\(\int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)} \, dx\) [47]

Optimal result

Integrand size = 20, antiderivative size = 46 \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)} \, dx=\frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {b \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.20 \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+c d x)} \, dx=\frac {2 b \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )+2 a \log (x)-2 a \log (1+c x)-b \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )}{2 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 46, 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 = {6494, 2897}

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.

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 2897

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

rule 6494

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x 
_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - 
Simp[b*c*(p/d)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))] 
/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c 
^2*d^2 - e^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(114\) vs. \(2(44)=88\).

Time = 0.23 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.50

method	result	size
parts	\(\frac {a \left (-\ln \left (c x +1\right )+\ln \left (x \right )\right )}{d}+\frac {b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {\ln \left (c x +1\right )^{2}}{4}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )}{d}\)	\(115\)
derivativedivides	\(\frac {a \left (\ln \left (c x \right )-\ln \left (c x +1\right )\right )}{d}+\frac {b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {\ln \left (c x +1\right )^{2}}{4}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )}{d}\)	\(117\)
default	\(\frac {a \left (\ln \left (c x \right )-\ln \left (c x +1\right )\right )}{d}+\frac {b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {\ln \left (c x +1\right )^{2}}{4}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )}{d}\)	\(117\)
risch	\(-\frac {\ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) b}{2 d}+\frac {\ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right ) b}{2 d}-\frac {a \ln \left (-c x -1\right )}{d}+\frac {a \ln \left (-c x \right )}{d}+\frac {\operatorname {dilog}\left (-c x +1\right ) b}{2 d}-\frac {\operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right ) b}{2 d}-\frac {b \ln \left (c x +1\right )^{2}}{4 d}-\frac {b \operatorname {dilog}\left (c x +1\right )}{2 d}\)	\(117\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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