\(\int \frac {\log (f x^p) (a+b \log (c (d+e x^m)^n))}{x} \, dx\) [623]

Optimal result

Integrand size = 26, antiderivative size = 102 \[ \int \frac {\log \left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{2 p}-\frac {b n \log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{2 p}-\frac {b n \log \left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}+\frac {b n p \operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(265\) vs. \(2(102)=204\).

Time = 0.25 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.60 \[ \int \frac {\log \left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=-\frac {1}{6} b m n p \log ^3(x)+\frac {a \log ^2\left (f x^p\right )}{2 p}-\frac {1}{2} b n p \log ^2(x) \log \left (1+\frac {d x^{-m}}{e}\right )+b n p \log ^2(x) \log \left (d+e x^m\right )-\frac {b n p \log (x) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )}{m}-b n \log (x) \log \left (f x^p\right ) \log \left (d+e x^m\right )+\frac {b n \log \left (-\frac {e x^m}{d}\right ) \log \left (f x^p\right ) \log \left (d+e x^m\right )}{m}-\frac {1}{2} b p \log ^2(x) \log \left (c \left (d+e x^m\right )^n\right )+b \log (x) \log \left (f x^p\right ) \log \left (c \left (d+e x^m\right )^n\right )+\frac {b n p \log (x) \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{m}-\frac {b n \left (p \log (x)-\log \left (f x^p\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^m}{d}\right )}{m}+\frac {b n p \operatorname {PolyLog}\left (3,-\frac {d x^{-m}}{e}\right )}{m^2} \] Input:

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

Output:

-1/6*(b*m*n*p*Log[x]^3) + (a*Log[f*x^p]^2)/(2*p) - (b*n*p*Log[x]^2*Log[1 + 
 d/(e*x^m)])/2 + b*n*p*Log[x]^2*Log[d + e*x^m] - (b*n*p*Log[x]*Log[-((e*x^ 
m)/d)]*Log[d + e*x^m])/m - b*n*Log[x]*Log[f*x^p]*Log[d + e*x^m] + (b*n*Log 
[-((e*x^m)/d)]*Log[f*x^p]*Log[d + e*x^m])/m - (b*p*Log[x]^2*Log[c*(d + e*x 
^m)^n])/2 + b*Log[x]*Log[f*x^p]*Log[c*(d + e*x^m)^n] + (b*n*p*Log[x]*PolyL 
og[2, -(d/(e*x^m))])/m - (b*n*(p*Log[x] - Log[f*x^p])*PolyLog[2, 1 + (e*x^ 
m)/d])/m + (b*n*p*PolyLog[3, -(d/(e*x^m))])/m^2
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2931, 2775, 2821, 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.

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

\(\Big \downarrow \) 2931

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

\(\Big \downarrow \) 2775

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.58 \[ \int \frac {\log \left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\frac {2 \, b n p {\rm polylog}\left (3, -\frac {e x^{m}}{d}\right ) + 2 \, {\left (b m^{2} \log \left (c\right ) + a m^{2}\right )} \log \left (f\right ) \log \left (x\right ) + {\left (b m^{2} p \log \left (c\right ) + a m^{2} p\right )} \log \left (x\right )^{2} - 2 \, {\left (b m n p \log \left (x\right ) + b m n \log \left (f\right )\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) + {\left (b m^{2} n p \log \left (x\right )^{2} + 2 \, b m^{2} n \log \left (f\right ) \log \left (x\right )\right )} \log \left (e x^{m} + d\right ) - {\left (b m^{2} n p \log \left (x\right )^{2} + 2 \, b m^{2} n \log \left (f\right ) \log \left (x\right )\right )} \log \left (\frac {e x^{m} + d}{d}\right )}{2 \, m^{2}} \] Input:

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

Output:

1/2*(2*b*n*p*polylog(3, -e*x^m/d) + 2*(b*m^2*log(c) + a*m^2)*log(f)*log(x) 
 + (b*m^2*p*log(c) + a*m^2*p)*log(x)^2 - 2*(b*m*n*p*log(x) + b*m*n*log(f)) 
*dilog(-(e*x^m + d)/d + 1) + (b*m^2*n*p*log(x)^2 + 2*b*m^2*n*log(f)*log(x) 
)*log(e*x^m + d) - (b*m^2*n*p*log(x)^2 + 2*b*m^2*n*log(f)*log(x))*log((e*x 
^m + d)/d))/m^2
 

Sympy [F(-2)]

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

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

Output:

Exception raised: TypeError >> Invalid comparison of non-real zoo
 

Maxima [F]

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

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

Output:

-1/2*(b*p*log(x)^2 - 2*b*log(f)*log(x) - 2*b*log(x)*log(x^p))*log((e*x^m + 
 d)^n) - integrate(-1/2*(2*b*d*log(c)*log(f) + 2*a*d*log(f) + (b*e*m*n*p*l 
og(x)^2 - 2*b*e*m*n*log(f)*log(x) + 2*b*e*log(c)*log(f) + 2*a*e*log(f))*x^ 
m + 2*(b*d*log(c) + a*d - (b*e*m*n*log(x) - b*e*log(c) - a*e)*x^m)*log(x^p 
))/(e*x*x^m + d*x), x)
 

Giac [F]

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

integrate(log(f*x^p)*(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)/x, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\log \left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (\left (x^{m} e +d \right )^{n} c \right ) \mathrm {log}\left (x^{p} f \right )}{x}d x \right ) b p +\mathrm {log}\left (x^{p} f \right )^{2} a}{2 p} \] Input:

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

Output:

(2*int((log((x**m*e + d)**n*c)*log(x**p*f))/x,x)*b*p + log(x**p*f)**2*a)/( 
2*p)