3.2.79 \(\int \frac {(a+b \text {arctanh}(\frac {c}{x^2}))^2}{x^2} \, dx\) [179]

3.2.79.1 Optimal result

Integrand size = 16, antiderivative size = 1117 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2}{x^2} \, dx =\text {Too large to display} \]

output

2*b^2*arctan(x/c^(1/2))*ln(2-2*c^(1/2)/(-I*x+c^(1/2)))/c^(1/2)-2*b^2*arcta 
nh(x/c^(1/2))*ln(2-2*c^(1/2)/(x+c^(1/2)))/c^(1/2)+1/2*b^2*ln(1-c/x^2)*ln(1 
+c/x^2)/x-2*a*b*arccot(x/c^(1/2))/c^(1/2)+2*b^2*arccot(x/c^(1/2))*ln(2/(1- 
I*c^(1/2)/x))/c^(1/2)+2*b^2*arccoth(x/c^(1/2))*ln(2/(1+1/x*c^(1/2)))/c^(1/ 
2)-I*b^2*arctan(x/c^(1/2))^2/c^(1/2)-I*b^2*polylog(2,-1+2*c^(1/2)/(-I*x+c^ 
(1/2)))/c^(1/2)-I*b^2*polylog(2,1-2/(1-I*c^(1/2)/x))/c^(1/2)+1/2*I*b^2*pol 
ylog(2,1-(1+I)*(1-1/x*c^(1/2))/(1-I*c^(1/2)/x))/c^(1/2)+1/2*I*b^2*polylog( 
2,1+(-1+I)*(1+1/x*c^(1/2))/(1-I*c^(1/2)/x))/c^(1/2)-1/4*b^2*ln(1+c/x^2)^2/ 
x-2*b^2*arccot(x/c^(1/2))/c^(1/2)-2*b^2*arccoth(x/c^(1/2))/c^(1/2)-2*b^2*a 
rctan(x/c^(1/2))/c^(1/2)+2*b^2*arctanh(x/c^(1/2))/c^(1/2)+1/2*b^2*polylog( 
2,1+2*(1-(-c)^(1/2)/x)*c^(1/2)/((-c)^(1/2)-c^(1/2))/(1+1/x*c^(1/2)))/c^(1/ 
2)+1/2*b^2*polylog(2,1-2*(1+(-c)^(1/2)/x)*c^(1/2)/((-c)^(1/2)+c^(1/2))/(1+ 
1/x*c^(1/2)))/c^(1/2)+2*a*b/x-1/4*(2*a-b*ln(1-c/x^2))^2/x+b^2*arccoth(x/c^ 
(1/2))*ln(1+c/x^2)/c^(1/2)+b^2*arctan(x/c^(1/2))*ln(1+c/x^2)/c^(1/2)-b^2*a 
rccot(x/c^(1/2))*ln((1+I)*(1-1/x*c^(1/2))/(1-I*c^(1/2)/x))/c^(1/2)+b^2*pol 
ylog(2,-1+2*c^(1/2)/(x+c^(1/2)))/c^(1/2)-b^2*polylog(2,1-2/(1+1/x*c^(1/2)) 
)/c^(1/2)-b^2*ln(1-c/x^2)/x-b*(2*a-b*ln(1-c/x^2))/x-b^2*arctanh(x/c^(1/2)) 
^2/c^(1/2)-b^2*arccoth(x/c^(1/2))*ln(-2*(1-(-c)^(1/2)/x)*c^(1/2)/((-c)^(1/ 
2)-c^(1/2))/(1+1/x*c^(1/2)))/c^(1/2)-b^2*arccoth(x/c^(1/2))*ln(2*(1+(-c)^( 
1/2)/x)*c^(1/2)/((-c)^(1/2)+c^(1/2))/(1+1/x*c^(1/2)))/c^(1/2)-b^2*arcco...
 

3.2.79.2 Mathematica [A] (verified)

Time = 2.13 (sec) , antiderivative size = 568, normalized size of antiderivative = 0.51 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2}{x^2} \, dx=\frac {-2 a^2-\frac {4 a b \left (\arctan \left (\sqrt {\frac {c}{x^2}}\right )-\text {arctanh}\left (\sqrt {\frac {c}{x^2}}\right )\right )}{\sqrt {\frac {c}{x^2}}}-4 a b \text {arctanh}\left (\frac {c}{x^2}\right )+\frac {b^2 \left (2 i \arctan \left (\sqrt {\frac {c}{x^2}}\right )^2-4 \arctan \left (\sqrt {\frac {c}{x^2}}\right ) \text {arctanh}\left (\frac {c}{x^2}\right )-2 \sqrt {\frac {c}{x^2}} \text {arctanh}\left (\frac {c}{x^2}\right )^2-2 \arctan \left (\sqrt {\frac {c}{x^2}}\right ) \log \left (1+e^{4 i \arctan \left (\sqrt {\frac {c}{x^2}}\right )}\right )-2 \text {arctanh}\left (\frac {c}{x^2}\right ) \log \left (1-\sqrt {\frac {c}{x^2}}\right )+\log (2) \log \left (1-\sqrt {\frac {c}{x^2}}\right )-\frac {1}{2} \log ^2\left (1-\sqrt {\frac {c}{x^2}}\right )+\log \left (1-\sqrt {\frac {c}{x^2}}\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i+\sqrt {\frac {c}{x^2}}\right )\right )+2 \text {arctanh}\left (\frac {c}{x^2}\right ) \log \left (1+\sqrt {\frac {c}{x^2}}\right )-\log (2) \log \left (1+\sqrt {\frac {c}{x^2}}\right )-\log \left (\frac {1}{2} \left ((1+i)-(1-i) \sqrt {\frac {c}{x^2}}\right )\right ) \log \left (1+\sqrt {\frac {c}{x^2}}\right )-\log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\sqrt {\frac {c}{x^2}}\right )\right ) \log \left (1+\sqrt {\frac {c}{x^2}}\right )+\frac {1}{2} \log ^2\left (1+\sqrt {\frac {c}{x^2}}\right )+\log \left (1-\sqrt {\frac {c}{x^2}}\right ) \log \left (\frac {1}{2} \left ((1+i)+(1-i) \sqrt {\frac {c}{x^2}}\right )\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{4 i \arctan \left (\sqrt {\frac {c}{x^2}}\right )}\right )-\operatorname {PolyLog}\left (2,\frac {1}{2} \left (1-\sqrt {\frac {c}{x^2}}\right )\right )+\operatorname {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\sqrt {\frac {c}{x^2}}\right )\right )+\operatorname {PolyLog}\left (2,\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\sqrt {\frac {c}{x^2}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\sqrt {\frac {c}{x^2}}\right )\right )-\operatorname {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\sqrt {\frac {c}{x^2}}\right )\right )-\operatorname {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\sqrt {\frac {c}{x^2}}\right )\right )\right )}{\sqrt {\frac {c}{x^2}}}}{2 x} \]

input

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

output

(-2*a^2 - (4*a*b*(ArcTan[Sqrt[c/x^2]] - ArcTanh[Sqrt[c/x^2]]))/Sqrt[c/x^2] 
 - 4*a*b*ArcTanh[c/x^2] + (b^2*((2*I)*ArcTan[Sqrt[c/x^2]]^2 - 4*ArcTan[Sqr 
t[c/x^2]]*ArcTanh[c/x^2] - 2*Sqrt[c/x^2]*ArcTanh[c/x^2]^2 - 2*ArcTan[Sqrt[ 
c/x^2]]*Log[1 + E^((4*I)*ArcTan[Sqrt[c/x^2]])] - 2*ArcTanh[c/x^2]*Log[1 - 
Sqrt[c/x^2]] + Log[2]*Log[1 - Sqrt[c/x^2]] - Log[1 - Sqrt[c/x^2]]^2/2 + Lo 
g[1 - Sqrt[c/x^2]]*Log[(1/2 + I/2)*(-I + Sqrt[c/x^2])] + 2*ArcTanh[c/x^2]* 
Log[1 + Sqrt[c/x^2]] - Log[2]*Log[1 + Sqrt[c/x^2]] - Log[((1 + I) - (1 - I 
)*Sqrt[c/x^2])/2]*Log[1 + Sqrt[c/x^2]] - Log[(-1/2 - I/2)*(I + Sqrt[c/x^2] 
)]*Log[1 + Sqrt[c/x^2]] + Log[1 + Sqrt[c/x^2]]^2/2 + Log[1 - Sqrt[c/x^2]]* 
Log[((1 + I) + (1 - I)*Sqrt[c/x^2])/2] + (I/2)*PolyLog[2, -E^((4*I)*ArcTan 
[Sqrt[c/x^2]])] - PolyLog[2, (1 - Sqrt[c/x^2])/2] + PolyLog[2, (-1/2 - I/2 
)*(-1 + Sqrt[c/x^2])] + PolyLog[2, (-1/2 + I/2)*(-1 + Sqrt[c/x^2])] + Poly 
Log[2, (1 + Sqrt[c/x^2])/2] - PolyLog[2, (1/2 - I/2)*(1 + Sqrt[c/x^2])] - 
PolyLog[2, (1/2 + I/2)*(1 + Sqrt[c/x^2])]))/Sqrt[c/x^2])/(2*x)
 

3.2.79.3 Rubi [A] (verified)

Time = 2.44 (sec) , antiderivative size = 1117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6460, 6457, 2009}

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.

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

\(\Big \downarrow \) 6460

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

\(\Big \downarrow \) 6457

\(\displaystyle \int \left (\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{4 x^2}-\frac {b \log \left (\frac {c}{x^2}+1\right ) \left (b \log \left (1-\frac {c}{x^2}\right )-2 a\right )}{2 x^2}+\frac {b^2 \log ^2\left (\frac {c}{x^2}+1\right )}{4 x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {i \arctan \left (\frac {x}{\sqrt {c}}\right )^2 b^2}{\sqrt {c}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {c}}\right )^2 b^2}{\sqrt {c}}-\frac {\log ^2\left (\frac {c}{x^2}+1\right ) b^2}{4 x}-\frac {2 \cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) b^2}{\sqrt {c}}-\frac {2 \coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) b^2}{\sqrt {c}}-\frac {2 \arctan \left (\frac {x}{\sqrt {c}}\right ) b^2}{\sqrt {c}}+\frac {2 \text {arctanh}\left (\frac {x}{\sqrt {c}}\right ) b^2}{\sqrt {c}}+\frac {2 \arctan \left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right ) b^2}{\sqrt {c}}+\frac {\cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right ) b^2}{\sqrt {c}}-\frac {\log \left (1-\frac {c}{x^2}\right ) b^2}{x}+\frac {\coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right ) b^2}{\sqrt {c}}+\frac {\arctan \left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right ) b^2}{\sqrt {c}}+\frac {\log \left (1-\frac {c}{x^2}\right ) \log \left (\frac {c}{x^2}+1\right ) b^2}{2 x}+\frac {2 \cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2}{1-\frac {i \sqrt {c}}{x}}\right ) b^2}{\sqrt {c}}-\frac {\cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (1-\frac {\sqrt {c}}{x}\right )}{1-\frac {i \sqrt {c}}{x}}\right ) b^2}{\sqrt {c}}+\frac {2 \coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2}{\frac {\sqrt {c}}{x}+1}\right ) b^2}{\sqrt {c}}-\frac {\coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\frac {\sqrt {-c}}{x}\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\frac {\sqrt {c}}{x}+1\right )}\right ) b^2}{\sqrt {c}}-\frac {\coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\frac {\sqrt {-c}}{x}+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\frac {\sqrt {c}}{x}+1\right )}\right ) b^2}{\sqrt {c}}-\frac {\cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\frac {\sqrt {c}}{x}+1\right )}{1-\frac {i \sqrt {c}}{x}}\right ) b^2}{\sqrt {c}}-\frac {2 \text {arctanh}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{x+\sqrt {c}}\right ) b^2}{\sqrt {c}}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {i \sqrt {c}}{x}}\right ) b^2}{\sqrt {c}}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {(1+i) \left (1-\frac {\sqrt {c}}{x}\right )}{1-\frac {i \sqrt {c}}{x}}\right ) b^2}{2 \sqrt {c}}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{\frac {\sqrt {c}}{x}+1}\right ) b^2}{\sqrt {c}}+\frac {\operatorname {PolyLog}\left (2,\frac {2 \sqrt {c} \left (1-\frac {\sqrt {-c}}{x}\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\frac {\sqrt {c}}{x}+1\right )}+1\right ) b^2}{2 \sqrt {c}}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {c} \left (\frac {\sqrt {-c}}{x}+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\frac {\sqrt {c}}{x}+1\right )}\right ) b^2}{2 \sqrt {c}}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {(1-i) \left (\frac {\sqrt {c}}{x}+1\right )}{1-\frac {i \sqrt {c}}{x}}\right ) b^2}{2 \sqrt {c}}-\frac {i \operatorname {PolyLog}\left (2,\frac {2 \sqrt {c}}{\sqrt {c}-i x}-1\right ) b^2}{\sqrt {c}}+\frac {\operatorname {PolyLog}\left (2,\frac {2 \sqrt {c}}{x+\sqrt {c}}-1\right ) b^2}{\sqrt {c}}-\frac {2 a \cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) b}{\sqrt {c}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right ) b}{\sqrt {c}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right ) b}{x}-\frac {a \log \left (\frac {c}{x^2}+1\right ) b}{x}+\frac {2 a b}{x}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{4 x}\)

input

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

output

(2*a*b)/x - (2*a*b*ArcCot[x/Sqrt[c]])/Sqrt[c] - (2*b^2*ArcCot[x/Sqrt[c]])/ 
Sqrt[c] - (2*b^2*ArcCoth[x/Sqrt[c]])/Sqrt[c] - (2*b^2*ArcTan[x/Sqrt[c]])/S 
qrt[c] - (I*b^2*ArcTan[x/Sqrt[c]]^2)/Sqrt[c] + (2*b^2*ArcTanh[x/Sqrt[c]])/ 
Sqrt[c] - (b^2*ArcTanh[x/Sqrt[c]]^2)/Sqrt[c] + (2*b^2*ArcTan[x/Sqrt[c]]*Lo 
g[2 - (2*Sqrt[c])/(Sqrt[c] - I*x)])/Sqrt[c] - (b^2*Log[1 - c/x^2])/x + (b^ 
2*ArcCot[x/Sqrt[c]]*Log[1 - c/x^2])/Sqrt[c] - (b*(2*a - b*Log[1 - c/x^2])) 
/x + (b*ArcTanh[x/Sqrt[c]]*(2*a - b*Log[1 - c/x^2]))/Sqrt[c] - (2*a - b*Lo 
g[1 - c/x^2])^2/(4*x) - (a*b*Log[1 + c/x^2])/x + (b^2*ArcCoth[x/Sqrt[c]]*L 
og[1 + c/x^2])/Sqrt[c] + (b^2*ArcTan[x/Sqrt[c]]*Log[1 + c/x^2])/Sqrt[c] + 
(b^2*Log[1 - c/x^2]*Log[1 + c/x^2])/(2*x) - (b^2*Log[1 + c/x^2]^2)/(4*x) + 
 (2*b^2*ArcCot[x/Sqrt[c]]*Log[2/(1 - (I*Sqrt[c])/x)])/Sqrt[c] - (b^2*ArcCo 
t[x/Sqrt[c]]*Log[((1 + I)*(1 - Sqrt[c]/x))/(1 - (I*Sqrt[c])/x)])/Sqrt[c] + 
 (2*b^2*ArcCoth[x/Sqrt[c]]*Log[2/(1 + Sqrt[c]/x)])/Sqrt[c] - (b^2*ArcCoth[ 
x/Sqrt[c]]*Log[(-2*Sqrt[c]*(1 - Sqrt[-c]/x))/((Sqrt[-c] - Sqrt[c])*(1 + Sq 
rt[c]/x))])/Sqrt[c] - (b^2*ArcCoth[x/Sqrt[c]]*Log[(2*Sqrt[c]*(1 + Sqrt[-c] 
/x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]/x))])/Sqrt[c] - (b^2*ArcCot[x/Sqrt 
[c]]*Log[((1 - I)*(1 + Sqrt[c]/x))/(1 - (I*Sqrt[c])/x)])/Sqrt[c] - (2*b^2* 
ArcTanh[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] + x)])/Sqrt[c] - (I*b^2*Po 
lyLog[2, 1 - 2/(1 - (I*Sqrt[c])/x)])/Sqrt[c] + ((I/2)*b^2*PolyLog[2, 1 - ( 
(1 + I)*(1 - Sqrt[c]/x))/(1 - (I*Sqrt[c])/x)])/Sqrt[c] - (b^2*PolyLog[2...
 

3.2.79.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6457

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> 
Int[ExpandIntegrand[x^m*(a + b*(Log[1 + 1/(x^n*c)]/2) - b*(Log[1 - 1/(x^n*c 
)]/2))^p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0] && Inte 
gerQ[m]
 

rule 6460

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

3.2.79.4 Maple [F]

\[\int \frac {{\left (a +b \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )\right )}^{2}}{x^{2}}d x\]

input

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

output

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

3.2.79.5 Fricas [F]

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

input

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

output

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

3.2.79.6 Sympy [F]

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

input

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

output

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

3.2.79.7 Maxima [F]

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

input

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

output

(c*(2*arctan(x/sqrt(c))/c^(3/2) - log((x - sqrt(c))/(x + sqrt(c)))/c^(3/2) 
) - 2*arctanh(c/x^2)/x)*a*b - 1/4*b^2*(log(x^2 - c)^2/x + integrate(-((x^2 
 - c)*log(x^2 + c)^2 + 2*(2*x^2 - (x^2 - c)*log(x^2 + c))*log(x^2 - c))/(x 
^4 - c*x^2), x)) - a^2/x
 

3.2.79.8 Giac [F]

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

input

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

output

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

3.2.79.9 Mupad [F(-1)]

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

input

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

output

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