\(\int \frac {(a+b \log (c x^n)) (d+e \log (f (g+h x)^q))}{x} \, dx\) [407]

Optimal result

Integrand size = 28, antiderivative size = 130 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )}{x} \, dx=a \log \left (-\frac {h x}{g}\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )+\frac {b \log ^2\left (c x^n\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )}{2 n}-\frac {b e q \log ^2\left (c x^n\right ) \log \left (1+\frac {h x}{g}\right )}{2 n}-b e q \log \left (c x^n\right ) \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )+a e q \operatorname {PolyLog}\left (2,1+\frac {h x}{g}\right )+b e n q \operatorname {PolyLog}\left (3,-\frac {h x}{g}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )}{x} \, dx=\frac {1}{2} \log (x) \left (-b n \log (x)+2 \left (a+b \log \left (c x^n\right )\right )\right ) \left (d+e \log \left (f (g+h x)^q\right )-e q \log \left (1+\frac {h x}{g}\right )\right )-e q \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {h x}{g}\right )+b e n q \operatorname {PolyLog}\left (3,-\frac {h x}{g}\right ) \] Input:

Integrate[((a + b*Log[c*x^n])*(d + e*Log[f*(g + h*x)^q]))/x,x]
 

Output:

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

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2888, 2841, 2752, 2872, 2754, 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.

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

\(\Big \downarrow \) 2888

\(\displaystyle a \int \frac {d+e \log \left (f (g+h x)^q\right )}{x}dx+b \int \frac {\log \left (c x^n\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )}{x}dx\)

\(\Big \downarrow \) 2841

\(\displaystyle a \left (\log \left (-\frac {h x}{g}\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )-e h q \int \frac {\log \left (-\frac {h x}{g}\right )}{g+h x}dx\right )+b \int \frac {\log \left (c x^n\right ) \left (d+e \log \left (f (g+h x)^q\right )\right )}{x}dx\)

\(\Big \downarrow \) 2752

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

\(\Big \downarrow \) 2872

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

\(\Big \downarrow \) 2754

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

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

Input:

Int[((a + b*Log[c*x^n])*(d + e*Log[f*(g + h*x)^q]))/x,x]
 

Output:

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

Defintions of rubi rules used

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2754

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

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 2841

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

rule 2872

Int[(Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b 
_.)))/(x_), x_Symbol] :> Simp[Log[f*x^m]^2*((a + b*Log[c*(d + e*x)^n])/(2*m 
)), x] - Simp[b*e*(n/(2*m))   Int[Log[f*x^m]^2/(d + e*x), x], x] /; FreeQ[{ 
a, b, c, d, e, f, m, n}, x]
 

rule 2888

Int[(((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*(Log[(h_.)*((i_.) 
 + (j_.)*(x_))^(m_.)]*(g_.) + (f_)))/(x_), x_Symbol] :> Simp[f   Int[(a + b 
*Log[c*(d + e*x)^n])/x, x], x] + Simp[g   Int[Log[h*(i + j*x)^m]*((a + b*Lo 
g[c*(d + e*x)^n])/x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, 
 x] && NeQ[e*i - d*j, 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 = 24.99 (sec) , antiderivative size = 828, normalized size of antiderivative = 6.37

method	result	size
risch	\(\text {Expression too large to display}\)	\(828\)

Input:

int((a+b*ln(c*x^n))*(d+e*ln(f*(h*x+g)^q))/x,x,method=_RETURNVERBOSE)
 

Output:

(b*e*ln(x)*ln(x^n)-1/2*b*e*n*ln(x)^2+1/2*I*ln(x)*Pi*b*e*csgn(I*x^n)*csgn(I 
*c*x^n)^2-1/2*I*ln(x)*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I*ln( 
x)*Pi*b*e*csgn(I*c*x^n)^3+1/2*I*ln(x)*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)+ln( 
x)*ln(c)*b*e+ln(x)*a*e)*ln((h*x+g)^q)+(1/4*I*e*Pi*csgn(I*(h*x+g)^q)*csgn(I 
*f*(h*x+g)^q)^2-1/4*I*e*Pi*csgn(I*(h*x+g)^q)*csgn(I*f*(h*x+g)^q)*csgn(I*f) 
-1/4*I*e*Pi*csgn(I*f*(h*x+g)^q)^3+1/4*I*e*Pi*csgn(I*f*(h*x+g)^q)^2*csgn(I* 
f)+1/2*ln(f)*e+1/2*d)*(I*Pi*ln(x)*b*csgn(I*x^n)*csgn(I*c*x^n)^2+I*Pi*ln(x) 
*b*csgn(I*c*x^n)^2*csgn(I*c)+2*ln(x)*a+2*ln(c)*b*ln(x)+b/n*ln(x^n)^2-I*ln( 
x)*Pi*b*csgn(I*c*x^n)^3-I*ln(x)*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c))- 
1/2*I*q*e*dilog((h*x+g)/g)*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)-1/2*I*q*e*ln(x)* 
ln((h*x+g)/g)*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+1/2*I*q*e*ln(x)*ln((h*x+g)/g) 
*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/2*I*q*e*ln(x)*ln((h*x+g)/g)*Pi 
*b*csgn(I*c*x^n)^3+1/2*I*q*e*dilog((h*x+g)/g)*Pi*b*csgn(I*x^n)*csgn(I*c*x^ 
n)*csgn(I*c)-1/2*I*q*e*dilog((h*x+g)/g)*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1 
/2*I*q*e*ln(x)*ln((h*x+g)/g)*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*q*e*di 
log((h*x+g)/g)*Pi*b*csgn(I*c*x^n)^3+q*e*ln(x)^2*ln((h*x+g)/g)*b*n-1/2*q*b* 
e*n*ln(x)^2*ln(1+h*x/g)-q*e*ln(x)*ln((h*x+g)/g)*b*ln(x^n)-q*e*ln(x)*ln((h* 
x+g)/g)*b*ln(c)+q*e*dilog((h*x+g)/g)*b*n*ln(x)-q*b*e*n*ln(x)*polylog(2,-h* 
x/g)-q*e*dilog((h*x+g)/g)*b*ln(x^n)-q*e*ln(x)*ln((h*x+g)/g)*a-q*e*dilog((h 
*x+g)/g)*b*ln(c)+b*e*n*q*polylog(3,-h*x/g)-q*e*dilog((h*x+g)/g)*a
 

Fricas [F]

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

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

Output:

integral((b*d*log(c*x^n) + a*d + (b*e*log(c*x^n) + a*e)*log((h*x + g)^q*f) 
)/x, x)
 

Sympy [F(-1)]

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

integrate((a+b*ln(c*x**n))*(d+e*ln(f*(h*x+g)**q))/x,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

integrate((a+b*log(c*x^n))*(d+e*log(f*(h*x+g)^q))/x,x, algorithm="giac")
 

Output:

integrate((e*log((h*x + g)^q*f) + d)*(b*log(c*x^n) + a)/x, x)
 

Mupad [F(-1)]

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

int(((d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n)))/x,x)
 

Output:

int(((d + e*log(f*(g + h*x)^q))*(a + b*log(c*x^n)))/x, x)
 

Reduce [F]

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

int((a+b*log(c*x^n))*(d+e*log(f*(h*x+g)^q))/x,x)
 

Output:

(2*int(log((g + h*x)**q*f)/(g*x + h*x**2),x)*a*e*g*n*q + 2*int((log((g + h 
*x)**q*f)*log(x**n*c))/x,x)*b*e*n*q + log((g + h*x)**q*f)**2*a*e*n + log(x 
**n*c)**2*b*d*q + 2*log(x)*a*d*n*q)/(2*n*q)