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

Optimal result

Integrand size = 16, antiderivative size = 1080 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2}{x^2} \, dx =\text {Too large to display} \] Output:

-1/4*(2*a-b*ln(1-c/x^2))^2/x-b^2*arctan(x/c^(1/2))*ln(1-c/x^2)/c^(1/2)+I*b 
^2*polylog(2,1-2*c^(1/2)/(c^(1/2)-I*x))/c^(1/2)+I*b^2*polylog(2,I*x/c^(1/2 
))/c^(1/2)-b^2*arctanh(x/c^(1/2))*ln(2*c^(1/2)*((-c)^(1/2)+x)/((-c)^(1/2)+ 
c^(1/2))/(c^(1/2)+x))/c^(1/2)-b^2*arctanh(x/c^(1/2))*ln(2*c^(1/2)*((-c)^(1 
/2)-x)/((-c)^(1/2)-c^(1/2))/(c^(1/2)+x))/c^(1/2)+b^2*arctanh(x/c^(1/2))*ln 
(1+c/x^2)/c^(1/2)+b^2*arctan(x/c^(1/2))*ln((1+I)*(c^(1/2)-x)/(c^(1/2)-I*x) 
)/c^(1/2)+b^2*arctan(x/c^(1/2))*ln((1-I)*(c^(1/2)+x)/(c^(1/2)-I*x))/c^(1/2 
)+2*b^2*arctanh(x/c^(1/2))*ln(2*c^(1/2)/(c^(1/2)+x))/c^(1/2)+2*a*b*arctan( 
x/c^(1/2))/c^(1/2)+2*b^2*arctan(x/c^(1/2))*ln(2-2*c^(1/2)/(c^(1/2)-I*x))/c 
^(1/2)-2*b^2*arctan(x/c^(1/2))*ln(2*c^(1/2)/(c^(1/2)-I*x))/c^(1/2)-2*b^2*a 
rctanh(x/c^(1/2))*ln(2-2*c^(1/2)/(c^(1/2)+x))/c^(1/2)-I*b^2*arctan(x/c^(1/ 
2))^2/c^(1/2)-1/2*I*b^2*polylog(2,1+(-1+I)*(c^(1/2)+x)/(c^(1/2)-I*x))/c^(1 
/2)-1/2*I*b^2*polylog(2,1-(1+I)*(c^(1/2)-x)/(c^(1/2)-I*x))/c^(1/2)-I*b^2*p 
olylog(2,-1+2*c^(1/2)/(c^(1/2)-I*x))/c^(1/2)-I*b^2*polylog(2,-I*x/c^(1/2)) 
/c^(1/2)+b^2*arctan(x/c^(1/2))*ln(1+c/x^2)/c^(1/2)+b*arctanh(x/c^(1/2))*(2 
*a-b*ln(1-c/x^2))/c^(1/2)+1/2*b^2*polylog(2,1-2*c^(1/2)*((-c)^(1/2)+x)/((- 
c)^(1/2)+c^(1/2))/(c^(1/2)+x))/c^(1/2)+1/2*b^2*polylog(2,1-2*c^(1/2)*((-c) 
^(1/2)-x)/((-c)^(1/2)-c^(1/2))/(c^(1/2)+x))/c^(1/2)-b^2*polylog(2,1-2*c^(1 
/2)/(c^(1/2)+x))/c^(1/2)+b^2*polylog(2,-1+2*c^(1/2)/(c^(1/2)+x))/c^(1/2)-b 
^2*polylog(2,-x/c^(1/2))/c^(1/2)+b^2*polylog(2,x/c^(1/2))/c^(1/2)-b^2*a...
 

Mathematica [A] (verified)

Time = 1.97 (sec) , antiderivative size = 568, normalized size of antiderivative = 0.53 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2}{x^2} \, dx =\text {Too large to display} \] 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)
 

Rubi [A] (verified)

Time = 2.43 (sec) , antiderivative size = 1117, normalized size of antiderivative = 1.03, 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...
 

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]
 

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)
 

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)
 

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)
 

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
 

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)
 

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)
 

Reduce [F]

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

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

Output:

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