\(\int \frac {a+b \text {arctanh}(\frac {c}{x})}{x} \, dx\) [139]

Optimal result

Integrand size = 14, antiderivative size = 30 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{x} \, dx=a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {c}{x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {c}{x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{x} \, dx=a \log (x)+\frac {1}{2} b \left (\operatorname {PolyLog}\left (2,-\frac {c}{x}\right )-\operatorname {PolyLog}\left (2,\frac {c}{x}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6450, 6446}

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.

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

\(\Big \downarrow \) 6450

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

\(\Big \downarrow \) 6446

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 6446

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x 
] + (-Simp[(b/2)*PolyLog[2, (-c)*x], x] + Simp[(b/2)*PolyLog[2, c*x], x]) / 
; FreeQ[{a, b, c}, x]
 

rule 6450

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[ 
1/n   Subst[Int[(a + b*ArcTanh[c*x])^p/x, x], x, x^n], x] /; FreeQ[{a, b, c 
, n}, x] && IGtQ[p, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(26)=52\).

Time = 0.40 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.90

method	result	size
parts	\(a \ln \left (x \right )+b \left (-\ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )+\frac {\operatorname {dilog}\left (\frac {c}{x}\right )}{2}+\frac {\operatorname {dilog}\left (1+\frac {c}{x}\right )}{2}+\frac {\ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}\right )\)	\(57\)
derivativedivides	\(-a \ln \left (\frac {c}{x}\right )-b \left (\ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )-\frac {\operatorname {dilog}\left (\frac {c}{x}\right )}{2}-\frac {\operatorname {dilog}\left (1+\frac {c}{x}\right )}{2}-\frac {\ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}\right )\)	\(62\)
default	\(-a \ln \left (\frac {c}{x}\right )-b \left (\ln \left (\frac {c}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x}\right )-\frac {\operatorname {dilog}\left (\frac {c}{x}\right )}{2}-\frac {\operatorname {dilog}\left (1+\frac {c}{x}\right )}{2}-\frac {\ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}\right )\)	\(62\)
risch	\(\frac {b \ln \left (x \right ) \ln \left (x +c \right )}{2}-\frac {\left (-2 i b \pi \operatorname {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{2}-i b \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (c -x \right )\right ) \operatorname {csgn}\left (\frac {i \left (c -x \right )}{x}\right )+i b \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (x +c \right )\right ) \operatorname {csgn}\left (\frac {i \left (x +c \right )}{x}\right )+i b \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{2}-i b \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (x +c \right )}{x}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i \left (c -x \right )\right ) \operatorname {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i \left (x +c \right )\right ) \operatorname {csgn}\left (\frac {i \left (x +c \right )}{x}\right )^{2}+i b \pi \operatorname {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{3}+i b \pi \operatorname {csgn}\left (\frac {i \left (x +c \right )}{x}\right )^{3}+2 i b \pi -4 a \right ) \ln \left (-x \right )}{4}-\frac {\ln \left (\frac {x}{c}\right ) \ln \left (c -x \right ) b}{2}-\frac {\operatorname {dilog}\left (\frac {x}{c}\right ) b}{2}-\frac {\ln \left (x \right ) \ln \left (\frac {x +c}{c}\right ) b}{2}-\frac {\operatorname {dilog}\left (\frac {x +c}{c}\right ) b}{2}\)	\(293\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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