Integrand size = 17, antiderivative size = 156 \[ \int \frac {x (a+b \text {arctanh}(c x))}{d+e x} \, dx=\frac {a x}{e}+\frac {b x \text {arctanh}(c x)}{e}+\frac {d (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{e^2}-\frac {d (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^2}+\frac {b \log \left (1-c^2 x^2\right )}{2 c e}-\frac {b d \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 e^2}+\frac {b d \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^2} \]
a*x/e+b*x*arctanh(c*x)/e+d*(a+b*arctanh(c*x))*ln(2/(c*x+1))/e^2-d*(a+b*arc tanh(c*x))*ln(2*c*(e*x+d)/(c*d+e)/(c*x+1))/e^2+1/2*b*ln(-c^2*x^2+1)/c/e-1/ 2*b*d*polylog(2,1-2/(c*x+1))/e^2+1/2*b*d*polylog(2,1-2*c*(e*x+d)/(c*d+e)/( c*x+1))/e^2
Result contains complex when optimal does not.
Time = 1.91 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.02 \[ \int \frac {x (a+b \text {arctanh}(c x))}{d+e x} \, dx=\frac {2 a e x-2 a d \log (d+e x)+\frac {b \left (-i c d \pi \text {arctanh}(c x)+2 c e x \text {arctanh}(c x)-2 c d \text {arctanh}\left (\frac {c d}{e}\right ) \text {arctanh}(c x)+c d \text {arctanh}(c x)^2-e \text {arctanh}(c x)^2+\sqrt {1-\frac {c^2 d^2}{e^2}} e e^{-\text {arctanh}\left (\frac {c d}{e}\right )} \text {arctanh}(c x)^2+2 c d \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+i c d \pi \log \left (1+e^{2 \text {arctanh}(c x)}\right )-2 c d \text {arctanh}\left (\frac {c d}{e}\right ) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )-2 c d \text {arctanh}(c x) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )+e \log \left (1-c^2 x^2\right )+\frac {1}{2} i c d \pi \log \left (1-c^2 x^2\right )+2 c d \text {arctanh}\left (\frac {c d}{e}\right ) \log \left (i \sinh \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )\right )-c d \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+c d \operatorname {PolyLog}\left (2,e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )\right )}{c}}{2 e^2} \]
(2*a*e*x - 2*a*d*Log[d + e*x] + (b*((-I)*c*d*Pi*ArcTanh[c*x] + 2*c*e*x*Arc Tanh[c*x] - 2*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x] + c*d*ArcTanh[c*x]^2 - e*A rcTanh[c*x]^2 + (Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^2)/E^ArcTanh[(c*d) /e] + 2*c*d*ArcTanh[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]))] + e*Log[1 - c^2*x^2] + (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*Po lyLog[2, -E^(-2*ArcTanh[c*x])] + c*d*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/c)/(2*e^2)
Time = 0.38 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.
| \(\displaystyle \int \frac {x (a+b \text {arctanh}(c x))}{d+e x} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {a+b \text {arctanh}(c x)}{e}-\frac {d (a+b \text {arctanh}(c x))}{e (d+e x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{e^2}-\frac {d (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^2}+\frac {a x}{e}+\frac {b x \text {arctanh}(c x)}{e}+\frac {b \log \left (1-c^2 x^2\right )}{2 c e}-\frac {b d \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 e^2}+\frac {b d \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e^2}\) |
(a*x)/e + (b*x*ArcTanh[c*x])/e + (d*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)]) /e^2 - (d*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))]) /e^2 + (b*Log[1 - c^2*x^2])/(2*c*e) - (b*d*PolyLog[2, 1 - 2/(1 + c*x)])/(2 *e^2) + (b*d*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e^2 )
3.2.49.3.1 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 = 1.00 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.37
| method | result | size |
| parts | \(\frac {a x}{e}-\frac {a d \ln \left (e x +d \right )}{e^{2}}+\frac {b \left (\frac {c^{2} \operatorname {arctanh}\left (c x \right ) x}{e}-\frac {c^{2} \operatorname {arctanh}\left (c x \right ) d \ln \left (e c x +c d \right )}{e^{2}}-\frac {c \left (-\frac {\ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )-e^{2}+\left (e c x +c d \right )^{2}\right )}{2}+c d \left (-\frac {\operatorname {dilog}\left (\frac {e c x +e}{-c d +e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x +e}{-c d +e}\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {e c x -e}{-c d -e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x -e}{-c d -e}\right )}{2 e}\right )\right )}{e}\right )}{c^{2}}\) | \(214\) |
| derivativedivides | \(\frac {\frac {a \,c^{2} x}{e}-\frac {a \,c^{2} d \ln \left (e c x +c d \right )}{e^{2}}+b c \left (\frac {\operatorname {arctanh}\left (c x \right ) c x}{e}-\frac {\operatorname {arctanh}\left (c x \right ) c d \ln \left (e c x +c d \right )}{e^{2}}-\frac {-\frac {\ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )-e^{2}+\left (e c x +c d \right )^{2}\right )}{2}+c d \left (-\frac {\operatorname {dilog}\left (\frac {e c x +e}{-c d +e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x +e}{-c d +e}\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {e c x -e}{-c d -e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x -e}{-c d -e}\right )}{2 e}\right )}{e}\right )}{c^{2}}\) | \(220\) |
| default | \(\frac {\frac {a \,c^{2} x}{e}-\frac {a \,c^{2} d \ln \left (e c x +c d \right )}{e^{2}}+b c \left (\frac {\operatorname {arctanh}\left (c x \right ) c x}{e}-\frac {\operatorname {arctanh}\left (c x \right ) c d \ln \left (e c x +c d \right )}{e^{2}}-\frac {-\frac {\ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )-e^{2}+\left (e c x +c d \right )^{2}\right )}{2}+c d \left (-\frac {\operatorname {dilog}\left (\frac {e c x +e}{-c d +e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x +e}{-c d +e}\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {e c x -e}{-c d -e}\right )+\ln \left (e c x +c d \right ) \ln \left (\frac {e c x -e}{-c d -e}\right )}{2 e}\right )}{e}\right )}{c^{2}}\) | \(220\) |
| risch | \(-\frac {b \ln \left (-c x +1\right ) x}{2 e}+\frac {b \ln \left (-c x +1\right )}{2 c e}-\frac {b}{c e}+\frac {b d \operatorname {dilog}\left (\frac {e \left (-c x +1\right )-c d -e}{-c d -e}\right )}{2 e^{2}}+\frac {b d \ln \left (-c x +1\right ) \ln \left (\frac {e \left (-c x +1\right )-c d -e}{-c d -e}\right )}{2 e^{2}}+\frac {a x}{e}-\frac {a}{c e}-\frac {a d \ln \left (e \left (-c x +1\right )-c d -e \right )}{e^{2}}+\frac {b \ln \left (c x +1\right ) x}{2 e}+\frac {b \ln \left (c x +1\right )}{2 c e}-\frac {b d \operatorname {dilog}\left (\frac {e \left (c x +1\right )+c d -e}{c d -e}\right )}{2 e^{2}}-\frac {b d \ln \left (c x +1\right ) \ln \left (\frac {e \left (c x +1\right )+c d -e}{c d -e}\right )}{2 e^{2}}\) | \(255\) |
a*x/e-a*d/e^2*ln(e*x+d)+b/c^2*(c^2*arctanh(c*x)*x/e-c^2*arctanh(c*x)/e^2*d *ln(c*e*x+c*d)-c/e*(-1/2*ln(c^2*d^2-2*c*d*(c*e*x+c*d)-e^2+(c*e*x+c*d)^2)+c *d*(-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))))) )
\[ \int \frac {x (a+b \text {arctanh}(c x))}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x}{e x + d} \,d x } \]
\[ \int \frac {x (a+b \text {arctanh}(c x))}{d+e x} \, dx=\int \frac {x \left (a + b \operatorname {atanh}{\left (c x \right )}\right )}{d + e x}\, dx \]
\[ \int \frac {x (a+b \text {arctanh}(c x))}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x}{e x + d} \,d x } \]
a*(x/e - d*log(e*x + d)/e^2) + 1/2*b*integrate(x*(log(c*x + 1) - log(-c*x + 1))/(e*x + d), x)
\[ \int \frac {x (a+b \text {arctanh}(c x))}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x (a+b \text {arctanh}(c x))}{d+e x} \, dx=\int \frac {x\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{d+e\,x} \,d x \]