Integrand size = 19, antiderivative size = 148 \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\frac {a \log (x)}{d}+\frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{d}-\frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d}-\frac {b \operatorname {PolyLog}(2,-c x)}{2 d}+\frac {b \operatorname {PolyLog}(2,c x)}{2 d}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 d}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d} \]
a*ln(x)/d+(a+b*arctanh(c*x))*ln(2/(c*x+1))/d-(a+b*arctanh(c*x))*ln(2*c*(e* x+d)/(c*d+e)/(c*x+1))/d-1/2*b*polylog(2,-c*x)/d+1/2*b*polylog(2,c*x)/d-1/2 *b*polylog(2,1-2/(c*x+1))/d+1/2*b*polylog(2,1-2*c*(e*x+d)/(c*d+e)/(c*x+1)) /d
Result contains complex when optimal does not.
Time = 1.27 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.99 \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\frac {2 a d \log (x)-2 a d \log (d+e x)+\frac {b \left (-i c d \pi \text {arctanh}(c x)-2 c d \text {arctanh}\left (\frac {c d}{e}\right ) \text {arctanh}(c x)+c d \text {arctanh}(c x)^2-e \text {arctanh}(c x)^2+\sqrt {1-\frac {c^2 d^2}{e^2}} e e^{-\text {arctanh}\left (\frac {c d}{e}\right )} \text {arctanh}(c x)^2+2 c d \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )+i c d \pi \log \left (1+e^{2 \text {arctanh}(c x)}\right )-2 c d \text {arctanh}\left (\frac {c d}{e}\right ) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )-2 c d \text {arctanh}(c x) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )+\frac {1}{2} i c d \pi \log \left (1-c^2 x^2\right )+2 c d \text {arctanh}\left (\frac {c d}{e}\right ) \log \left (i \sinh \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )\right )-c d \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+c d \operatorname {PolyLog}\left (2,e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )\right )}{c}}{2 d^2} \]
(2*a*d*Log[x] - 2*a*d*Log[d + e*x] + (b*((-I)*c*d*Pi*ArcTanh[c*x] - 2*c*d* ArcTanh[(c*d)/e]*ArcTanh[c*x] + c*d*ArcTanh[c*x]^2 - e*ArcTanh[c*x]^2 + (S qrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^2)/E^ArcTanh[(c*d)/e] + 2*c*d*ArcTan h[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] + I*c*d*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 2*c*d*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x])) ] - 2*c*d*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + (I/2)*c*d*Pi*Log[1 - c^2*x^2] + 2*c*d*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh [(c*d)/e] + ArcTanh[c*x]]] - c*d*PolyLog[2, E^(-2*ArcTanh[c*x])] + c*d*Pol yLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/c)/(2*d^2)
Time = 0.42 (sec) , antiderivative size = 148, 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.
\(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx\) |
\(\Big \downarrow \) 6502 |
\(\displaystyle \int \left (\frac {a+b \text {arctanh}(c x)}{d x}-\frac {e (a+b \text {arctanh}(c x))}{d (d+e x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d}+\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d}+\frac {a \log (x)}{d}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 d}-\frac {b \operatorname {PolyLog}(2,-c x)}{2 d}+\frac {b \operatorname {PolyLog}(2,c x)}{2 d}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 d}\) |
(a*Log[x])/d + ((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d - ((a + b*ArcTanh [c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d - (b*PolyLog[2, -(c*x )])/(2*d) + (b*PolyLog[2, c*x])/(2*d) - (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2 *d) + (b*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*d)
3.2.51.3.1 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 = 0.97 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.37
method | result | size |
risch | \(\frac {\operatorname {dilog}\left (-c x +1\right ) b}{2 d}+\frac {b \operatorname {dilog}\left (\frac {e \left (-c x +1\right )-c d -e}{-c d -e}\right )}{2 d}+\frac {b \ln \left (-c x +1\right ) \ln \left (\frac {e \left (-c x +1\right )-c d -e}{-c d -e}\right )}{2 d}+\frac {\ln \left (-c x \right ) a}{d}-\frac {a \ln \left (e \left (-c x +1\right )-c d -e \right )}{d}-\frac {b \operatorname {dilog}\left (c x +1\right )}{2 d}-\frac {b \operatorname {dilog}\left (\frac {e \left (c x +1\right )+c d -e}{c d -e}\right )}{2 d}-\frac {b \ln \left (c x +1\right ) \ln \left (\frac {e \left (c x +1\right )+c d -e}{c d -e}\right )}{2 d}\) | \(203\) |
parts | \(-\frac {a \ln \left (e x +d \right )}{d}+\frac {a \ln \left (x \right )}{d}+b \left (-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (e c x +c d \right )}{d}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{d}-c \left (\frac {-\frac {e \left (\operatorname {dilog}\left (\frac {e c x +e}{-c d +e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x +e}{-c d +e}\right )\right )}{2}+\frac {e \left (\operatorname {dilog}\left (\frac {e c x -e}{-c d -e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x -e}{-c d -e}\right )\right )}{2}}{d c e}-\frac {-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}}{d c}\right )\right )\) | \(204\) |
derivativedivides | \(\frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (e c x +c d \right )}{d}+b c \left (\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{d c}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (e c x +c d \right )}{d c}-\frac {-\frac {e \left (\operatorname {dilog}\left (\frac {e c x +e}{-c d +e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x +e}{-c d +e}\right )\right )}{2}+\frac {e \left (\operatorname {dilog}\left (\frac {e c x -e}{-c d -e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x -e}{-c d -e}\right )\right )}{2}}{d c e}+\frac {-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}}{d c}\right )\) | \(212\) |
default | \(\frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (e c x +c d \right )}{d}+b c \left (\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{d c}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (e c x +c d \right )}{d c}-\frac {-\frac {e \left (\operatorname {dilog}\left (\frac {e c x +e}{-c d +e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x +e}{-c d +e}\right )\right )}{2}+\frac {e \left (\operatorname {dilog}\left (\frac {e c x -e}{-c d -e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x -e}{-c d -e}\right )\right )}{2}}{d c e}+\frac {-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}}{d c}\right )\) | \(212\) |
1/2/d*dilog(-c*x+1)*b+1/2*b/d*dilog((e*(-c*x+1)-c*d-e)/(-c*d-e))+1/2*b/d*l n(-c*x+1)*ln((e*(-c*x+1)-c*d-e)/(-c*d-e))+1/d*ln(-c*x)*a-a/d*ln(e*(-c*x+1) -c*d-e)-1/2*b/d*dilog(c*x+1)-1/2*b/d*dilog((e*(c*x+1)+c*d-e)/(c*d-e))-1/2* b/d*ln(c*x+1)*ln((e*(c*x+1)+c*d-e)/(c*d-e))
\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x} \,d x } \]
\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\int \frac {a + b \operatorname {atanh}{\left (c x \right )}}{x \left (d + e x\right )}\, dx \]
\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x} \,d x } \]
-a*(log(e*x + d)/d - log(x)/d) + 1/2*b*integrate((log(c*x + 1) - log(-c*x + 1))/(e*x^2 + d*x), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x\,\left (d+e\,x\right )} \,d x \]