\(\int \frac {(a+b \text {arctanh}(c x^2))^2}{x^2} \, dx\) [74]

Optimal result

Integrand size = 16, antiderivative size = 942 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^2} \, dx =\text {Too large to display} \] Output:

-1/4*b^2*ln(c*x^2+1)^2/x-b^2*c^(1/2)*polylog(2,1-2/(1+c^(1/2)*x))-b^2*c^(1 
/2)*polylog(2,1-2/(1-c^(1/2)*x))+b^2*c^(1/2)*arctanh(c^(1/2)*x)^2+1/2*b^2* 
c^(1/2)*polylog(2,1-2*c^(1/2)*(1+(-c)^(1/2)*x)/((-c)^(1/2)+c^(1/2))/(1+c^( 
1/2)*x))+1/2*b^2*c^(1/2)*polylog(2,1+2*c^(1/2)*(1-(-c)^(1/2)*x)/((-c)^(1/2 
)-c^(1/2))/(1+c^(1/2)*x))-a*b*ln(c*x^2+1)/x+1/2*b^2*ln(-c*x^2+1)*ln(c*x^2+ 
1)/x-1/4*(2*a-b*ln(-c*x^2+1))^2/x+I*b^2*c^(1/2)*polylog(2,1-2/(1-I*c^(1/2) 
*x))-b^2*c^(1/2)*arctanh(c^(1/2)*x)*ln(2*c^(1/2)*(1+(-c)^(1/2)*x)/((-c)^(1 
/2)+c^(1/2))/(1+c^(1/2)*x))-b^2*c^(1/2)*arctanh(c^(1/2)*x)*ln(-2*c^(1/2)*( 
1-(-c)^(1/2)*x)/((-c)^(1/2)-c^(1/2))/(1+c^(1/2)*x))+b^2*c^(1/2)*arctan(c^( 
1/2)*x)*ln((1+I)*(1-c^(1/2)*x)/(1-I*c^(1/2)*x))+b^2*c^(1/2)*arctan(c^(1/2) 
*x)*ln((1-I)*(1+c^(1/2)*x)/(1-I*c^(1/2)*x))+I*b^2*c^(1/2)*arctan(c^(1/2)*x 
)^2+b^2*c^(1/2)*arctanh(c^(1/2)*x)*ln(c*x^2+1)+b^2*c^(1/2)*arctan(c^(1/2)* 
x)*ln(c*x^2+1)-b^2*c^(1/2)*arctan(c^(1/2)*x)*ln(-c*x^2+1)+b*c^(1/2)*arctan 
h(c^(1/2)*x)*(2*a-b*ln(-c*x^2+1))+I*b^2*c^(1/2)*polylog(2,1-2/(1+I*c^(1/2) 
*x))+2*b^2*c^(1/2)*arctanh(c^(1/2)*x)*ln(2/(1+c^(1/2)*x))-2*b^2*c^(1/2)*ar 
ctanh(c^(1/2)*x)*ln(2/(1-c^(1/2)*x))+2*b^2*c^(1/2)*arctan(c^(1/2)*x)*ln(2/ 
(1+I*c^(1/2)*x))-2*b^2*c^(1/2)*arctan(c^(1/2)*x)*ln(2/(1-I*c^(1/2)*x))+2*a 
*b*c^(1/2)*arctan(c^(1/2)*x)-1/2*I*b^2*c^(1/2)*polylog(2,1+(-1+I)*(1+c^(1/ 
2)*x)/(1-I*c^(1/2)*x))-1/2*I*b^2*c^(1/2)*polylog(2,1-(1+I)*(1-c^(1/2)*x)/( 
1-I*c^(1/2)*x))
 

Mathematica [A] (verified)

Time = 2.32 (sec) , antiderivative size = 566, normalized size of antiderivative = 0.60 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^2} \, dx=\frac {-2 a^2-4 a b \text {arctanh}\left (c x^2\right )+4 a b \sqrt {c x^2} \left (\arctan \left (\sqrt {c x^2}\right )+\text {arctanh}\left (\sqrt {c x^2}\right )\right )+b^2 \sqrt {c x^2} \left (-2 i \arctan \left (\sqrt {c x^2}\right )^2+4 \arctan \left (\sqrt {c x^2}\right ) \text {arctanh}\left (c x^2\right )-\frac {2 \text {arctanh}\left (c x^2\right )^2}{\sqrt {c x^2}}+2 \arctan \left (\sqrt {c x^2}\right ) \log \left (1+e^{4 i \arctan \left (\sqrt {c x^2}\right )}\right )-2 \text {arctanh}\left (c x^2\right ) \log \left (1-\sqrt {c x^2}\right )+\log (2) \log \left (1-\sqrt {c x^2}\right )-\frac {1}{2} \log ^2\left (1-\sqrt {c x^2}\right )+\log \left (1-\sqrt {c x^2}\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i+\sqrt {c x^2}\right )\right )+2 \text {arctanh}\left (c x^2\right ) \log \left (1+\sqrt {c x^2}\right )-\log (2) \log \left (1+\sqrt {c x^2}\right )-\log \left (\frac {1}{2} \left ((1+i)-(1-i) \sqrt {c x^2}\right )\right ) \log \left (1+\sqrt {c x^2}\right )-\log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\sqrt {c x^2}\right )\right ) \log \left (1+\sqrt {c x^2}\right )+\frac {1}{2} \log ^2\left (1+\sqrt {c x^2}\right )+\log \left (1-\sqrt {c x^2}\right ) \log \left (\frac {1}{2} \left ((1+i)+(1-i) \sqrt {c x^2}\right )\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{4 i \arctan \left (\sqrt {c x^2}\right )}\right )-\operatorname {PolyLog}\left (2,\frac {1}{2} \left (1-\sqrt {c x^2}\right )\right )+\operatorname {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\sqrt {c x^2}\right )\right )+\operatorname {PolyLog}\left (2,\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\sqrt {c x^2}\right )\right )+\operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\sqrt {c x^2}\right )\right )-\operatorname {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\sqrt {c x^2}\right )\right )-\operatorname {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\sqrt {c x^2}\right )\right )\right )}{2 x} \] Input:

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

Output:

(-2*a^2 - 4*a*b*ArcTanh[c*x^2] + 4*a*b*Sqrt[c*x^2]*(ArcTan[Sqrt[c*x^2]] + 
ArcTanh[Sqrt[c*x^2]]) + b^2*Sqrt[c*x^2]*((-2*I)*ArcTan[Sqrt[c*x^2]]^2 + 4* 
ArcTan[Sqrt[c*x^2]]*ArcTanh[c*x^2] - (2*ArcTanh[c*x^2]^2)/Sqrt[c*x^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 + Log[1 - Sqrt[c*x^2]]*Log[(1/2 + I/2)*(-I + Sqrt[c*x^2])] + 2*Arc 
Tanh[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 - S 
qrt[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])] + 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])]))/(2*x)
 

Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 942, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6456, 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.

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

\(\Big \downarrow \) 6456

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

2*a*b*Sqrt[c]*ArcTan[Sqrt[c]*x] + I*b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]^2 + b^2* 
Sqrt[c]*ArcTanh[Sqrt[c]*x]^2 - 2*b^2*Sqrt[c]*ArcTanh[Sqrt[c]*x]*Log[2/(1 - 
 Sqrt[c]*x)] - 2*b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)] + 
b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c] 
*x)] + 2*b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)] + 2*b^2*Sq 
rt[c]*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)] - b^2*Sqrt[c]*ArcTanh[Sqrt 
[c]*x]*Log[(-2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c 
]*x))] - b^2*Sqrt[c]*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/( 
(Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))] + b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log 
[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] - b^2*Sqrt[c]*ArcTan[Sqrt[c] 
*x]*Log[1 - c*x^2] + b*Sqrt[c]*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]) 
 - (2*a - b*Log[1 - c*x^2])^2/(4*x) - (a*b*Log[1 + c*x^2])/x + b^2*Sqrt[c] 
*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2] + b^2*Sqrt[c]*ArcTanh[Sqrt[c]*x]*Log[1 + 
 c*x^2] + (b^2*Log[1 - c*x^2]*Log[1 + c*x^2])/(2*x) - (b^2*Log[1 + c*x^2]^ 
2)/(4*x) - b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)] + I*b^2*Sqrt[c]*P 
olyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)] - (I/2)*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 
+ I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] + I*b^2*Sqrt[c]*PolyLog[2, 1 - 2/ 
(1 + I*Sqrt[c]*x)] - b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)] + (b^2* 
Sqrt[c]*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])* 
(1 + Sqrt[c]*x))])/2 + (b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt...
 

Defintions of rubi rules used

rule 2009

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

rule 6456

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

Maple [F]

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

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]

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

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

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

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

Output:

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

Giac [F]

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

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{x^2} \, dx=\frac {2 \sqrt {c}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}}\right ) a b x -2 \sqrt {c}\, \mathit {atanh} \left (c \,x^{2}\right ) a b x -2 \mathit {atanh} \left (c \,x^{2}\right ) a b -2 \sqrt {c}\, \mathrm {log}\left (\sqrt {c}\, x -1\right ) a b x +\sqrt {c}\, \mathrm {log}\left (c \,x^{2}+1\right ) a b x +\left (\int \frac {\mathit {atanh} \left (c \,x^{2}\right )^{2}}{x^{2}}d x \right ) b^{2} x -a^{2}}{x} \] Input:

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

Output:

(2*sqrt(c)*atan((c*x)/sqrt(c))*a*b*x - 2*sqrt(c)*atanh(c*x**2)*a*b*x - 2*a 
tanh(c*x**2)*a*b - 2*sqrt(c)*log(sqrt(c)*x - 1)*a*b*x + sqrt(c)*log(c*x**2 
 + 1)*a*b*x + int(atanh(c*x**2)**2/x**2,x)*b**2*x - a**2)/x