Integrand size = 26, antiderivative size = 40 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f x^m\right )}{m}+\frac {b n \operatorname {PolyLog}\left (3,-d f x^m\right )}{m^2} \] Output:
-(a+b*ln(c*x^n))*polylog(2,-d*f*x^m)/m+b*n*polylog(3,-d*f*x^m)/m^2
Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=-\frac {a \operatorname {PolyLog}\left (2,-d f x^m\right )}{m}-\frac {b \log \left (c x^n\right ) \operatorname {PolyLog}\left (2,-d f x^m\right )}{m}+\frac {b n \operatorname {PolyLog}\left (3,-d f x^m\right )}{m^2} \] Input:
Integrate[((a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^m)])/x,x]
Output:
-((a*PolyLog[2, -(d*f*x^m)])/m) - (b*Log[c*x^n]*PolyLog[2, -(d*f*x^m)])/m + (b*n*PolyLog[3, -(d*f*x^m)])/m^2
Time = 0.27 (sec) , antiderivative size = 40, 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.
\(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f x^m\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 2821 |
\(\displaystyle \frac {b n \int \frac {\operatorname {PolyLog}\left (2,-d f x^m\right )}{x}dx}{m}-\frac {\operatorname {PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {b n \operatorname {PolyLog}\left (3,-d f x^m\right )}{m^2}-\frac {\operatorname {PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m}\) |
Input:
Int[((a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^m)])/x,x]
Output:
-(((a + b*Log[c*x^n])*PolyLog[2, -(d*f*x^m)])/m) + (b*n*PolyLog[3, -(d*f*x ^m)])/m^2
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 25.57 (sec) , antiderivative size = 248, normalized size of antiderivative = 6.20
method | result | size |
risch | \(-\frac {b \ln \left (x \right )^{2} n \ln \left (d \left (\frac {1}{d}+f \,x^{m}\right )\right )}{2}+b \ln \left (x \right ) \ln \left (d \left (\frac {1}{d}+f \,x^{m}\right )\right ) \ln \left (x^{n}\right )+\frac {b n \ln \left (x \right )^{2} \ln \left (d f \,x^{m}+1\right )}{2}-\frac {b n \ln \left (x \right ) \operatorname {polylog}\left (2, -d f \,x^{m}\right )}{m}+\frac {b n \operatorname {polylog}\left (3, -d f \,x^{m}\right )}{m^{2}}+\frac {b \operatorname {dilog}\left (d f \,x^{m}+1\right ) n \ln \left (x \right )}{m}-\frac {b \operatorname {dilog}\left (d f \,x^{m}+1\right ) \ln \left (x^{n}\right )}{m}-b \ln \left (d f \,x^{m}+1\right ) \ln \left (x \right ) \ln \left (x^{n}\right )-\frac {\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \operatorname {dilog}\left (d f \,x^{m}+1\right )}{m}\) | \(248\) |
Input:
int((a+b*ln(c*x^n))*ln(d*(1/d+f*x^m))/x,x,method=_RETURNVERBOSE)
Output:
-1/2*b*ln(x)^2*n*ln(d*(1/d+f*x^m))+b*ln(x)*ln(d*(1/d+f*x^m))*ln(x^n)+1/2*b *n*ln(x)^2*ln(d*f*x^m+1)-b*n/m*ln(x)*polylog(2,-d*f*x^m)+b*n*polylog(3,-d* f*x^m)/m^2+b/m*dilog(d*f*x^m+1)*n*ln(x)-b/m*dilog(d*f*x^m+1)*ln(x^n)-b*ln( d*f*x^m+1)*ln(x)*ln(x^n)-(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)/m*dilog(d*f*x^m+1)
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\frac {b n {\rm polylog}\left (3, -d f x^{m}\right ) - {\left (b m n \log \left (x\right ) + b m \log \left (c\right ) + a m\right )} {\rm Li}_2\left (-d f x^{m}\right )}{m^{2}} \] Input:
integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^m))/x,x, algorithm="fricas")
Output:
(b*n*polylog(3, -d*f*x^m) - (b*m*n*log(x) + b*m*log(c) + a*m)*dilog(-d*f*x ^m))/m^2
Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*ln(c*x**n))*ln(d*(1/d+f*x**m))/x,x)
Output:
Exception raised: TypeError >> Invalid comparison of non-real zoo
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{m} + \frac {1}{d}\right )} d\right )}{x} \,d x } \] Input:
integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^m))/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^m + 1) - integrate(1/2*(2*b*d*f*m*x^m*log(x)*log(x^n) - (b*d*f*m*n*log (x)^2 - 2*(b*d*f*m*log(c) + a*d*f*m)*log(x))*x^m)/(d*f*x*x^m + x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{m} + \frac {1}{d}\right )} d\right )}{x} \,d x } \] Input:
integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^m))/x,x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)*log((f*x^m + 1/d)*d)/x, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\int \frac {\ln \left (d\,\left (f\,x^m+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \] Input:
int((log(d*(f*x^m + 1/d))*(a + b*log(c*x^n)))/x,x)
Output:
int((log(d*(f*x^m + 1/d))*(a + b*log(c*x^n)))/x, x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (x^{m} d f +1\right )}{x^{m} d f x +x}d x \right ) a m +2 \left (\int \frac {\mathrm {log}\left (x^{m} d f +1\right ) \mathrm {log}\left (x^{n} c \right )}{x}d x \right ) b m +\mathrm {log}\left (x^{m} d f +1\right )^{2} a}{2 m} \] Input:
int((a+b*log(c*x^n))*log(d*(1/d+f*x^m))/x,x)
Output:
(2*int(log(x**m*d*f + 1)/(x**m*d*f*x + x),x)*a*m + 2*int((log(x**m*d*f + 1 )*log(x**n*c))/x,x)*b*m + log(x**m*d*f + 1)**2*a)/(2*m)