\(\int \frac {(a+b \text {arctanh}(c x))^2}{x^3} \, dx\) [21]

Optimal result

Integrand size = 14, antiderivative size = 80 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3} \, dx=-\frac {b c (a+b \text {arctanh}(c x))}{x}+\frac {1}{2} c^2 (a+b \text {arctanh}(c x))^2-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}+b^2 c^2 \log (x)-\frac {1}{2} b^2 c^2 \log \left (1-c^2 x^2\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3} \, dx=-\frac {a^2+2 a b c x+2 b (a+b c x) \text {arctanh}(c x)-b^2 \left (-1+c^2 x^2\right ) \text {arctanh}(c x)^2-2 b^2 c^2 x^2 \log (x)+b (a+b) c^2 x^2 \log (1-c x)-(a-b) b c^2 x^2 \log (1+c x)}{2 x^2} \] Input:

Integrate[(a + b*ArcTanh[c*x])^2/x^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6452, 6544, 6452, 243, 47, 14, 16, 6510}

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.

\(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{x^3} \, dx\)

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 6510

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

Input:

Int[(a + b*ArcTanh[c*x])^2/x^3,x]
 

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

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

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 6452

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x 
], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 
] && IntegerQ[m])) && NeQ[m, -1]
 

rule 6510

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

rule 6544

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( 
e_.)*(x_)^2), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x 
], x] - Simp[e/(d*f^2)   Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d + e*x 
^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.68

method	result	size
parallelrisch	\(\frac {b^{2} \operatorname {arctanh}\left (c x \right )^{2} x^{2} c^{2}+2 b^{2} c^{2} \ln \left (x \right ) x^{2}-2 \ln \left (c x -1\right ) x^{2} b^{2} c^{2}+2 x^{2} \operatorname {arctanh}\left (c x \right ) a b \,c^{2}-2 x^{2} \operatorname {arctanh}\left (c x \right ) b^{2} c^{2}-a^{2} c^{2} x^{2}-2 b^{2} \operatorname {arctanh}\left (c x \right ) x c -2 a b c x -b^{2} \operatorname {arctanh}\left (c x \right )^{2}-2 \,\operatorname {arctanh}\left (c x \right ) a b -a^{2}}{2 x^{2}}\)	\(134\)
parts	\(-\frac {a^{2}}{2 x^{2}}+b^{2} c^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\ln \left (c x -1\right )^{2}}{8}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (c x +1\right )}{2}-\frac {\ln \left (c x -1\right )}{2}+\ln \left (c x \right )-\frac {\ln \left (c x +1\right )^{2}}{8}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{4}\right )+2 a b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {1}{2 c x}-\frac {\ln \left (c x -1\right )}{4}+\frac {\ln \left (c x +1\right )}{4}\right )\)	\(192\)
derivativedivides	\(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\ln \left (c x -1\right )^{2}}{8}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (c x +1\right )}{2}-\frac {\ln \left (c x -1\right )}{2}+\ln \left (c x \right )-\frac {\ln \left (c x +1\right )^{2}}{8}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{4}\right )+2 a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {1}{2 c x}-\frac {\ln \left (c x -1\right )}{4}+\frac {\ln \left (c x +1\right )}{4}\right )\right )\)	\(193\)
default	\(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\ln \left (c x -1\right )^{2}}{8}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (c x +1\right )}{2}-\frac {\ln \left (c x -1\right )}{2}+\ln \left (c x \right )-\frac {\ln \left (c x +1\right )^{2}}{8}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{4}\right )+2 a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {1}{2 c x}-\frac {\ln \left (c x -1\right )}{4}+\frac {\ln \left (c x +1\right )}{4}\right )\right )\)	\(193\)
risch	\(\frac {b^{2} \left (c^{2} x^{2}-1\right ) \ln \left (c x +1\right )^{2}}{8 x^{2}}-\frac {b \left (x^{2} b \ln \left (-c x +1\right ) c^{2}+2 b c x -b \ln \left (-c x +1\right )+2 a \right ) \ln \left (c x +1\right )}{4 x^{2}}+\frac {b^{2} c^{2} x^{2} \ln \left (-c x +1\right )^{2}+4 b \,c^{2} \ln \left (-c x -1\right ) x^{2} a -4 b^{2} c^{2} \ln \left (-c x -1\right ) x^{2}-4 a b \,c^{2} x^{2} \ln \left (-c x +1\right )-4 b^{2} c^{2} \ln \left (-c x +1\right ) x^{2}+8 b^{2} c^{2} \ln \left (x \right ) x^{2}+4 b^{2} c x \ln \left (-c x +1\right )-8 a b c x -b^{2} \ln \left (-c x +1\right )^{2}+4 b \ln \left (-c x +1\right ) a -4 a^{2}}{8 x^{2}}\)	\(231\)

Input:

int((a+b*arctanh(c*x))^2/x^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.69 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3} \, dx=\frac {8 \, b^{2} c^{2} x^{2} \log \left (x\right ) + 4 \, {\left (a b - b^{2}\right )} c^{2} x^{2} \log \left (c x + 1\right ) - 4 \, {\left (a b + b^{2}\right )} c^{2} x^{2} \log \left (c x - 1\right ) - 8 \, a b c x + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} - 4 \, a^{2} - 4 \, {\left (b^{2} c x + a b\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{8 \, x^{2}} \] Input:

integrate((a+b*arctanh(c*x))^2/x^3,x, algorithm="fricas")
 

Output:

1/8*(8*b^2*c^2*x^2*log(x) + 4*(a*b - b^2)*c^2*x^2*log(c*x + 1) - 4*(a*b + 
b^2)*c^2*x^2*log(c*x - 1) - 8*a*b*c*x + (b^2*c^2*x^2 - b^2)*log(-(c*x + 1) 
/(c*x - 1))^2 - 4*a^2 - 4*(b^2*c*x + a*b)*log(-(c*x + 1)/(c*x - 1)))/x^2
 

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3} \, dx=\begin {cases} - \frac {a^{2}}{2 x^{2}} + a b c^{2} \operatorname {atanh}{\left (c x \right )} - \frac {a b c}{x} - \frac {a b \operatorname {atanh}{\left (c x \right )}}{x^{2}} + b^{2} c^{2} \log {\left (x \right )} - b^{2} c^{2} \log {\left (x - \frac {1}{c} \right )} + \frac {b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{2} - b^{2} c^{2} \operatorname {atanh}{\left (c x \right )} - \frac {b^{2} c \operatorname {atanh}{\left (c x \right )}}{x} - \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{2 x^{2}} & \text {for}\: c \neq 0 \\- \frac {a^{2}}{2 x^{2}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-a**2/(2*x**2) + a*b*c**2*atanh(c*x) - a*b*c/x - a*b*atanh(c*x) 
/x**2 + b**2*c**2*log(x) - b**2*c**2*log(x - 1/c) + b**2*c**2*atanh(c*x)** 
2/2 - b**2*c**2*atanh(c*x) - b**2*c*atanh(c*x)/x - b**2*atanh(c*x)**2/(2*x 
**2), Ne(c, 0)), (-a**2/(2*x**2), True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (74) = 148\).

Time = 0.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.89 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3} \, dx=\frac {1}{2} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} a b + \frac {1}{8} \, {\left ({\left (2 \, {\left (\log \left (c x - 1\right ) - 2\right )} \log \left (c x + 1\right ) - \log \left (c x + 1\right )^{2} - \log \left (c x - 1\right )^{2} - 4 \, \log \left (c x - 1\right ) + 8 \, \log \left (x\right )\right )} c^{2} + 4 \, {\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c \operatorname {artanh}\left (c x\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c x\right )^{2}}{2 \, x^{2}} - \frac {a^{2}}{2 \, x^{2}} \] Input:

integrate((a+b*arctanh(c*x))^2/x^3,x, algorithm="maxima")
 

Output:

1/2*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*a*b + 
 1/8*((2*(log(c*x - 1) - 2)*log(c*x + 1) - log(c*x + 1)^2 - log(c*x - 1)^2 
 - 4*log(c*x - 1) + 8*log(x))*c^2 + 4*(c*log(c*x + 1) - c*log(c*x - 1) - 2 
/x)*c*arctanh(c*x))*b^2 - 1/2*b^2*arctanh(c*x)^2/x^2 - 1/2*a^2/x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (74) = 148\).

Time = 0.13 (sec) , antiderivative size = 278, normalized size of antiderivative = 3.48 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3} \, dx=\frac {1}{2} \, {\left (2 \, b^{2} c \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - 2 \, b^{2} c \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {{\left (c x + 1\right )} b^{2} c \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x - 1\right )} {\left (\frac {{\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {2 \, {\left (c x + 1\right )}}{c x - 1} + 1\right )}} + \frac {2 \, {\left (\frac {2 \, {\left (c x + 1\right )} a b c}{c x - 1} + \frac {{\left (c x + 1\right )} b^{2} c}{c x - 1} + b^{2} c\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {2 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {4 \, {\left (\frac {{\left (c x + 1\right )} a^{2} c}{c x - 1} + \frac {{\left (c x + 1\right )} a b c}{c x - 1} + a b c\right )}}{\frac {{\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {2 \, {\left (c x + 1\right )}}{c x - 1} + 1}\right )} c \] Input:

integrate((a+b*arctanh(c*x))^2/x^3,x, algorithm="giac")
 

Output:

1/2*(2*b^2*c*log(-(c*x + 1)/(c*x - 1) - 1) - 2*b^2*c*log(-(c*x + 1)/(c*x - 
 1)) + (c*x + 1)*b^2*c*log(-(c*x + 1)/(c*x - 1))^2/((c*x - 1)*((c*x + 1)^2 
/(c*x - 1)^2 + 2*(c*x + 1)/(c*x - 1) + 1)) + 2*(2*(c*x + 1)*a*b*c/(c*x - 1 
) + (c*x + 1)*b^2*c/(c*x - 1) + b^2*c)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1 
)^2/(c*x - 1)^2 + 2*(c*x + 1)/(c*x - 1) + 1) + 4*((c*x + 1)*a^2*c/(c*x - 1 
) + (c*x + 1)*a*b*c/(c*x - 1) + a*b*c)/((c*x + 1)^2/(c*x - 1)^2 + 2*(c*x + 
 1)/(c*x - 1) + 1))*c
 

Mupad [B] (verification not implemented)

Time = 4.21 (sec) , antiderivative size = 246, normalized size of antiderivative = 3.08 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3} \, dx=\frac {b^2\,c^2\,{\ln \left (c\,x+1\right )}^2}{8}-\frac {a^2}{2\,x^2}+\frac {b^2\,c^2\,{\ln \left (1-c\,x\right )}^2}{8}-\frac {b^2\,{\ln \left (c\,x+1\right )}^2}{8\,x^2}-\frac {b^2\,{\ln \left (1-c\,x\right )}^2}{8\,x^2}+b^2\,c^2\,\ln \left (x\right )-\frac {b^2\,c^2\,\ln \left (c\,x-1\right )}{2}-\frac {b^2\,c^2\,\ln \left (c\,x+1\right )}{2}-\frac {a\,b\,\ln \left (c\,x+1\right )}{2\,x^2}+\frac {a\,b\,\ln \left (1-c\,x\right )}{2\,x^2}+\frac {b^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )}{4\,x^2}-\frac {a\,b\,c}{x}-\frac {b^2\,c\,\ln \left (c\,x+1\right )}{2\,x}+\frac {b^2\,c\,\ln \left (1-c\,x\right )}{2\,x}-\frac {a\,b\,c^2\,\ln \left (c\,x-1\right )}{2}+\frac {a\,b\,c^2\,\ln \left (c\,x+1\right )}{2}-\frac {b^2\,c^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )}{4} \] Input:

int((a + b*atanh(c*x))^2/x^3,x)
 

Output:

(b^2*c^2*log(c*x + 1)^2)/8 - a^2/(2*x^2) + (b^2*c^2*log(1 - c*x)^2)/8 - (b 
^2*log(c*x + 1)^2)/(8*x^2) - (b^2*log(1 - c*x)^2)/(8*x^2) + b^2*c^2*log(x) 
 - (b^2*c^2*log(c*x - 1))/2 - (b^2*c^2*log(c*x + 1))/2 - (a*b*log(c*x + 1) 
)/(2*x^2) + (a*b*log(1 - c*x))/(2*x^2) + (b^2*log(c*x + 1)*log(1 - c*x))/( 
4*x^2) - (a*b*c)/x - (b^2*c*log(c*x + 1))/(2*x) + (b^2*c*log(1 - c*x))/(2* 
x) - (a*b*c^2*log(c*x - 1))/2 + (a*b*c^2*log(c*x + 1))/2 - (b^2*c^2*log(c* 
x + 1)*log(1 - c*x))/4
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3} \, dx=\frac {\mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-\mathit {atanh} \left (c x \right )^{2} b^{2}+2 \mathit {atanh} \left (c x \right ) a b \,c^{2} x^{2}-2 \mathit {atanh} \left (c x \right ) a b -2 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}-2 \mathit {atanh} \left (c x \right ) b^{2} c x -2 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2} c^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) b^{2} c^{2} x^{2}-a^{2}-2 a b c x}{2 x^{2}} \] Input:

int((a+b*atanh(c*x))^2/x^3,x)
 

Output:

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