Integrand size = 26, antiderivative size = 39 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=-\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f x^2\right )+\frac {1}{4} b n \operatorname {PolyLog}\left (3,-d f x^2\right ) \]
Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=-\frac {1}{2} a \operatorname {PolyLog}\left (2,-d f x^2\right )-\frac {1}{2} b \log \left (c x^n\right ) \operatorname {PolyLog}\left (2,-d f x^2\right )+\frac {1}{4} b n \operatorname {PolyLog}\left (3,-d f x^2\right ) \]
-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
Time = 0.24 (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 |
\(\displaystyle \frac {1}{4} b n \operatorname {PolyLog}\left (3,-d f x^2\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )\) |
3.1.26.3.1 Defintions of rubi rules used
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 = 4.44 (sec) , antiderivative size = 385, normalized size of antiderivative = 9.87
method | result | size |
risch | \(-\ln \left (d f \,x^{2}+1\right ) \ln \left (x \right )^{2} b n +\ln \left (x \right )^{2} \ln \left (1+x \sqrt {-d f}\right ) b n +\ln \left (x \right )^{2} \ln \left (1-x \sqrt {-d f}\right ) b n +\ln \left (d f \,x^{2}+1\right ) \ln \left (x \right ) \ln \left (x^{n}\right ) b +\ln \left (x \right ) \operatorname {dilog}\left (1+x \sqrt {-d f}\right ) b n +\ln \left (x \right ) \operatorname {dilog}\left (1-x \sqrt {-d f}\right ) b n -\ln \left (x \right ) \ln \left (1+x \sqrt {-d f}\right ) \ln \left (x^{n}\right ) b -\ln \left (x \right ) \ln \left (1-x \sqrt {-d f}\right ) \ln \left (x^{n}\right ) b -\operatorname {dilog}\left (1+x \sqrt {-d f}\right ) \ln \left (x^{n}\right ) b -\operatorname {dilog}\left (1-x \sqrt {-d f}\right ) \ln \left (x^{n}\right ) b -\frac {\ln \left (x \right ) \operatorname {Li}_{2}\left (-d f \,x^{2}\right ) b n}{2}+\frac {b n \,\operatorname {Li}_{3}\left (-d f \,x^{2}\right )}{4}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\ln \left (x \right ) \ln \left (d f \,x^{2}+1\right )-2 d f \left (\frac {\ln \left (x \right ) \left (\ln \left (1+x \sqrt {-d f}\right )+\ln \left (1-x \sqrt {-d f}\right )\right )}{2 d f}+\frac {\operatorname {dilog}\left (1+x \sqrt {-d f}\right )+\operatorname {dilog}\left (1-x \sqrt {-d f}\right )}{2 d f}\right )\right )\) | \(385\) |
-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)*dilog(1+x*(-d*f)^(1 /2))*b*n+ln(x)*dilog(1-x*(-d*f)^(1/2))*b*n-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-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*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+ 1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^ 2-1/2*I*b*Pi*csgn(I*c*x^n)^3+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))
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x} \,d x } \]
-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)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx=\int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]