\(\int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)^2} \, dx\) [55]

Optimal result

Integrand size = 20, antiderivative size = 124 \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)^2} \, dx=\frac {b}{2 d^2 (1+c x)}-\frac {b \text {arctanh}(c x)}{2 d^2}+\frac {a+b \text {arctanh}(c x)}{d^2 (1+c x)}+\frac {a \log (x)}{d^2}+\frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b \operatorname {PolyLog}(2,-c x)}{2 d^2}+\frac {b \operatorname {PolyLog}(2,c x)}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.81 \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)^2} \, dx=\frac {\frac {4 a}{1+c x}+4 a \log (x)-4 a \log (1+c x)+b \left (\cosh (2 \text {arctanh}(c x))-2 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+2 \text {arctanh}(c x) \left (\cosh (2 \text {arctanh}(c x))+2 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-\sinh (2 \text {arctanh}(c x))\right )-\sinh (2 \text {arctanh}(c x))\right )}{4 d^2} \] Input:

Integrate[(a + b*ArcTanh[c*x])/(x*(d + c*d*x)^2),x]
 

Output:

((4*a)/(1 + c*x) + 4*a*Log[x] - 4*a*Log[1 + c*x] + b*(Cosh[2*ArcTanh[c*x]] 
 - 2*PolyLog[2, E^(-2*ArcTanh[c*x])] + 2*ArcTanh[c*x]*(Cosh[2*ArcTanh[c*x] 
] + 2*Log[1 - E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]]) - Sinh[2*ArcTan 
h[c*x]]))/(4*d^2)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.

\(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{x (c d x+d)^2} \, dx\)

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*ArcTanh[c*x])/(x*(d + c*d*x)^2),x]
 

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 6502

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])
 

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.28

method	result	size
parts	\(\frac {a \left (\frac {1}{c x +1}-\ln \left (c x +1\right )+\ln \left (x \right )\right )}{d^{2}}+\frac {b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )+\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}-\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {\ln \left (c x +1\right )^{2}}{4}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (c x -1\right )}{4}+\frac {1}{2 c x +2}-\frac {\ln \left (c x +1\right )}{4}-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )}{d^{2}}\)	\(159\)
derivativedivides	\(\frac {a \left (\ln \left (c x \right )+\frac {1}{c x +1}-\ln \left (c x +1\right )\right )}{d^{2}}+\frac {b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )+\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}-\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {\ln \left (c x +1\right )^{2}}{4}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (c x -1\right )}{4}+\frac {1}{2 c x +2}-\frac {\ln \left (c x +1\right )}{4}-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )}{d^{2}}\)	\(161\)
default	\(\frac {a \left (\ln \left (c x \right )+\frac {1}{c x +1}-\ln \left (c x +1\right )\right )}{d^{2}}+\frac {b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )+\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}-\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {\ln \left (c x +1\right )^{2}}{4}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (c x -1\right )}{4}+\frac {1}{2 c x +2}-\frac {\ln \left (c x +1\right )}{4}-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )}{d^{2}}\)	\(161\)
risch	\(\frac {a \ln \left (-c x \right )}{d^{2}}-\frac {a}{d^{2} \left (-c x -1\right )}-\frac {a \ln \left (-c x -1\right )}{d^{2}}-\frac {b \ln \left (-c x -1\right )}{4 d^{2}}-\frac {b \ln \left (-c x +1\right ) c x}{4 d^{2} \left (-c x -1\right )}+\frac {b \ln \left (-c x +1\right )}{4 d^{2} \left (-c x -1\right )}+\frac {b \operatorname {dilog}\left (-c x +1\right )}{2 d^{2}}-\frac {b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}+\frac {b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{2 d^{2}}-\frac {b \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}-\frac {b \ln \left (c x +1\right )^{2}}{4 d^{2}}-\frac {b \operatorname {dilog}\left (c x +1\right )}{2 d^{2}}+\frac {b \ln \left (c x +1\right )}{2 d^{2} \left (c x +1\right )}+\frac {b}{2 d^{2} \left (c x +1\right )}\)	\(220\)

Input:

int((a+b*arctanh(c*x))/x/(c*d*x+d)^2,x,method=_RETURNVERBOSE)
 

Output:

a/d^2*(1/(c*x+1)-ln(c*x+1)+ln(x))+b/d^2*(arctanh(c*x)*ln(c*x)+1/(c*x+1)*ar 
ctanh(c*x)-arctanh(c*x)*ln(c*x+1)+1/4*ln(c*x+1)^2-1/2*(ln(c*x+1)-ln(1/2*c* 
x+1/2))*ln(-1/2*c*x+1/2)+1/2*dilog(1/2*c*x+1/2)+1/4*ln(c*x-1)+1/2/(c*x+1)- 
1/4*ln(c*x+1)-1/2*dilog(c*x)-1/2*dilog(c*x+1)-1/2*ln(c*x)*ln(c*x+1))
 

Fricas [F]

\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)^2} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )}^{2} x} \,d x } \] Input:

integrate((a+b*arctanh(c*x))/x/(c*d*x+d)^2,x, algorithm="fricas")
 

Output:

integral((b*arctanh(c*x) + a)/(c^2*d^2*x^3 + 2*c*d^2*x^2 + d^2*x), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)^2} \, dx=\frac {\int \frac {a}{c^{2} x^{3} + 2 c x^{2} + x}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{3} + 2 c x^{2} + x}\, dx}{d^{2}} \] Input:

integrate((a+b*atanh(c*x))/x/(c*d*x+d)**2,x)
 

Output:

(Integral(a/(c**2*x**3 + 2*c*x**2 + x), x) + Integral(b*atanh(c*x)/(c**2*x 
**3 + 2*c*x**2 + x), x))/d**2
 

Maxima [F]

\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)^2} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )}^{2} x} \,d x } \] Input:

integrate((a+b*arctanh(c*x))/x/(c*d*x+d)^2,x, algorithm="maxima")
 

Output:

a*(1/(c*d^2*x + d^2) - log(c*x + 1)/d^2 + log(x)/d^2) + 1/2*b*integrate((l 
og(c*x + 1) - log(-c*x + 1))/(c^2*d^2*x^3 + 2*c*d^2*x^2 + d^2*x), x)
 

Giac [F]

\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)^2} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )}^{2} x} \,d x } \] Input:

integrate((a+b*arctanh(c*x))/x/(c*d*x+d)^2,x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x) + a)/((c*d*x + d)^2*x), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)^2} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x\,{\left (d+c\,d\,x\right )}^2} \,d x \] Input:

int((a + b*atanh(c*x))/(x*(d + c*d*x)^2),x)
 

Output:

int((a + b*atanh(c*x))/(x*(d + c*d*x)^2), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)^2} \, dx=\frac {-2 \mathit {atanh} \left (c x \right )^{2} b c x -2 \mathit {atanh} \left (c x \right )^{2} b -4 \mathit {atanh} \left (c x \right ) b c x -8 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{3} x^{4}+c^{2} x^{3}-c \,x^{2}-x}d x \right ) b c x -8 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{3} x^{4}+c^{2} x^{3}-c \,x^{2}-x}d x \right ) b -\mathrm {log}\left (c x -1\right ) b c x -\mathrm {log}\left (c x -1\right ) b -8 \,\mathrm {log}\left (c x +1\right ) a c x -8 \,\mathrm {log}\left (c x +1\right ) a +\mathrm {log}\left (c x +1\right ) b c x +\mathrm {log}\left (c x +1\right ) b +8 \,\mathrm {log}\left (x \right ) a c x +8 \,\mathrm {log}\left (x \right ) a -8 a c x -2 b c x}{8 d^{2} \left (c x +1\right )} \] Input:

int((a+b*atanh(c*x))/x/(c*d*x+d)^2,x)
 

Output:

( - 2*atanh(c*x)**2*b*c*x - 2*atanh(c*x)**2*b - 4*atanh(c*x)*b*c*x - 8*int 
(atanh(c*x)/(c**3*x**4 + c**2*x**3 - c*x**2 - x),x)*b*c*x - 8*int(atanh(c* 
x)/(c**3*x**4 + c**2*x**3 - c*x**2 - x),x)*b - log(c*x - 1)*b*c*x - log(c* 
x - 1)*b - 8*log(c*x + 1)*a*c*x - 8*log(c*x + 1)*a + log(c*x + 1)*b*c*x + 
log(c*x + 1)*b + 8*log(x)*a*c*x + 8*log(x)*a - 8*a*c*x - 2*b*c*x)/(8*d**2* 
(c*x + 1))