\(\int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x}} \, dx\) [56]

Optimal result

Integrand size = 16, antiderivative size = 186 \[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x}} \, dx=\frac {(1-a-b x) \log (1-a-b x)}{2 b c}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}-\frac {d \log (1+a+b x) \log \left (-\frac {b (d+c x)}{c+a c-b d}\right )}{2 c^2}+\frac {d \log (1-a-b x) \log \left (\frac {b (d+c x)}{c-a c+b d}\right )}{2 c^2}+\frac {d \operatorname {PolyLog}\left (2,\frac {c (1-a-b x)}{c-a c+b d}\right )}{2 c^2}-\frac {d \operatorname {PolyLog}\left (2,\frac {c (1+a+b x)}{c+a c-b d}\right )}{2 c^2} \] Output:

1/2*(-b*x-a+1)*ln(-b*x-a+1)/b/c+1/2*(b*x+a+1)*ln(b*x+a+1)/b/c-1/2*d*ln(b*x 
+a+1)*ln(-b*(c*x+d)/(a*c-b*d+c))/c^2+1/2*d*ln(-b*x-a+1)*ln(b*(c*x+d)/(-a*c 
+b*d+c))/c^2+1/2*d*polylog(2,c*(-b*x-a+1)/(-a*c+b*d+c))/c^2-1/2*d*polylog( 
2,c*(b*x+a+1)/(a*c-b*d+c))/c^2
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 8.77 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.12 \[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x}} \, dx=\frac {2 c (a+b x) \text {arctanh}(a+b x)+\frac {b c d \text {arctanh}(a+b x)^2}{a c-b d}-2 c \log \left (\frac {1}{\sqrt {1-(a+b x)^2}}\right )+\frac {b d \left (c \sqrt {1-a^2+\frac {2 a b d}{c}-\frac {b^2 d^2}{c^2}} e^{\text {arctanh}\left (a-\frac {b d}{c}\right )} \text {arctanh}(a+b x)^2+(a c-b d) \text {arctanh}(a+b x) \left (i \pi -2 \text {arctanh}\left (a-\frac {b d}{c}\right )+2 \log \left (1-e^{2 \left (\text {arctanh}\left (a-\frac {b d}{c}\right )-\text {arctanh}(a+b x)\right )}\right )\right )-(a c-b d) \left (i \pi \left (\log \left (1+e^{2 \text {arctanh}(a+b x)}\right )-\log \left (\frac {1}{\sqrt {1-(a+b x)^2}}\right )\right )+2 \text {arctanh}\left (a-\frac {b d}{c}\right ) \left (\log \left (1-e^{2 \left (\text {arctanh}\left (a-\frac {b d}{c}\right )-\text {arctanh}(a+b x)\right )}\right )-\log \left (-i \sinh \left (\text {arctanh}\left (a-\frac {b d}{c}\right )-\text {arctanh}(a+b x)\right )\right )\right )\right )+(-a c+b d) \operatorname {PolyLog}\left (2,e^{2 \left (\text {arctanh}\left (a-\frac {b d}{c}\right )-\text {arctanh}(a+b x)\right )}\right )\right )}{-a c+b d}+b d \left (\text {arctanh}(a+b x) \left (\text {arctanh}(a+b x)+2 \log \left (1+e^{-2 \text {arctanh}(a+b x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a+b x)}\right )\right )}{2 b c^2} \] Input:

Integrate[ArcTanh[a + b*x]/(c + d/x),x]
 

Output:

(2*c*(a + b*x)*ArcTanh[a + b*x] + (b*c*d*ArcTanh[a + b*x]^2)/(a*c - b*d) - 
 2*c*Log[1/Sqrt[1 - (a + b*x)^2]] + (b*d*(c*Sqrt[1 - a^2 + (2*a*b*d)/c - ( 
b^2*d^2)/c^2]*E^ArcTanh[a - (b*d)/c]*ArcTanh[a + b*x]^2 + (a*c - b*d)*ArcT 
anh[a + b*x]*(I*Pi - 2*ArcTanh[a - (b*d)/c] + 2*Log[1 - E^(2*(ArcTanh[a - 
(b*d)/c] - ArcTanh[a + b*x]))]) - (a*c - b*d)*(I*Pi*(Log[1 + E^(2*ArcTanh[ 
a + b*x])] - Log[1/Sqrt[1 - (a + b*x)^2]]) + 2*ArcTanh[a - (b*d)/c]*(Log[1 
 - E^(2*(ArcTanh[a - (b*d)/c] - ArcTanh[a + b*x]))] - Log[(-I)*Sinh[ArcTan 
h[a - (b*d)/c] - ArcTanh[a + b*x]]])) + (-(a*c) + b*d)*PolyLog[2, E^(2*(Ar 
cTanh[a - (b*d)/c] - ArcTanh[a + b*x]))]))/(-(a*c) + b*d) + b*d*(ArcTanh[a 
 + b*x]*(ArcTanh[a + b*x] + 2*Log[1 + E^(-2*ArcTanh[a + b*x])]) - PolyLog[ 
2, -E^(-2*ArcTanh[a + b*x])]))/(2*b*c^2)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6665, 2856, 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.

\(\displaystyle \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x}} \, dx\)

\(\Big \downarrow \) 6665

\(\displaystyle \frac {1}{2} \int \frac {\log (a+b x+1)}{c+\frac {d}{x}}dx-\frac {1}{2} \int \frac {\log (-a-b x+1)}{c+\frac {d}{x}}dx\)

\(\Big \downarrow \) 2856

\(\displaystyle \frac {1}{2} \int \left (\frac {\log (a+b x+1)}{c}-\frac {d \log (a+b x+1)}{c (d+c x)}\right )dx-\frac {1}{2} \int \left (\frac {\log (-a-b x+1)}{c}-\frac {d \log (-a-b x+1)}{c (d+c x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {d \operatorname {PolyLog}\left (2,\frac {c (-a-b x+1)}{-a c+c+b d}\right )}{c^2}+\frac {d \log (-a-b x+1) \log \left (\frac {b (c x+d)}{-a c+b d+c}\right )}{c^2}+\frac {(-a-b x+1) \log (-a-b x+1)}{b c}+\frac {x}{c}\right )+\frac {1}{2} \left (-\frac {d \operatorname {PolyLog}\left (2,\frac {c (a+b x+1)}{a c+c-b d}\right )}{c^2}-\frac {d \log (a+b x+1) \log \left (-\frac {b (c x+d)}{a c-b d+c}\right )}{c^2}+\frac {(a+b x+1) \log (a+b x+1)}{b c}-\frac {x}{c}\right )\)

Input:

Int[ArcTanh[a + b*x]/(c + d/x),x]
 

Output:

(x/c + ((1 - a - b*x)*Log[1 - a - b*x])/(b*c) + (d*Log[1 - a - b*x]*Log[(b 
*(d + c*x))/(c - a*c + b*d)])/c^2 + (d*PolyLog[2, (c*(1 - a - b*x))/(c - a 
*c + b*d)])/c^2)/2 + (-(x/c) + ((1 + a + b*x)*Log[1 + a + b*x])/(b*c) - (d 
*Log[1 + a + b*x]*Log[-((b*(d + c*x))/(c + a*c - b*d))])/c^2 - (d*PolyLog[ 
2, (c*(1 + a + b*x))/(c + a*c - b*d)])/c^2)/2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2856

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. 
)*(x_)^(r_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x) 
^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && I 
GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
 

rule 6665

Int[ArcTanh[(c_) + (d_.)*(x_)]/((e_) + (f_.)*(x_)^(n_.)), x_Symbol] :> Simp 
[1/2   Int[Log[1 + c + d*x]/(e + f*x^n), x], x] - Simp[1/2   Int[Log[1 - c 
- d*x]/(e + f*x^n), x], x] /; FreeQ[{c, d, e, f}, x] && RationalQ[n]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.39

method	result	size
parts	\(\frac {\operatorname {arctanh}\left (b x +a \right ) x}{c}-\frac {\operatorname {arctanh}\left (b x +a \right ) d \ln \left (c x +d \right )}{c^{2}}-\frac {b \left (\frac {\left (a -1\right ) \ln \left (a c -b d +b \left (c x +d \right )-c \right )}{2 b^{2}}+\frac {\left (-1-a \right ) \ln \left (a c -b d +b \left (c x +d \right )+c \right )}{2 b^{2}}+d \left (\frac {\frac {\operatorname {dilog}\left (\frac {a c -b d +b \left (c x +d \right )-c}{a c -b d -c}\right )}{b}+\frac {\ln \left (c x +d \right ) \ln \left (\frac {a c -b d +b \left (c x +d \right )-c}{a c -b d -c}\right )}{b}}{2 c}-\frac {\frac {\operatorname {dilog}\left (\frac {a c -b d +b \left (c x +d \right )+c}{a c -b d +c}\right )}{b}+\frac {\ln \left (c x +d \right ) \ln \left (\frac {a c -b d +b \left (c x +d \right )+c}{a c -b d +c}\right )}{b}}{2 c}\right )\right )}{c}\)	\(259\)
risch	\(\frac {\ln \left (b x +a +1\right ) x}{2 c}+\frac {\ln \left (b x +a +1\right ) a}{2 b c}+\frac {\ln \left (b x +a +1\right )}{2 b c}-\frac {1}{b c}-\frac {d \operatorname {dilog}\left (\frac {\left (b x +a +1\right ) c -a c +b d -c}{-a c +b d -c}\right )}{2 c^{2}}-\frac {d \ln \left (b x +a +1\right ) \ln \left (\frac {\left (b x +a +1\right ) c -a c +b d -c}{-a c +b d -c}\right )}{2 c^{2}}-\frac {\ln \left (-b x -a +1\right ) x}{2 c}-\frac {\ln \left (-b x -a +1\right ) a}{2 b c}+\frac {\ln \left (-b x -a +1\right )}{2 b c}+\frac {d \operatorname {dilog}\left (\frac {\left (-b x -a +1\right ) c +a c -b d -c}{a c -b d -c}\right )}{2 c^{2}}+\frac {d \ln \left (-b x -a +1\right ) \ln \left (\frac {\left (-b x -a +1\right ) c +a c -b d -c}{a c -b d -c}\right )}{2 c^{2}}\)	\(290\)
derivativedivides	\(\frac {\frac {\operatorname {arctanh}\left (b x +a \right ) \left (b x +a \right )}{c}-\frac {\operatorname {arctanh}\left (b x +a \right ) d b \ln \left (a c -b d -c \left (b x +a \right )\right )}{c^{2}}+\frac {b d \left (-\frac {\operatorname {dilog}\left (\frac {-c \left (b x +a \right )+c}{-a c +b d +c}\right )+\ln \left (a c -b d -c \left (b x +a \right )\right ) \ln \left (\frac {-c \left (b x +a \right )+c}{-a c +b d +c}\right )}{2 c}+\frac {\operatorname {dilog}\left (\frac {-c \left (b x +a \right )-c}{-a c +b d -c}\right )+\ln \left (a c -b d -c \left (b x +a \right )\right ) \ln \left (\frac {-c \left (b x +a \right )-c}{-a c +b d -c}\right )}{2 c}\right )+\frac {\ln \left (a^{2} c^{2}-2 a b c d +b^{2} d^{2}-2 a c \left (a c -b d -c \left (b x +a \right )\right )+2 b d \left (a c -b d -c \left (b x +a \right )\right )-c^{2}+\left (a c -b d -c \left (b x +a \right )\right )^{2}\right )}{2}}{c}}{b}\)	\(295\)
default	\(\frac {\frac {\operatorname {arctanh}\left (b x +a \right ) \left (b x +a \right )}{c}-\frac {\operatorname {arctanh}\left (b x +a \right ) d b \ln \left (a c -b d -c \left (b x +a \right )\right )}{c^{2}}+\frac {b d \left (-\frac {\operatorname {dilog}\left (\frac {-c \left (b x +a \right )+c}{-a c +b d +c}\right )+\ln \left (a c -b d -c \left (b x +a \right )\right ) \ln \left (\frac {-c \left (b x +a \right )+c}{-a c +b d +c}\right )}{2 c}+\frac {\operatorname {dilog}\left (\frac {-c \left (b x +a \right )-c}{-a c +b d -c}\right )+\ln \left (a c -b d -c \left (b x +a \right )\right ) \ln \left (\frac {-c \left (b x +a \right )-c}{-a c +b d -c}\right )}{2 c}\right )+\frac {\ln \left (a^{2} c^{2}-2 a b c d +b^{2} d^{2}-2 a c \left (a c -b d -c \left (b x +a \right )\right )+2 b d \left (a c -b d -c \left (b x +a \right )\right )-c^{2}+\left (a c -b d -c \left (b x +a \right )\right )^{2}\right )}{2}}{c}}{b}\)	\(295\)

Input:

int(arctanh(b*x+a)/(c+d/x),x,method=_RETURNVERBOSE)
 

Output:

arctanh(b*x+a)*x/c-arctanh(b*x+a)/c^2*d*ln(c*x+d)-b/c*(1/2*(a-1)/b^2*ln(a* 
c-b*d+b*(c*x+d)-c)+1/2*(-1-a)/b^2*ln(a*c-b*d+b*(c*x+d)+c)+d*(1/2/c*(dilog( 
(a*c-b*d+b*(c*x+d)-c)/(a*c-b*d-c))/b+ln(c*x+d)*ln((a*c-b*d+b*(c*x+d)-c)/(a 
*c-b*d-c))/b)-1/2/c*(dilog((a*c-b*d+b*(c*x+d)+c)/(a*c-b*d+c))/b+ln(c*x+d)* 
ln((a*c-b*d+b*(c*x+d)+c)/(a*c-b*d+c))/b)))
 

Fricas [F]

\[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x}} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )}{c + \frac {d}{x}} \,d x } \] Input:

integrate(arctanh(b*x+a)/(c+d/x),x, algorithm="fricas")
 

Output:

integral(x*arctanh(b*x + a)/(c*x + d), x)
 

Sympy [F]

\[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x}} \, dx=\int \frac {x \operatorname {atanh}{\left (a + b x \right )}}{c x + d}\, dx \] Input:

integrate(atanh(b*x+a)/(c+d/x),x)
 

Output:

Integral(x*atanh(a + b*x)/(c*x + d), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.03 \[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x}} \, dx=\frac {1}{2} \, b {\left (\frac {{\left (\log \left (c x + d\right ) \log \left (\frac {b c x + b d}{a c - b d + c} + 1\right ) + {\rm Li}_2\left (-\frac {b c x + b d}{a c - b d + c}\right )\right )} d}{b c^{2}} - \frac {{\left (\log \left (c x + d\right ) \log \left (\frac {b c x + b d}{a c - b d - c} + 1\right ) + {\rm Li}_2\left (-\frac {b c x + b d}{a c - b d - c}\right )\right )} d}{b c^{2}} + \frac {{\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{2} c} - \frac {{\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{2} c}\right )} + {\left (\frac {x}{c} - \frac {d \log \left (c x + d\right )}{c^{2}}\right )} \operatorname {artanh}\left (b x + a\right ) \] Input:

integrate(arctanh(b*x+a)/(c+d/x),x, algorithm="maxima")
 

Output:

1/2*b*((log(c*x + d)*log((b*c*x + b*d)/(a*c - b*d + c) + 1) + dilog(-(b*c* 
x + b*d)/(a*c - b*d + c)))*d/(b*c^2) - (log(c*x + d)*log((b*c*x + b*d)/(a* 
c - b*d - c) + 1) + dilog(-(b*c*x + b*d)/(a*c - b*d - c)))*d/(b*c^2) + (a 
+ 1)*log(b*x + a + 1)/(b^2*c) - (a - 1)*log(b*x + a - 1)/(b^2*c)) + (x/c - 
 d*log(c*x + d)/c^2)*arctanh(b*x + a)
 

Giac [F]

\[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x}} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )}{c + \frac {d}{x}} \,d x } \] Input:

integrate(arctanh(b*x+a)/(c+d/x),x, algorithm="giac")
 

Output:

integrate(arctanh(b*x + a)/(c + d/x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x}} \, dx=\int \frac {\mathrm {atanh}\left (a+b\,x\right )}{c+\frac {d}{x}} \,d x \] Input:

int(atanh(a + b*x)/(c + d/x),x)
 

Output:

int(atanh(a + b*x)/(c + d/x), x)
 

Reduce [F]

\[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x}} \, dx=\int \frac {\mathit {atanh} \left (b x +a \right ) x}{c x +d}d x \] Input:

int(atanh(b*x+a)/(c+d/x),x)
                                                                                    
                                                                                    
 

Output:

int((atanh(a + b*x)*x)/(c*x + d),x)