Integrand size = 13, antiderivative size = 173 \[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx=\frac {b^2 c \log (x)}{2 a (1-a c)}-\frac {b^2 c \log (1-a c-b c x)}{2 a (1-a c)}+\frac {b \log (1-a c-b c x)}{2 a x}+\frac {b^2 \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{2 a^2}+\frac {b^2 \operatorname {PolyLog}(2,c (a+b x))}{2 a^2}-\frac {\operatorname {PolyLog}(2,c (a+b x))}{2 x^2}+\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {b c x}{1-a c}\right )}{2 a^2} \]
1/2*b^2*c*ln(x)/a/(-a*c+1)-1/2*b^2*c*ln(-b*c*x-a*c+1)/a/(-a*c+1)+1/2*b*ln( -b*c*x-a*c+1)/a/x+1/2*b^2*ln(b*c*x/(-a*c+1))*ln(-b*c*x-a*c+1)/a^2+1/2*b^2* polylog(2,c*(b*x+a))/a^2-1/2*polylog(2,c*(b*x+a))/x^2+1/2*b^2*polylog(2,1- b*c*x/(-a*c+1))/a^2
Time = 0.13 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.76 \[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx=\frac {-\left ((-1+a c) \left (a^2-b^2 x^2\right ) \operatorname {PolyLog}(2,c (a+b x))\right )+b x \left (-a b c x \log (x)+\left (a (-1+a c+b c x)+b (-1+a c) x \log \left (\frac {b c x}{1-a c}\right )\right ) \log (1-a c-b c x)+b (-1+a c) x \operatorname {PolyLog}\left (2,\frac {-1+a c+b c x}{-1+a c}\right )\right )}{2 a^2 (-1+a c) x^2} \]
(-((-1 + a*c)*(a^2 - b^2*x^2)*PolyLog[2, c*(a + b*x)]) + b*x*(-(a*b*c*x*Lo g[x]) + (a*(-1 + a*c + b*c*x) + b*(-1 + a*c)*x*Log[(b*c*x)/(1 - a*c)])*Log [1 - a*c - b*c*x] + b*(-1 + a*c)*x*PolyLog[2, (-1 + a*c + b*c*x)/(-1 + a*c )]))/(2*a^2*(-1 + a*c)*x^2)
Time = 0.44 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.90, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {7152, 2863, 2009}
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.
\(\displaystyle \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx\) |
\(\Big \downarrow \) 7152 |
\(\displaystyle -\frac {1}{2} b \int \frac {\log (-a c-b x c+1)}{x^2 (a+b x)}dx-\frac {\operatorname {PolyLog}(2,c (a+b x))}{2 x^2}\) |
\(\Big \downarrow \) 2863 |
\(\displaystyle -\frac {1}{2} b \int \left (\frac {\log (-a c-b x c+1) b^2}{a^2 (a+b x)}-\frac {\log (-a c-b x c+1) b}{a^2 x}+\frac {\log (-a c-b x c+1)}{a x^2}\right )dx-\frac {\operatorname {PolyLog}(2,c (a+b x))}{2 x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{2} b \left (-\frac {b \operatorname {PolyLog}(2,c (a+b x))}{a^2}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {b c x}{1-a c}\right )}{a^2}-\frac {b \log \left (\frac {b c x}{1-a c}\right ) \log (-a c-b c x+1)}{a^2}-\frac {b c \log (x)}{a (1-a c)}+\frac {b c \log (-a c-b c x+1)}{a (1-a c)}-\frac {\log (-a c-b c x+1)}{a x}\right )-\frac {\operatorname {PolyLog}(2,c (a+b x))}{2 x^2}\) |
-1/2*PolyLog[2, c*(a + b*x)]/x^2 - (b*(-((b*c*Log[x])/(a*(1 - a*c))) + (b* c*Log[1 - a*c - b*c*x])/(a*(1 - a*c)) - Log[1 - a*c - b*c*x]/(a*x) - (b*Lo g[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/a^2 - (b*PolyLog[2, c*(a + b*x) ])/a^2 - (b*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/a^2))/2
3.2.29.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Int[((d_.) + (e_.)*(x_))^(m_.)*PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))], x_Sy mbol] :> Simp[(d + e*x)^(m + 1)*(PolyLog[2, c*(a + b*x)]/(e*(m + 1))), x] + Simp[b/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(Log[1 - a*c - b*c*x]/(a + b*x) ), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Time = 2.76 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.87
method | result | size |
parts | \(-\frac {\operatorname {polylog}\left (2, c \left (b x +a \right )\right )}{2 x^{2}}+\frac {b^{2} c \left (\frac {\operatorname {dilog}\left (-\frac {b c x}{a c -1}\right )+\ln \left (-b c x -a c +1\right ) \ln \left (-\frac {b c x}{a c -1}\right )}{a^{2} c}+\frac {\operatorname {dilog}\left (-b c x -a c +1\right )}{a^{2} c}+\frac {-\frac {\ln \left (-b c x \right )}{a c -1}-\frac {\ln \left (-b c x -a c +1\right ) \left (-b c x -a c +1\right )}{\left (a c -1\right ) b c x}}{a}\right )}{2}\) | \(150\) |
derivativedivides | \(b^{2} c^{2} \left (-\frac {\operatorname {polylog}\left (2, b c x +a c \right )}{2 b^{2} c^{2} x^{2}}+\frac {\operatorname {dilog}\left (-b c x -a c +1\right )}{2 a^{2} c^{2}}+\frac {-\frac {\ln \left (-b c x \right )}{a c -1}-\frac {\ln \left (-b c x -a c +1\right ) \left (-b c x -a c +1\right )}{\left (a c -1\right ) b c x}}{2 a c}+\frac {\operatorname {dilog}\left (-\frac {b c x}{a c -1}\right )+\ln \left (-b c x -a c +1\right ) \ln \left (-\frac {b c x}{a c -1}\right )}{2 a^{2} c^{2}}\right )\) | \(163\) |
default | \(b^{2} c^{2} \left (-\frac {\operatorname {polylog}\left (2, b c x +a c \right )}{2 b^{2} c^{2} x^{2}}+\frac {\operatorname {dilog}\left (-b c x -a c +1\right )}{2 a^{2} c^{2}}+\frac {-\frac {\ln \left (-b c x \right )}{a c -1}-\frac {\ln \left (-b c x -a c +1\right ) \left (-b c x -a c +1\right )}{\left (a c -1\right ) b c x}}{2 a c}+\frac {\operatorname {dilog}\left (-\frac {b c x}{a c -1}\right )+\ln \left (-b c x -a c +1\right ) \ln \left (-\frac {b c x}{a c -1}\right )}{2 a^{2} c^{2}}\right )\) | \(163\) |
-1/2*polylog(2,c*(b*x+a))/x^2+1/2*b^2*c*(1/a^2/c*(dilog(-b*c*x/(a*c-1))+ln (-b*c*x-a*c+1)*ln(-b*c*x/(a*c-1)))+1/a^2/c*dilog(-b*c*x-a*c+1)+1/a*(-1/(a* c-1)*ln(-b*c*x)-ln(-b*c*x-a*c+1)*(-b*c*x-a*c+1)/(a*c-1)/b/c/x))
\[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx=\int { \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x^{3}} \,d x } \]
\[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx=\int \frac {\operatorname {Li}_{2}\left (a c + b c x\right )}{x^{3}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.12 \[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx=-\frac {b^{2} c \log \left (x\right )}{2 \, {\left (a^{2} c - a\right )}} - \frac {{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )} b^{2}}{2 \, a^{2}} + \frac {{\left (\log \left (-b c x - a c + 1\right ) \log \left (-\frac {b c x + a c - 1}{a c - 1} + 1\right ) + {\rm Li}_2\left (\frac {b c x + a c - 1}{a c - 1}\right )\right )} b^{2}}{2 \, a^{2}} - \frac {{\left (a^{2} c - a\right )} {\rm Li}_2\left (b c x + a c\right ) - {\left (b^{2} c x^{2} + {\left (a b c - b\right )} x\right )} \log \left (-b c x - a c + 1\right )}{2 \, {\left (a^{2} c - a\right )} x^{2}} \]
-1/2*b^2*c*log(x)/(a^2*c - a) - 1/2*(log(b*c*x + a*c)*log(-b*c*x - a*c + 1 ) + dilog(-b*c*x - a*c + 1))*b^2/a^2 + 1/2*(log(-b*c*x - a*c + 1)*log(-(b* c*x + a*c - 1)/(a*c - 1) + 1) + dilog((b*c*x + a*c - 1)/(a*c - 1)))*b^2/a^ 2 - 1/2*((a^2*c - a)*dilog(b*c*x + a*c) - (b^2*c*x^2 + (a*b*c - b)*x)*log( -b*c*x - a*c + 1))/((a^2*c - a)*x^2)
\[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx=\int { \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx=\int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{x^3} \,d x \]