Integrand size = 19, antiderivative size = 200 \[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=-\frac {a+b \text {arctanh}(c x)}{d x}+\frac {b c \log (x)}{d}-\frac {a e \log (x)}{d^2}-\frac {e (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{d^2}+\frac {e (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d}+\frac {b e \operatorname {PolyLog}(2,-c x)}{2 d^2}-\frac {b e \operatorname {PolyLog}(2,c x)}{2 d^2}+\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 d^2}-\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d^2} \]
(-a-b*arctanh(c*x))/d/x+b*c*ln(x)/d-a*e*ln(x)/d^2-e*(a+b*arctanh(c*x))*ln( 2/(c*x+1))/d^2+e*(a+b*arctanh(c*x))*ln(2*c*(e*x+d)/(c*d+e)/(c*x+1))/d^2-1/ 2*b*c*ln(-c^2*x^2+1)/d+1/2*b*e*polylog(2,-c*x)/d^2-1/2*b*e*polylog(2,c*x)/ d^2+1/2*b*e*polylog(2,1-2/(c*x+1))/d^2-1/2*b*e*polylog(2,1-2*c*(e*x+d)/(c* d+e)/(c*x+1))/d^2
Result contains complex when optimal does not.
Time = 6.34 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.66 \[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=\frac {-\frac {2 a d^2}{x}-2 a d e \log (x)+2 a d e \log (d+e x)+\frac {b \left (-\frac {2 c d^2 \text {arctanh}(c x)}{x}+e^2 \text {arctanh}(c x)^2-\sqrt {1-\frac {c^2 d^2}{e^2}} e^2 e^{-\text {arctanh}\left (\frac {c d}{e}\right )} \text {arctanh}(c x)^2-c d e \text {arctanh}(c x) \left (\text {arctanh}(c x)+2 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )+c d e \text {arctanh}(c x) \left (i \pi +2 \text {arctanh}\left (\frac {c d}{e}\right )+2 \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )\right )+c^2 d^2 \left (2 \log (c x)-\log \left (1-c^2 x^2\right )\right )-\frac {1}{2} i c d e \pi \left (2 \log \left (1+e^{2 \text {arctanh}(c x)}\right )+\log \left (1-c^2 x^2\right )\right )+2 c d e \text {arctanh}\left (\frac {c d}{e}\right ) \left (\log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )-\log \left (i \sinh \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )\right )\right )+c d e \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )-c d e \operatorname {PolyLog}\left (2,e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )\right )}{c}}{2 d^3} \]
((-2*a*d^2)/x - 2*a*d*e*Log[x] + 2*a*d*e*Log[d + e*x] + (b*((-2*c*d^2*ArcT anh[c*x])/x + e^2*ArcTanh[c*x]^2 - (Sqrt[1 - (c^2*d^2)/e^2]*e^2*ArcTanh[c* x]^2)/E^ArcTanh[(c*d)/e] - c*d*e*ArcTanh[c*x]*(ArcTanh[c*x] + 2*Log[1 - E^ (-2*ArcTanh[c*x])]) + c*d*e*ArcTanh[c*x]*(I*Pi + 2*ArcTanh[(c*d)/e] + 2*Lo g[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]) + c^2*d^2*(2*Log[c*x] - L og[1 - c^2*x^2]) - (I/2)*c*d*e*Pi*(2*Log[1 + E^(2*ArcTanh[c*x])] + Log[1 - c^2*x^2]) + 2*c*d*e*ArcTanh[(c*d)/e]*(Log[1 - E^(-2*(ArcTanh[(c*d)/e] + A rcTanh[c*x]))] - Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) + c*d*e*Pol yLog[2, E^(-2*ArcTanh[c*x])] - c*d*e*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/c)/(2*d^3)
Time = 0.49 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6502, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {e^2 (a+b \text {arctanh}(c x))}{d^2 (d+e x)}-\frac {e (a+b \text {arctanh}(c x))}{d^2 x}+\frac {a+b \text {arctanh}(c x)}{d x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {e \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^2}+\frac {e (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d^2}-\frac {a+b \text {arctanh}(c x)}{d x}-\frac {a e \log (x)}{d^2}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d}+\frac {b e \operatorname {PolyLog}(2,-c x)}{2 d^2}-\frac {b e \operatorname {PolyLog}(2,c x)}{2 d^2}+\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 d^2}-\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 d^2}+\frac {b c \log (x)}{d}\) |
-((a + b*ArcTanh[c*x])/(d*x)) + (b*c*Log[x])/d - (a*e*Log[x])/d^2 - (e*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^2 + (e*(a + b*ArcTanh[c*x])*Log[(2*c *(d + e*x))/((c*d + e)*(1 + c*x))])/d^2 - (b*c*Log[1 - c^2*x^2])/(2*d) + ( b*e*PolyLog[2, -(c*x)])/(2*d^2) - (b*e*PolyLog[2, c*x])/(2*d^2) + (b*e*Pol yLog[2, 1 - 2/(1 + c*x)])/(2*d^2) - (b*e*PolyLog[2, 1 - (2*c*(d + e*x))/(( c*d + e)*(1 + c*x))])/(2*d^2)
3.2.52.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, ( f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
Time = 0.96 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.34
| method | result | size |
| parts | \(a \left (\frac {e \ln \left (e x +d \right )}{d^{2}}-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}\right )+b c \left (\frac {\operatorname {arctanh}\left (c x \right ) e \ln \left (e c x +c d \right )}{c \,d^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{d c x}-\frac {\operatorname {arctanh}\left (c x \right ) e \ln \left (c x \right )}{c \,d^{2}}-c \left (\frac {\frac {\ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right )}{2}-\ln \left (c x \right )}{d c}+\frac {e \left (-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}\right )}{d^{2} c^{2}}-\frac {-\frac {e \left (\operatorname {dilog}\left (\frac {e c x +e}{-c d +e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x +e}{-c d +e}\right )\right )}{2}+\frac {e \left (\operatorname {dilog}\left (\frac {e c x -e}{-c d -e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x -e}{-c d -e}\right )\right )}{2}}{d^{2} c^{2}}\right )\right )\) | \(267\) |
| derivativedivides | \(c \left (-\frac {a}{d c x}-\frac {a e \ln \left (c x \right )}{c \,d^{2}}+\frac {a e \ln \left (e c x +c d \right )}{c \,d^{2}}+b c \left (-\frac {\operatorname {arctanh}\left (c x \right )}{d \,c^{2} x}-\frac {\operatorname {arctanh}\left (c x \right ) e \ln \left (c x \right )}{d^{2} c^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) e \ln \left (e c x +c d \right )}{d^{2} c^{2}}-\frac {\frac {\ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right )}{2}-\ln \left (c x \right )}{d c}-\frac {e \left (-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}\right )}{d^{2} c^{2}}+\frac {-\frac {e \left (\operatorname {dilog}\left (\frac {e c x +e}{-c d +e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x +e}{-c d +e}\right )\right )}{2}+\frac {e \left (\operatorname {dilog}\left (\frac {e c x -e}{-c d -e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x -e}{-c d -e}\right )\right )}{2}}{d^{2} c^{2}}\right )\right )\) | \(280\) |
| default | \(c \left (-\frac {a}{d c x}-\frac {a e \ln \left (c x \right )}{c \,d^{2}}+\frac {a e \ln \left (e c x +c d \right )}{c \,d^{2}}+b c \left (-\frac {\operatorname {arctanh}\left (c x \right )}{d \,c^{2} x}-\frac {\operatorname {arctanh}\left (c x \right ) e \ln \left (c x \right )}{d^{2} c^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) e \ln \left (e c x +c d \right )}{d^{2} c^{2}}-\frac {\frac {\ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right )}{2}-\ln \left (c x \right )}{d c}-\frac {e \left (-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}\right )}{d^{2} c^{2}}+\frac {-\frac {e \left (\operatorname {dilog}\left (\frac {e c x +e}{-c d +e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x +e}{-c d +e}\right )\right )}{2}+\frac {e \left (\operatorname {dilog}\left (\frac {e c x -e}{-c d -e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x -e}{-c d -e}\right )\right )}{2}}{d^{2} c^{2}}\right )\right )\) | \(280\) |
| risch | \(\frac {c b \ln \left (-c x \right )}{2 d}-\frac {c b \ln \left (-c x +1\right )}{2 d}+\frac {b \ln \left (-c x +1\right )}{2 d x}-\frac {b e \operatorname {dilog}\left (-c x +1\right )}{2 d^{2}}-\frac {b e \operatorname {dilog}\left (\frac {e \left (-c x +1\right )-c d -e}{-c d -e}\right )}{2 d^{2}}-\frac {b e \ln \left (-c x +1\right ) \ln \left (\frac {e \left (-c x +1\right )-c d -e}{-c d -e}\right )}{2 d^{2}}-\frac {a}{d x}-\frac {a e \ln \left (-c x \right )}{d^{2}}+\frac {a e \ln \left (e \left (-c x +1\right )-c d -e \right )}{d^{2}}+\frac {b c \ln \left (c x \right )}{2 d}-\frac {b c \ln \left (c x +1\right )}{2 d}-\frac {b \ln \left (c x +1\right )}{2 d x}+\frac {b e \operatorname {dilog}\left (c x +1\right )}{2 d^{2}}+\frac {b e \operatorname {dilog}\left (\frac {e \left (c x +1\right )+c d -e}{c d -e}\right )}{2 d^{2}}+\frac {b e \ln \left (c x +1\right ) \ln \left (\frac {e \left (c x +1\right )+c d -e}{c d -e}\right )}{2 d^{2}}\) | \(301\) |
a*(e/d^2*ln(e*x+d)-1/d/x-e/d^2*ln(x))+b*c*(1/c*arctanh(c*x)*e/d^2*ln(c*e*x +c*d)-arctanh(c*x)/d/c/x-1/c*arctanh(c*x)*e/d^2*ln(c*x)-c*(1/d/c*(1/2*ln(c *x+1)+1/2*ln(c*x-1)-ln(c*x))+1/d^2/c^2*e*(-1/2*dilog(c*x+1)-1/2*ln(c*x)*ln (c*x+1)-1/2*dilog(c*x))-1/d^2/c^2*(-1/2*e*(dilog((c*e*x+e)/(-c*d+e))+ln(c* e*x+c*d)*ln((c*e*x+e)/(-c*d+e)))+1/2*e*(dilog((c*e*x-e)/(-c*d-e))+ln(c*e*x +c*d)*ln((c*e*x-e)/(-c*d-e))))))
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \]
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=\int \frac {a + b \operatorname {atanh}{\left (c x \right )}}{x^{2} \left (d + e x\right )}\, dx \]
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \]
a*(e*log(e*x + d)/d^2 - e*log(x)/d^2 - 1/(d*x)) + 1/2*b*integrate((log(c*x + 1) - log(-c*x + 1))/(e*x^3 + d*x^2), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x^2\,\left (d+e\,x\right )} \,d x \]