\(\int \frac {(a+b \log (c x^n)) \log (d (\frac {1}{d}+f x^2))}{x} \, dx\) [32]

Optimal result

Integrand size = 26, antiderivative size = 39 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=-\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f x^2\right )+\frac {1}{4} b n \operatorname {PolyLog}\left (3,-d f x^2\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=-\frac {1}{2} a \operatorname {PolyLog}\left (2,-d f x^2\right )-\frac {1}{2} b \log \left (c x^n\right ) \operatorname {PolyLog}\left (2,-d f x^2\right )+\frac {1}{4} b n \operatorname {PolyLog}\left (3,-d f x^2\right ) \] Input:

Integrate[((a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^2)])/x,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {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.

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

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

Input:

Int[((a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^2)])/x,x]
 

Output:

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

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 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 = 7.91 (sec) , antiderivative size = 385, normalized size of antiderivative = 9.87

method	result	size
risch	\(-\ln \left (d f \,x^{2}+1\right ) \ln \left (x \right )^{2} b n +\ln \left (x \right )^{2} \ln \left (1+x \sqrt {-d f}\right ) b n +\ln \left (x \right )^{2} \ln \left (1-x \sqrt {-d f}\right ) b n +\ln \left (d f \,x^{2}+1\right ) \ln \left (x \right ) \ln \left (x^{n}\right ) b -\ln \left (x \right ) \ln \left (1+x \sqrt {-d f}\right ) \ln \left (x^{n}\right ) b -\ln \left (x \right ) \ln \left (1-x \sqrt {-d f}\right ) \ln \left (x^{n}\right ) b +\ln \left (x \right ) \operatorname {dilog}\left (1+x \sqrt {-d f}\right ) b n +\ln \left (x \right ) \operatorname {dilog}\left (1-x \sqrt {-d f}\right ) b n -\operatorname {dilog}\left (1+x \sqrt {-d f}\right ) \ln \left (x^{n}\right ) b -\operatorname {dilog}\left (1-x \sqrt {-d f}\right ) \ln \left (x^{n}\right ) b -\frac {\ln \left (x \right ) \operatorname {polylog}\left (2, -d f \,x^{2}\right ) b n}{2}+\frac {b n \operatorname {polylog}\left (3, -d f \,x^{2}\right )}{4}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\ln \left (x \right ) \ln \left (d f \,x^{2}+1\right )-2 d f \left (\frac {\ln \left (x \right ) \left (\ln \left (1+x \sqrt {-d f}\right )+\ln \left (1-x \sqrt {-d f}\right )\right )}{2 d f}+\frac {\operatorname {dilog}\left (1+x \sqrt {-d f}\right )+\operatorname {dilog}\left (1-x \sqrt {-d f}\right )}{2 d f}\right )\right )\)	\(385\)

Input:

int((a+b*ln(c*x^n))*ln(d*(1/d+f*x^2))/x,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x} \,d x } \] Input:

integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^2))/x,x, algorithm="fricas")
 

Output:

integral((b*log(d*f*x^2 + 1)*log(c*x^n) + a*log(d*f*x^2 + 1))/x, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\text {Timed out} \] Input:

integrate((a+b*ln(c*x**n))*ln(d*(1/d+f*x**2))/x,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x} \,d x } \] Input:

integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^2))/x,x, algorithm="maxima")
 

Output:

-1/2*(b*n*log(x)^2 - 2*b*log(x)*log(x^n) - 2*(b*log(c) + a)*log(x))*log(d* 
f*x^2 + 1) - integrate(-(b*d*f*n*x*log(x)^2 - 2*b*d*f*x*log(x)*log(x^n) - 
2*(b*d*f*log(c) + a*d*f)*x*log(x))/(d*f*x^2 + 1), x)
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x} \,d x } \] Input:

integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^2))/x,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)*log((f*x^2 + 1/d)*d)/x, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \] Input:

int((log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n)))/x,x)
 

Output:

int((log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n)))/x, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\left (\int \frac {\mathrm {log}\left (d f \,x^{2}+1\right )}{d f \,x^{3}+x}d x \right ) a +\left (\int \frac {\mathrm {log}\left (d f \,x^{2}+1\right ) \mathrm {log}\left (x^{n} c \right )}{x}d x \right ) b +\frac {\mathrm {log}\left (d f \,x^{2}+1\right )^{2} a}{4} \] Input:

int((a+b*log(c*x^n))*log(d*(1/d+f*x^2))/x,x)
 

Output:

(4*int(log(d*f*x**2 + 1)/(d*f*x**3 + x),x)*a + 4*int((log(d*f*x**2 + 1)*lo 
g(x**n*c))/x,x)*b + log(d*f*x**2 + 1)**2*a)/4