\(\int \frac {a+b \text {arctanh}(\frac {c}{x^2})}{x} \, dx\) [161]

Optimal result

Integrand size = 14, antiderivative size = 30 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x} \, dx=a \log (x)+\frac {1}{4} b \operatorname {PolyLog}\left (2,-\frac {c}{x^2}\right )-\frac {1}{4} b \operatorname {PolyLog}\left (2,\frac {c}{x^2}\right ) \] Output:

a*ln(x)+1/4*b*polylog(2,-c/x^2)-1/4*b*polylog(2,c/x^2)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x} \, dx=a \log (x)+\frac {1}{4} b \left (\operatorname {PolyLog}\left (2,-\frac {c}{x^2}\right )-\operatorname {PolyLog}\left (2,\frac {c}{x^2}\right )\right ) \] Input:

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

Output:

a*Log[x] + (b*(PolyLog[2, -(c/x^2)] - PolyLog[2, c/x^2]))/4
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23, 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.

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

\(\Big \downarrow \) 6450

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

\(\Big \downarrow \) 6446

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

Input:

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

Output:

(-(a*Log[x^(-2)]) + (b*PolyLog[2, -(c/x^2)])/2 - (b*PolyLog[2, c/x^2])/2)/ 
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. \(148\) vs. \(2(26)=52\).

Time = 0.37 (sec) , antiderivative size = 149, normalized size of antiderivative = 4.97

method	result	size
parts	\(a \ln \left (x \right )+b \left (-\ln \left (\frac {1}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x^{2}}\right )+2 c \left (-\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1-\frac {\sqrt {c}}{x}\right )+\ln \left (1+\frac {\sqrt {c}}{x}\right )\right )}{4 c}-\frac {\operatorname {dilog}\left (1-\frac {\sqrt {c}}{x}\right )+\operatorname {dilog}\left (1+\frac {\sqrt {c}}{x}\right )}{4 c}+\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1+\frac {\sqrt {-c}}{x}\right )+\ln \left (1-\frac {\sqrt {-c}}{x}\right )\right )}{4 c}+\frac {\operatorname {dilog}\left (1+\frac {\sqrt {-c}}{x}\right )+\operatorname {dilog}\left (1-\frac {\sqrt {-c}}{x}\right )}{4 c}\right )\right )\)	\(149\)
derivativedivides	\(-a \ln \left (\frac {1}{x}\right )-b \left (\ln \left (\frac {1}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x^{2}}\right )-2 c \left (-\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1-\frac {\sqrt {c}}{x}\right )+\ln \left (1+\frac {\sqrt {c}}{x}\right )\right )}{4 c}-\frac {\operatorname {dilog}\left (1-\frac {\sqrt {c}}{x}\right )+\operatorname {dilog}\left (1+\frac {\sqrt {c}}{x}\right )}{4 c}+\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1+\frac {\sqrt {-c}}{x}\right )+\ln \left (1-\frac {\sqrt {-c}}{x}\right )\right )}{4 c}+\frac {\operatorname {dilog}\left (1+\frac {\sqrt {-c}}{x}\right )+\operatorname {dilog}\left (1-\frac {\sqrt {-c}}{x}\right )}{4 c}\right )\right )\)	\(152\)
default	\(-a \ln \left (\frac {1}{x}\right )-b \left (\ln \left (\frac {1}{x}\right ) \operatorname {arctanh}\left (\frac {c}{x^{2}}\right )-2 c \left (-\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1-\frac {\sqrt {c}}{x}\right )+\ln \left (1+\frac {\sqrt {c}}{x}\right )\right )}{4 c}-\frac {\operatorname {dilog}\left (1-\frac {\sqrt {c}}{x}\right )+\operatorname {dilog}\left (1+\frac {\sqrt {c}}{x}\right )}{4 c}+\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1+\frac {\sqrt {-c}}{x}\right )+\ln \left (1-\frac {\sqrt {-c}}{x}\right )\right )}{4 c}+\frac {\operatorname {dilog}\left (1+\frac {\sqrt {-c}}{x}\right )+\operatorname {dilog}\left (1-\frac {\sqrt {-c}}{x}\right )}{4 c}\right )\right )\)	\(152\)
risch	\(\frac {b \ln \left (x \right ) \ln \left (x^{2}+c \right )}{2}+\frac {\left (4 a -2 i b \pi -i b \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{3}+2 i b \pi {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}+i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )-i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )+i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}-i b \pi \,\operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}+i b \pi \,\operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}-i b \pi {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{3}-i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}\right ) \ln \left (x \right )}{4}+\frac {\ln \left (x \right ) \ln \left (\frac {\sqrt {c}+x}{\sqrt {c}}\right ) b}{2}-\frac {b \ln \left (x \right ) \ln \left (-x^{2}+c \right )}{2}+\frac {\ln \left (x \right ) \ln \left (\frac {\sqrt {c}-x}{\sqrt {c}}\right ) b}{2}+\frac {\operatorname {dilog}\left (\frac {\sqrt {c}-x}{\sqrt {c}}\right ) b}{2}+\frac {\operatorname {dilog}\left (\frac {\sqrt {c}+x}{\sqrt {c}}\right ) b}{2}-\frac {b \ln \left (x \right ) \ln \left (\frac {\sqrt {-c}-x}{\sqrt {-c}}\right )}{2}-\frac {b \ln \left (x \right ) \ln \left (\frac {\sqrt {-c}+x}{\sqrt {-c}}\right )}{2}-\frac {b \operatorname {dilog}\left (\frac {\sqrt {-c}-x}{\sqrt {-c}}\right )}{2}-\frac {b \operatorname {dilog}\left (\frac {\sqrt {-c}+x}{\sqrt {-c}}\right )}{2}\)	\(420\)

Input:

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

Output:

a*ln(x)+b*(-ln(1/x)*arctanh(c/x^2)+2*c*(-1/4*ln(1/x)*(ln(1-1/x*c^(1/2))+ln 
(1+1/x*c^(1/2)))/c-1/4*(dilog(1-1/x*c^(1/2))+dilog(1+1/x*c^(1/2)))/c+1/4*l 
n(1/x)*(ln(1+1/x*(-c)^(1/2))+ln(1-1/x*(-c)^(1/2)))/c+1/4*(dilog(1+1/x*(-c) 
^(1/2))+dilog(1-1/x*(-c)^(1/2)))/c))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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