[f*x^p]*PolyLog[4, -(d/(e*x^m))])/m^3 + (6*b*n*p^3*PolyLog[5, -(d/(e 
*x^m))])/m^4
 

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2931, 2775, 2821, 2830, 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.

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

\(\Big \downarrow \) 2931

\(\displaystyle \frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {b e m n \int \frac {x^{m-1} \log ^4\left (f x^p\right )}{e x^m+d}dx}{4 p}\)

\(\Big \downarrow \) 2775

\(\displaystyle \frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {b e m n \left (\frac {\log ^4\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{e m}-\frac {4 p \int \frac {\log ^3\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{x}dx}{e m}\right )}{4 p}\)

\(\Big \downarrow \) 2821

\(\displaystyle \frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {b e m n \left (\frac {\log ^4\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{e m}-\frac {4 p \left (\frac {3 p \int \frac {\log ^2\left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{x}dx}{m}-\frac {\log ^3\left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}\right )}{e m}\right )}{4 p}\)

\(\Big \downarrow \) 2830

\(\displaystyle \frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {b e m n \left (\frac {\log ^4\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{e m}-\frac {4 p \left (\frac {3 p \left (\frac {\log ^2\left (f x^p\right ) \operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m}-\frac {2 p \int \frac {\log \left (f x^p\right ) \operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{x}dx}{m}\right )}{m}-\frac {\log ^3\left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}\right )}{e m}\right )}{4 p}\)

\(\Big \downarrow \) 2830

\(\displaystyle \frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {b e m n \left (\frac {\log ^4\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{e m}-\frac {4 p \left (\frac {3 p \left (\frac {\log ^2\left (f x^p\right ) \operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m}-\frac {2 p \left (\frac {\log \left (f x^p\right ) \operatorname {PolyLog}\left (4,-\frac {e x^m}{d}\right )}{m}-\frac {p \int \frac {\operatorname {PolyLog}\left (4,-\frac {e x^m}{d}\right )}{x}dx}{m}\right )}{m}\right )}{m}-\frac {\log ^3\left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}\right )}{e m}\right )}{4 p}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {b e m n \left (\frac {\log ^4\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{e m}-\frac {4 p \left (\frac {3 p \left (\frac {\log ^2\left (f x^p\right ) \operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m}-\frac {2 p \left (\frac {\log \left (f x^p\right ) \operatorname {PolyLog}\left (4,-\frac {e x^m}{d}\right )}{m}-\frac {p \operatorname {PolyLog}\left (5,-\frac {e x^m}{d}\right )}{m^2}\right )}{m}\right )}{m}-\frac {\log ^3\left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}\right )}{e m}\right )}{4 p}\)

Input:

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

Output:

(Log[f*x^p]^4*(a + b*Log[c*(d + e*x^m)^n]))/(4*p) - (b*e*m*n*((Log[f*x^p]^ 
4*Log[1 + (e*x^m)/d])/(e*m) - (4*p*(-((Log[f*x^p]^3*PolyLog[2, -((e*x^m)/d 
)])/m) + (3*p*((Log[f*x^p]^2*PolyLog[3, -((e*x^m)/d)])/m - (2*p*((Log[f*x^ 
p]*PolyLog[4, -((e*x^m)/d)])/m - (p*PolyLog[5, -((e*x^m)/d)])/m^2))/m))/m) 
)/(e*m)))/(4*p)
 

Defintions of rubi rules used

rule 2775

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

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 2931

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

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 [F]

\[\int \frac {\ln \left (f \,x^{p}\right )^{3} \left (a +b \ln \left (c \left (d +e \,x^{m}\right )^{n}\right )\right )}{x}d x\]

Input:

int(ln(f*x^p)^3*(a+b*ln(c*(d+e*x^m)^n))/x,x)
 

Output:

int(ln(f*x^p)^3*(a+b*ln(c*(d+e*x^m)^n))/x,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (156) = 312\).

Time = 0.14 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.59 \[ \int \frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\frac {24 \, b n p^{3} {\rm polylog}\left (5, -\frac {e x^{m}}{d}\right ) + 4 \, {\left (b m^{4} \log \left (c\right ) + a m^{4}\right )} \log \left (f\right )^{3} \log \left (x\right ) + 6 \, {\left (b m^{4} p \log \left (c\right ) + a m^{4} p\right )} \log \left (f\right )^{2} \log \left (x\right )^{2} + 4 \, {\left (b m^{4} p^{2} \log \left (c\right ) + a m^{4} p^{2}\right )} \log \left (f\right ) \log \left (x\right )^{3} + {\left (b m^{4} p^{3} \log \left (c\right ) + a m^{4} p^{3}\right )} \log \left (x\right )^{4} - 4 \, {\left (b m^{3} n p^{3} \log \left (x\right )^{3} + 3 \, b m^{3} n p^{2} \log \left (f\right ) \log \left (x\right )^{2} + 3 \, b m^{3} n p \log \left (f\right )^{2} \log \left (x\right ) + b m^{3} n \log \left (f\right )^{3}\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) + {\left (b m^{4} n p^{3} \log \left (x\right )^{4} + 4 \, b m^{4} n p^{2} \log \left (f\right ) \log \left (x\right )^{3} + 6 \, b m^{4} n p \log \left (f\right )^{2} \log \left (x\right )^{2} + 4 \, b m^{4} n \log \left (f\right )^{3} \log \left (x\right )\right )} \log \left (e x^{m} + d\right ) - {\left (b m^{4} n p^{3} \log \left (x\right )^{4} + 4 \, b m^{4} n p^{2} \log \left (f\right ) \log \left (x\right )^{3} + 6 \, b m^{4} n p \log \left (f\right )^{2} \log \left (x\right )^{2} + 4 \, b m^{4} n \log \left (f\right )^{3} \log \left (x\right )\right )} \log \left (\frac {e x^{m} + d}{d}\right ) - 24 \, {\left (b m n p^{3} \log \left (x\right ) + b m n p^{2} \log \left (f\right )\right )} {\rm polylog}\left (4, -\frac {e x^{m}}{d}\right ) + 12 \, {\left (b m^{2} n p^{3} \log \left (x\right )^{2} + 2 \, b m^{2} n p^{2} \log \left (f\right ) \log \left (x\right ) + b m^{2} n p \log \left (f\right )^{2}\right )} {\rm polylog}\left (3, -\frac {e x^{m}}{d}\right )}{4 \, m^{4}} \] Input:

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

Output:

1/4*(24*b*n*p^3*polylog(5, -e*x^m/d) + 4*(b*m^4*log(c) + a*m^4)*log(f)^3*l 
og(x) + 6*(b*m^4*p*log(c) + a*m^4*p)*log(f)^2*log(x)^2 + 4*(b*m^4*p^2*log( 
c) + a*m^4*p^2)*log(f)*log(x)^3 + (b*m^4*p^3*log(c) + a*m^4*p^3)*log(x)^4 
- 4*(b*m^3*n*p^3*log(x)^3 + 3*b*m^3*n*p^2*log(f)*log(x)^2 + 3*b*m^3*n*p*lo 
g(f)^2*log(x) + b*m^3*n*log(f)^3)*dilog(-(e*x^m + d)/d + 1) + (b*m^4*n*p^3 
*log(x)^4 + 4*b*m^4*n*p^2*log(f)*log(x)^3 + 6*b*m^4*n*p*log(f)^2*log(x)^2 
+ 4*b*m^4*n*log(f)^3*log(x))*log(e*x^m + d) - (b*m^4*n*p^3*log(x)^4 + 4*b* 
m^4*n*p^2*log(f)*log(x)^3 + 6*b*m^4*n*p*log(f)^2*log(x)^2 + 4*b*m^4*n*log( 
f)^3*log(x))*log((e*x^m + d)/d) - 24*(b*m*n*p^3*log(x) + b*m*n*p^2*log(f)) 
*polylog(4, -e*x^m/d) + 12*(b*m^2*n*p^3*log(x)^2 + 2*b*m^2*n*p^2*log(f)*lo 
g(x) + b*m^2*n*p*log(f)^2)*polylog(3, -e*x^m/d))/m^4
 

Sympy [F(-1)]

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

integrate(ln(f*x**p)**3*(a+b*ln(c*(d+e*x**m)**n))/x,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/4*(b*p^3*log(x)^4 - 4*b*p^2*log(f)*log(x)^3 + 6*b*p*log(f)^2*log(x)^2 - 
 4*b*log(f)^3*log(x) - 4*b*log(x)*log(x^p)^3 + 6*(b*p*log(x)^2 - 2*b*log(f 
)*log(x))*log(x^p)^2 - 4*(b*p^2*log(x)^3 - 3*b*p*log(f)*log(x)^2 + 3*b*log 
(f)^2*log(x))*log(x^p))*log((e*x^m + d)^n) - integrate(-1/4*(4*b*d*log(c)* 
log(f)^3 + 4*a*d*log(f)^3 + 4*(b*d*log(c) + a*d - (b*e*m*n*log(x) - b*e*lo 
g(c) - a*e)*x^m)*log(x^p)^3 + 6*(2*b*d*log(c)*log(f) + 2*a*d*log(f) + (b*e 
*m*n*p*log(x)^2 - 2*b*e*m*n*log(f)*log(x) + 2*b*e*log(c)*log(f) + 2*a*e*lo 
g(f))*x^m)*log(x^p)^2 + (b*e*m*n*p^3*log(x)^4 - 4*b*e*m*n*p^2*log(f)*log(x 
)^3 + 6*b*e*m*n*p*log(f)^2*log(x)^2 - 4*b*e*m*n*log(f)^3*log(x) + 4*b*e*lo 
g(c)*log(f)^3 + 4*a*e*log(f)^3)*x^m + 4*(3*b*d*log(c)*log(f)^2 + 3*a*d*log 
(f)^2 - (b*e*m*n*p^2*log(x)^3 - 3*b*e*m*n*p*log(f)*log(x)^2 + 3*b*e*m*n*lo 
g(f)^2*log(x) - 3*b*e*log(c)*log(f)^2 - 3*a*e*log(f)^2)*x^m)*log(x^p))/(e* 
x*x^m + d*x), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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