Integrand size = 22, antiderivative size = 55 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx=-\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}(2,-e x)+2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,-e x)-2 b^2 n^2 \operatorname {PolyLog}(4,-e x) \]
-(a+b*ln(c*x^n))^2*polylog(2,-e*x)+2*b*n*(a+b*ln(c*x^n))*polylog(3,-e*x)-2 *b^2*n^2*polylog(4,-e*x)
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx=-\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}(2,-e x)+2 b n \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,-e x)-b n \operatorname {PolyLog}(4,-e x)\right ) \]
-((a + b*Log[c*x^n])^2*PolyLog[2, -(e*x)]) + 2*b*n*((a + b*Log[c*x^n])*Pol yLog[3, -(e*x)] - b*n*PolyLog[4, -(e*x)])
Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2821, 2830, 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.
\(\displaystyle \int \frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx\) |
\(\Big \downarrow \) 2821 |
\(\displaystyle 2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,-e x)}{x}dx-\operatorname {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2\) |
\(\Big \downarrow \) 2830 |
\(\displaystyle 2 b n \left (\operatorname {PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}(3,-e x)}{x}dx\right )-\operatorname {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle 2 b n \left (\operatorname {PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )-b n \operatorname {PolyLog}(4,-e x)\right )-\operatorname {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2\) |
-((a + b*Log[c*x^n])^2*PolyLog[2, -(e*x)]) + 2*b*n*((a + b*Log[c*x^n])*Pol yLog[3, -(e*x)] - b*n*PolyLog[4, -(e*x)])
3.1.14.3.1 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[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ .)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) , x] - Simp[b*n*(p/q) Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
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 = 3.85 (sec) , antiderivative size = 352, normalized size of antiderivative = 6.40
method | result | size |
risch | \(-\ln \left (x \right )^{2} \operatorname {dilog}\left (e x +1\right ) b^{2} n^{2}+\ln \left (x \right )^{2} \operatorname {Li}_{2}\left (-e x \right ) b^{2} n^{2}+2 \ln \left (x \right ) \ln \left (x^{n}\right ) \operatorname {dilog}\left (e x +1\right ) b^{2} n -2 \ln \left (x \right ) \ln \left (x^{n}\right ) \operatorname {Li}_{2}\left (-e x \right ) b^{2} n -\ln \left (x^{n}\right )^{2} \operatorname {dilog}\left (e x +1\right ) b^{2}+2 \ln \left (x^{n}\right ) \operatorname {Li}_{3}\left (-e x \right ) b^{2} n -2 b^{2} n^{2} \operatorname {Li}_{4}\left (-e x \right )+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (-\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) \operatorname {dilog}\left (e x +1\right )-\ln \left (x \right ) \operatorname {Li}_{2}\left (-e x \right ) n +\operatorname {Li}_{3}\left (-e x \right ) n \right )-\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2} \operatorname {dilog}\left (e x +1\right )}{4}\) | \(352\) |
-ln(x)^2*dilog(e*x+1)*b^2*n^2+ln(x)^2*polylog(2,-e*x)*b^2*n^2+2*ln(x)*ln(x ^n)*dilog(e*x+1)*b^2*n-2*ln(x)*ln(x^n)*polylog(2,-e*x)*b^2*n-ln(x^n)^2*dil og(e*x+1)*b^2+2*ln(x^n)*polylog(3,-e*x)*b^2*n-2*b^2*n^2*polylog(4,-e*x)+(- I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^ 2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a) *b*(-(ln(x^n)-n*ln(x))*dilog(e*x+1)-ln(x)*polylog(2,-e*x)*n+polylog(3,-e*x )*n)-1/4*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csg n(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b *ln(c)+2*a)^2*dilog(e*x+1)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left (e x + 1\right )}{x} \,d x } \]
integral((b^2*log(c*x^n)^2*log(e*x + 1) + 2*a*b*log(c*x^n)*log(e*x + 1) + a^2*log(e*x + 1))/x, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left (e x + 1\right )}{x} \,d x } \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left (e x + 1\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx=\int \frac {\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x} \,d x \]