\(\int \frac {(a+b \text {arctanh}(c \sqrt {x}))^2}{x^2} \, dx\) [199]

Optimal result

Integrand size = 18, antiderivative size = 85 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx=-\frac {2 b c \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{\sqrt {x}}+c^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x}+b^2 c^2 \log (x)-b^2 c^2 \log \left (1-c^2 x\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx=-\frac {a^2+2 a b c \sqrt {x}+2 b \left (a+b c \sqrt {x}\right ) \text {arctanh}\left (c \sqrt {x}\right )-b^2 \left (-1+c^2 x\right ) \text {arctanh}\left (c \sqrt {x}\right )^2+b (a+b) c^2 x \log \left (1-c \sqrt {x}\right )-a b c^2 x \log \left (1+c \sqrt {x}\right )+b^2 c^2 x \log \left (1+c \sqrt {x}\right )-b^2 c^2 x \log (x)}{x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6454, 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.

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

\(\Big \downarrow \) 6454

\(\displaystyle 2 \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^{3/2}}d\sqrt {x}\)

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 6510

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

Input:

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

Output:

2*(-1/2*(a + b*ArcTanh[c*Sqrt[x]])^2/x + b*c*(-((a + b*ArcTanh[c*Sqrt[x]]) 
/Sqrt[x]) + (c*(a + b*ArcTanh[c*Sqrt[x]])^2)/(2*b) + (b*c*(Log[x] - Log[1 
- c^2*x]))/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 6454

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(230\) vs. \(2(77)=154\).

Time = 0.46 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.72

method	result	size
parts	\(-\frac {a^{2}}{x}+2 b^{2} c^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{2 c^{2} x}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{c \sqrt {x}}+\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}-\frac {\ln \left (c \sqrt {x}-1\right )}{2}-\frac {\ln \left (1+c \sqrt {x}\right )}{2}+\ln \left (c \sqrt {x}\right )-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}\right )+4 a b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c^{2} x}-\frac {\ln \left (c \sqrt {x}-1\right )}{4}-\frac {1}{2 c \sqrt {x}}+\frac {\ln \left (1+c \sqrt {x}\right )}{4}\right )\)	\(231\)
derivativedivides	\(2 c^{2} \left (-\frac {a^{2}}{2 c^{2} x}+b^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{2 c^{2} x}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{c \sqrt {x}}+\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}-\frac {\ln \left (c \sqrt {x}-1\right )}{2}-\frac {\ln \left (1+c \sqrt {x}\right )}{2}+\ln \left (c \sqrt {x}\right )-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}\right )+2 a b \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c^{2} x}-\frac {\ln \left (c \sqrt {x}-1\right )}{4}-\frac {1}{2 c \sqrt {x}}+\frac {\ln \left (1+c \sqrt {x}\right )}{4}\right )\right )\)	\(232\)
default	\(2 c^{2} \left (-\frac {a^{2}}{2 c^{2} x}+b^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{2 c^{2} x}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{c \sqrt {x}}+\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}-\frac {\ln \left (c \sqrt {x}-1\right )}{2}-\frac {\ln \left (1+c \sqrt {x}\right )}{2}+\ln \left (c \sqrt {x}\right )-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}\right )+2 a b \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c^{2} x}-\frac {\ln \left (c \sqrt {x}-1\right )}{4}-\frac {1}{2 c \sqrt {x}}+\frac {\ln \left (1+c \sqrt {x}\right )}{4}\right )\right )\)	\(232\)

Input:

int((a+b*arctanh(c*x^(1/2)))^2/x^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (77) = 154\).

Time = 0.10 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx=\frac {8 \, b^{2} c^{2} x \log \left (\sqrt {x}\right ) + 4 \, {\left (a b - b^{2}\right )} c^{2} x \log \left (c \sqrt {x} + 1\right ) - 4 \, {\left (a b + b^{2}\right )} c^{2} x \log \left (c \sqrt {x} - 1\right ) - 8 \, a b c \sqrt {x} + {\left (b^{2} c^{2} x - b^{2}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )^{2} - 4 \, a^{2} - 4 \, {\left (b^{2} c \sqrt {x} + a b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )}{4 \, x} \] Input:

integrate((a+b*arctanh(c*x^(1/2)))^2/x^2,x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 680 vs. \(2 (78) = 156\).

Time = 2.62 (sec) , antiderivative size = 680, normalized size of antiderivative = 8.00 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((-a**2/x, Eq(c, 0)), (-a**2/x + 2*a*b*atanh(sqrt(x)*sqrt(1/x))/x 
 - b**2*atanh(sqrt(x)*sqrt(1/x))**2/x, Eq(c, -sqrt(1/x))), (-a**2/x - 2*a* 
b*atanh(sqrt(x)*sqrt(1/x))/x - b**2*atanh(sqrt(x)*sqrt(1/x))**2/x, Eq(c, s 
qrt(1/x))), (-a**2*c**2*x**(3/2)/(c**2*x**(5/2) - x**(3/2)) + a**2*sqrt(x) 
/(c**2*x**(5/2) - x**(3/2)) + 2*a*b*c**4*x**(5/2)*atanh(c*sqrt(x))/(c**2*x 
**(5/2) - x**(3/2)) - 2*a*b*c**3*x**2/(c**2*x**(5/2) - x**(3/2)) - 4*a*b*c 
**2*x**(3/2)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) + 2*a*b*c*x/(c**2 
*x**(5/2) - x**(3/2)) + 2*a*b*sqrt(x)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x* 
*(3/2)) + b**2*c**4*x**(5/2)*log(x)/(c**2*x**(5/2) - x**(3/2)) - 2*b**2*c* 
*4*x**(5/2)*log(sqrt(x) - 1/c)/(c**2*x**(5/2) - x**(3/2)) + b**2*c**4*x**( 
5/2)*atanh(c*sqrt(x))**2/(c**2*x**(5/2) - x**(3/2)) - 2*b**2*c**4*x**(5/2) 
*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) - 2*b**2*c**3*x**2*atanh(c*sq 
rt(x))/(c**2*x**(5/2) - x**(3/2)) - b**2*c**2*x**(3/2)*log(x)/(c**2*x**(5/ 
2) - x**(3/2)) + 2*b**2*c**2*x**(3/2)*log(sqrt(x) - 1/c)/(c**2*x**(5/2) - 
x**(3/2)) - 2*b**2*c**2*x**(3/2)*atanh(c*sqrt(x))**2/(c**2*x**(5/2) - x**( 
3/2)) + 2*b**2*c**2*x**(3/2)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) + 
 2*b**2*c*x*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) + b**2*sqrt(x)*ata 
nh(c*sqrt(x))**2/(c**2*x**(5/2) - x**(3/2)), True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (77) = 154\).

Time = 0.04 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx={\left ({\left (c \log \left (c \sqrt {x} + 1\right ) - c \log \left (c \sqrt {x} - 1\right ) - \frac {2}{\sqrt {x}}\right )} c - \frac {2 \, \operatorname {artanh}\left (c \sqrt {x}\right )}{x}\right )} a b + \frac {1}{4} \, {\left ({\left (2 \, {\left (\log \left (c \sqrt {x} - 1\right ) - 2\right )} \log \left (c \sqrt {x} + 1\right ) - \log \left (c \sqrt {x} + 1\right )^{2} - \log \left (c \sqrt {x} - 1\right )^{2} - 4 \, \log \left (c \sqrt {x} - 1\right ) + 4 \, \log \left (x\right )\right )} c^{2} + 4 \, {\left (c \log \left (c \sqrt {x} + 1\right ) - c \log \left (c \sqrt {x} - 1\right ) - \frac {2}{\sqrt {x}}\right )} c \operatorname {artanh}\left (c \sqrt {x}\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c \sqrt {x}\right )^{2}}{x} - \frac {a^{2}}{x} \] Input:

integrate((a+b*arctanh(c*x^(1/2)))^2/x^2,x, algorithm="maxima")
 

Output:

((c*log(c*sqrt(x) + 1) - c*log(c*sqrt(x) - 1) - 2/sqrt(x))*c - 2*arctanh(c 
*sqrt(x))/x)*a*b + 1/4*((2*(log(c*sqrt(x) - 1) - 2)*log(c*sqrt(x) + 1) - l 
og(c*sqrt(x) + 1)^2 - log(c*sqrt(x) - 1)^2 - 4*log(c*sqrt(x) - 1) + 4*log( 
x))*c^2 + 4*(c*log(c*sqrt(x) + 1) - c*log(c*sqrt(x) - 1) - 2/sqrt(x))*c*ar 
ctanh(c*sqrt(x)))*b^2 - b^2*arctanh(c*sqrt(x))^2/x - a^2/x
 

Giac [F]

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

integrate((a+b*arctanh(c*x^(1/2)))^2/x^2,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 4.43 (sec) , antiderivative size = 278, normalized size of antiderivative = 3.27 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx=2\,b^2\,c^2\,\ln \left (\sqrt {x}\right )-\frac {a^2}{x}-b^2\,c^2\,\ln \left (c\,\sqrt {x}-1\right )-b^2\,c^2\,\ln \left (c\,\sqrt {x}+1\right )+\frac {b^2\,c^2\,{\ln \left (c\,\sqrt {x}+1\right )}^2}{4}+\frac {b^2\,c^2\,{\ln \left (1-c\,\sqrt {x}\right )}^2}{4}-\frac {b^2\,{\ln \left (c\,\sqrt {x}+1\right )}^2}{4\,x}-\frac {b^2\,{\ln \left (1-c\,\sqrt {x}\right )}^2}{4\,x}-a\,b\,c^2\,\ln \left (c\,\sqrt {x}-1\right )+a\,b\,c^2\,\ln \left (c\,\sqrt {x}+1\right )-\frac {2\,a\,b\,c}{\sqrt {x}}-\frac {a\,b\,\ln \left (c\,\sqrt {x}+1\right )}{x}+\frac {a\,b\,\ln \left (1-c\,\sqrt {x}\right )}{x}-\frac {b^2\,c^2\,\ln \left (c\,\sqrt {x}+1\right )\,\ln \left (1-c\,\sqrt {x}\right )}{2}-\frac {b^2\,c\,\ln \left (c\,\sqrt {x}+1\right )}{\sqrt {x}}+\frac {b^2\,c\,\ln \left (1-c\,\sqrt {x}\right )}{\sqrt {x}}+\frac {b^2\,\ln \left (c\,\sqrt {x}+1\right )\,\ln \left (1-c\,\sqrt {x}\right )}{2\,x} \] Input:

int((a + b*atanh(c*x^(1/2)))^2/x^2,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx=\frac {\mathit {atanh} \left (\sqrt {x}\, c \right )^{2} b^{2} c^{2} x -\mathit {atanh} \left (\sqrt {x}\, c \right )^{2} b^{2}-2 \sqrt {x}\, \mathit {atanh} \left (\sqrt {x}\, c \right ) b^{2} c +2 \mathit {atanh} \left (\sqrt {x}\, c \right ) a b \,c^{2} x -2 \mathit {atanh} \left (\sqrt {x}\, c \right ) a b -2 \mathit {atanh} \left (\sqrt {x}\, c \right ) b^{2} c^{2} x -2 \sqrt {x}\, a b c -2 \,\mathrm {log}\left (\sqrt {x}\, c -1\right ) b^{2} c^{2} x +2 \,\mathrm {log}\left (\sqrt {x}\right ) b^{2} c^{2} x -a^{2}}{x} \] Input:

int((a+b*atanh(c*x^(1/2)))^2/x^2,x)
 

Output:

(atanh(sqrt(x)*c)**2*b**2*c**2*x - atanh(sqrt(x)*c)**2*b**2 - 2*sqrt(x)*at 
anh(sqrt(x)*c)*b**2*c + 2*atanh(sqrt(x)*c)*a*b*c**2*x - 2*atanh(sqrt(x)*c) 
*a*b - 2*atanh(sqrt(x)*c)*b**2*c**2*x - 2*sqrt(x)*a*b*c - 2*log(sqrt(x)*c 
- 1)*b**2*c**2*x + 2*log(sqrt(x))*b**2*c**2*x - a**2)/x