Integrand size = 19, antiderivative size = 142 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x^2} \, dx=2 b e n \log (x)-\frac {1}{2} b e n \log ^2(x)+e \log (x) \left (a+b \log \left (c x^n\right )\right )-2 b e n \log (1-e x)+\frac {2 b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-b e n \operatorname {PolyLog}(2,e x)-\frac {b n \operatorname {PolyLog}(2,e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x} \]
2*b*e*n*ln(x)-1/2*b*e*n*ln(x)^2+e*ln(x)*(a+b*ln(c*x^n))-2*b*e*n*ln(-e*x+1) +2*b*n*ln(-e*x+1)/x-e*(a+b*ln(c*x^n))*ln(-e*x+1)+(a+b*ln(c*x^n))*ln(-e*x+1 )/x-b*e*n*polylog(2,e*x)-b*n*polylog(2,e*x)/x-(a+b*ln(c*x^n))*polylog(2,e* x)/x
Time = 0.14 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x^2} \, dx=\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) (e x \log (x)+(1-e x) \log (1-e x)-\operatorname {PolyLog}(2,e x))}{x}+\frac {b n \left (e x \log ^2(x)-4 (-1+e x) \log (1-e x)+\log (x) (4 e x+(2-2 e x) \log (1-e x))-2 (1+e x+\log (x)) \operatorname {PolyLog}(2,e x)\right )}{2 x} \]
((a - b*n*Log[x] + b*Log[c*x^n])*(e*x*Log[x] + (1 - e*x)*Log[1 - e*x] - Po lyLog[2, e*x]))/x + (b*n*(e*x*Log[x]^2 - 4*(-1 + e*x)*Log[1 - e*x] + Log[x ]*(4*e*x + (2 - 2*e*x)*Log[1 - e*x]) - 2*(1 + e*x + Log[x])*PolyLog[2, e*x ]))/(2*x)
Time = 0.43 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2832, 25, 2823, 2009, 2842, 47, 14, 16}
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.
\(\displaystyle \int \frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 2832 |
\(\displaystyle \int -\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x^2}dx+b n \int -\frac {\log (1-e x)}{x^2}dx-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b n \operatorname {PolyLog}(2,e x)}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x^2}dx-b n \int \frac {\log (1-e x)}{x^2}dx-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b n \operatorname {PolyLog}(2,e x)}{x}\) |
\(\Big \downarrow \) 2823 |
\(\displaystyle -b n \int \frac {\log (1-e x)}{x^2}dx+b n \int \left (-\frac {e \log (x)}{x}+\frac {e \log (1-e x)}{x}-\frac {\log (1-e x)}{x^2}\right )dx-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b n \operatorname {PolyLog}(2,e x)}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -b n \int \frac {\log (1-e x)}{x^2}dx-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b n \operatorname {PolyLog}(2,e x)}{x}+b n \left (-e \operatorname {PolyLog}(2,e x)-\frac {1}{2} e \log ^2(x)+e \log (x)-e \log (1-e x)+\frac {\log (1-e x)}{x}\right )\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle -b n \left (-e \int \frac {1}{x (1-e x)}dx-\frac {\log (1-e x)}{x}\right )-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b n \operatorname {PolyLog}(2,e x)}{x}+b n \left (-e \operatorname {PolyLog}(2,e x)-\frac {1}{2} e \log ^2(x)+e \log (x)-e \log (1-e x)+\frac {\log (1-e x)}{x}\right )\) |
\(\Big \downarrow \) 47 |
\(\displaystyle -b n \left (-e \left (e \int \frac {1}{1-e x}dx+\int \frac {1}{x}dx\right )-\frac {\log (1-e x)}{x}\right )-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b n \operatorname {PolyLog}(2,e x)}{x}+b n \left (-e \operatorname {PolyLog}(2,e x)-\frac {1}{2} e \log ^2(x)+e \log (x)-e \log (1-e x)+\frac {\log (1-e x)}{x}\right )\) |
\(\Big \downarrow \) 14 |
\(\displaystyle -b n \left (-e \left (e \int \frac {1}{1-e x}dx+\log (x)\right )-\frac {\log (1-e x)}{x}\right )-\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b n \operatorname {PolyLog}(2,e x)}{x}+b n \left (-e \operatorname {PolyLog}(2,e x)-\frac {1}{2} e \log ^2(x)+e \log (x)-e \log (1-e x)+\frac {\log (1-e x)}{x}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle -\frac {\operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b n \operatorname {PolyLog}(2,e x)}{x}+b n \left (-e \operatorname {PolyLog}(2,e x)-\frac {1}{2} e \log ^2(x)+e \log (x)-e \log (1-e x)+\frac {\log (1-e x)}{x}\right )-b n \left (-e (\log (x)-\log (1-e x))-\frac {\log (1-e x)}{x}\right )\) |
e*Log[x]*(a + b*Log[c*x^n]) - e*(a + b*Log[c*x^n])*Log[1 - e*x] + ((a + b* Log[c*x^n])*Log[1 - e*x])/x - b*n*(-(e*(Log[x] - Log[1 - e*x])) - Log[1 - e*x]/x) - (b*n*PolyLog[2, e*x])/x - ((a + b*Log[c*x^n])*PolyLog[2, e*x])/x + b*n*(e*Log[x] - (e*Log[x]^2)/2 - e*Log[1 - e*x] + Log[1 - e*x]/x - e*Po lyLog[2, e*x])
3.3.12.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* (e + f*x^m)^r], x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] && RationalQ[q])) && NeQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.)*PolyLog[k_, (e _.)*(x_)^(q_.)], x_Symbol] :> Simp[(-b)*n*(d*x)^(m + 1)*(PolyLog[k, e*x^q]/ (d*(m + 1)^2)), x] + (Simp[(d*x)^(m + 1)*PolyLog[k, e*x^q]*((a + b*Log[c*x^ n])/(d*(m + 1))), x] - Simp[q/(m + 1) Int[(d*x)^m*PolyLog[k - 1, e*x^q]*( a + b*Log[c*x^n]), x], x] + Simp[b*n*(q/(m + 1)^2) Int[(d*x)^m*PolyLog[k - 1, e*x^q], x], x]) /; FreeQ[{a, b, c, d, e, m, n, q}, x] && IGtQ[k, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ ))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1))) Int[(f + g*x)^(q + 1)/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]
Time = 4.75 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.44
method | result | size |
parallelrisch | \(\frac {-2 b \,e^{2} \ln \left (-e x +1\right ) \ln \left (c \,x^{n}\right ) x n -2 x \,\operatorname {Li}_{2}\left (e x \right ) b \,e^{2} n^{2}-4 x \ln \left (-e x +1\right ) b \,e^{2} n^{2}+e^{2} b \ln \left (c \,x^{n}\right )^{2} x +4 x \ln \left (c \,x^{n}\right ) b \,e^{2} n -2 x \ln \left (-e x +1\right ) a \,e^{2} n +2 x \ln \left (c \,x^{n}\right ) a \,e^{2}-2 \ln \left (c \,x^{n}\right ) \operatorname {Li}_{2}\left (e x \right ) b e n +2 b \ln \left (-e x +1\right ) \ln \left (c \,x^{n}\right ) e n -2 \,\operatorname {Li}_{2}\left (e x \right ) b e \,n^{2}+4 \ln \left (-e x +1\right ) b e \,n^{2}-2 \,\operatorname {Li}_{2}\left (e x \right ) a e n +2 \ln \left (-e x +1\right ) a e n}{2 x e n}\) | \(204\) |
1/2*(-2*b*e^2*ln(-e*x+1)*ln(c*x^n)*x*n-2*x*polylog(2,e*x)*b*e^2*n^2-4*x*ln (-e*x+1)*b*e^2*n^2+e^2*b*ln(c*x^n)^2*x+4*x*ln(c*x^n)*b*e^2*n-2*x*ln(-e*x+1 )*a*e^2*n+2*x*ln(c*x^n)*a*e^2-2*ln(c*x^n)*polylog(2,e*x)*b*e*n+2*b*ln(-e*x +1)*ln(c*x^n)*e*n-2*polylog(2,e*x)*b*e*n^2+4*ln(-e*x+1)*b*e*n^2-2*polylog( 2,e*x)*a*e*n+2*ln(-e*x+1)*a*e*n)/x/e/n
Time = 0.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x^2} \, dx=\frac {b e n x \log \left (x\right )^{2} - 2 \, {\left (b e n x + b n + a\right )} {\rm Li}_2\left (e x\right ) + 2 \, {\left (2 \, b n - {\left (2 \, b e n + a e\right )} x + a\right )} \log \left (-e x + 1\right ) - 2 \, {\left (b {\rm Li}_2\left (e x\right ) + {\left (b e x - b\right )} \log \left (-e x + 1\right )\right )} \log \left (c\right ) + 2 \, {\left (b e x \log \left (c\right ) - b n {\rm Li}_2\left (e x\right ) + {\left (2 \, b e n + a e\right )} x - {\left (b e n x - b n\right )} \log \left (-e x + 1\right )\right )} \log \left (x\right )}{2 \, x} \]
1/2*(b*e*n*x*log(x)^2 - 2*(b*e*n*x + b*n + a)*dilog(e*x) + 2*(2*b*n - (2*b *e*n + a*e)*x + a)*log(-e*x + 1) - 2*(b*dilog(e*x) + (b*e*x - b)*log(-e*x + 1))*log(c) + 2*(b*e*x*log(c) - b*n*dilog(e*x) + (2*b*e*n + a*e)*x - (b*e *n*x - b*n)*log(-e*x + 1))*log(x))/x
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{2}\left (e x\right )}{x^{2}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} {\rm Li}_2\left (e x\right )}{x^{2}} \,d x } \]
(e*log(x) - ((e*x - 1)*log(-e*x + 1) + dilog(e*x))/x)*a - b*(((n + log(c) + log(x^n))*dilog(e*x) - (e*n*x*log(x) + 2*n + log(c))*log(-e*x + 1) - (e* x*log(x) - (e*x - 1)*log(-e*x + 1))*log(x^n))/x + integrate((2*e*n + e*log (c) + (2*e^2*n*x - e*n)*log(x))/(e*x^2 - x), x))
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} {\rm Li}_2\left (e x\right )}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)}{x^2} \, dx=\int \frac {\mathrm {polylog}\left (2,e\,x\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \]