\(\int \frac {a+b \text {arctanh}(c x^3)}{x} \, dx\) [103]

Optimal result

Integrand size = 14, antiderivative size = 30 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x} \, dx=a \log (x)-\frac {1}{6} b \operatorname {PolyLog}\left (2,-c x^3\right )+\frac {1}{6} b \operatorname {PolyLog}\left (2,c x^3\right ) \] Output:

a*ln(x)-1/6*b*polylog(2,-c*x^3)+1/6*b*polylog(2,c*x^3)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x} \, dx=a \log (x)+\frac {1}{6} b \left (-\operatorname {PolyLog}\left (2,-c x^3\right )+\operatorname {PolyLog}\left (2,c x^3\right )\right ) \] Input:

Integrate[(a + b*ArcTanh[c*x^3])/x,x]
 

Output:

a*Log[x] + (b*(-PolyLog[2, -(c*x^3)] + PolyLog[2, c*x^3]))/6
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20, 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.

\(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x} \, dx\)

\(\Big \downarrow \) 6450

\(\displaystyle \frac {1}{3} \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^3}dx^3\)

\(\Big \downarrow \) 6446

\(\displaystyle \frac {1}{3} \left (a \log \left (x^3\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,-c x^3\right )+\frac {1}{2} b \operatorname {PolyLog}\left (2,c x^3\right )\right )\)

Input:

Int[(a + b*ArcTanh[c*x^3])/x,x]
 

Output:

(a*Log[x^3] - (b*PolyLog[2, -(c*x^3)])/2 + (b*PolyLog[2, c*x^3])/2)/3
 

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 [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.07

method	result	size
default	\(a \ln \left (x \right )+b \ln \left (x \right ) \operatorname {arctanh}\left (c \,x^{3}\right )+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}-\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}+1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}\)	\(92\)
parts	\(a \ln \left (x \right )+b \ln \left (x \right ) \operatorname {arctanh}\left (c \,x^{3}\right )+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}-\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}+1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}\)	\(92\)
risch	\(a \ln \left (x \right )-\frac {\ln \left (x \right ) \ln \left (-c \,x^{3}+1\right ) b}{2}+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}+\frac {\ln \left (x \right ) \ln \left (c \,x^{3}+1\right ) b}{2}-\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}+1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}\)	\(109\)

Input:

int((a+b*arctanh(c*x^3))/x,x,method=_RETURNVERBOSE)
 

Output:

a*ln(x)+b*ln(x)*arctanh(c*x^3)+1/2*b*sum(ln(x)*ln((_R1-x)/_R1)+dilog((_R1- 
x)/_R1),_R1=RootOf(_Z^3*c-1))-1/2*b*sum(ln(x)*ln((_R1-x)/_R1)+dilog((_R1-x 
)/_R1),_R1=RootOf(_Z^3*c+1))
 

Fricas [F]

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

integrate((a+b*arctanh(c*x^3))/x,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((a+b*atanh(c*x**3))/x,x)
 

Output:

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

Maxima [F]

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

integrate((a+b*arctanh(c*x^3))/x,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((a+b*arctanh(c*x^3))/x,x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x^3) + a)/x, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((a + b*atanh(c*x^3))/x,x)
 

Output:

int((a + b*atanh(c*x^3))/x, x)
 

Reduce [F]

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

int((a+b*atanh(c*x^3))/x,x)
 

Output:

int(atanh(c*x**3)/x,x)*b + log(x)*a