3.2.78 \(\int (a+b \text {arctanh}(\frac {c}{x^2}))^2 \, dx\) [178]

3.2.78.1 Optimal result

Integrand size = 12, antiderivative size = 1549 \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2 \, dx =\text {Too large to display} \]

output

-b^2*polylog(2,1-x/(-c)^(1/2))*(-c)^(1/2)+b^2*polylog(2,1+x/(-c)^(1/2))*(- 
c)^(1/2)-b^2*polylog(2,1-x/c^(1/2))*c^(1/2)-b^2*polylog(2,x/c^(1/2))*c^(1/ 
2)+b^2*polylog(2,1+x/c^(1/2))*c^(1/2)+b^2*polylog(2,-x/c^(1/2))*c^(1/2)+b^ 
2*polylog(2,1-2*c^(1/2)/(x+c^(1/2)))*c^(1/2)-a*b*x*ln(1-c/x^2)-b^2*ln(x/(- 
c)^(1/2))*ln(-x+(-c)^(1/2))*(-c)^(1/2)+a*b*x*ln(1+c/x^2)+b^2*ln(-x/(-c)^(1 
/2))*ln(x+(-c)^(1/2))*(-c)^(1/2)-b^2*arctan(x/c^(1/2))*ln(1-c/x^2)*c^(1/2) 
-b^2*arctanh(x/c^(1/2))*ln(1+c/x^2)*c^(1/2)-b^2*ln(x/c^(1/2))*ln(-x+c^(1/2 
))*c^(1/2)+b^2*arctan(x/c^(1/2))*ln((1+I)*(-x+c^(1/2))/(-I*x+c^(1/2)))*c^( 
1/2)+b^2*arctanh(x/c^(1/2))*ln(2*(-x+(-c)^(1/2))*c^(1/2)/((-c)^(1/2)-c^(1/ 
2))/(x+c^(1/2)))*c^(1/2)+b^2*ln(-x/c^(1/2))*ln(x+c^(1/2))*c^(1/2)+b^2*arct 
an(x/c^(1/2))*ln((1-I)*(x+c^(1/2))/(-I*x+c^(1/2)))*c^(1/2)+b^2*arctanh(x/c 
^(1/2))*ln(2*(x+(-c)^(1/2))*c^(1/2)/(x+c^(1/2))/((-c)^(1/2)+c^(1/2)))*c^(1 
/2)+I*b^2*polylog(2,I*x/c^(1/2))*c^(1/2)+I*b^2*polylog(2,1-2*c^(1/2)/(-I*x 
+c^(1/2)))*c^(1/2)-1/2*b^2*x*ln(1-c/x^2)*ln(1+c/x^2)-1/2*b^2*ln(1+c/x^2)*l 
n(-x+(-c)^(1/2))*(-c)^(1/2)+1/2*b^2*ln(1+c/x^2)*ln(x+(-c)^(1/2))*(-c)^(1/2 
)-1/2*b^2*ln(1/2*(-x+(-c)^(1/2))/(-c)^(1/2))*ln(x+(-c)^(1/2))*(-c)^(1/2)+1 
/2*b^2*ln(-x+(-c)^(1/2))*ln(1/2*(x+(-c)^(1/2))/(-c)^(1/2))*(-c)^(1/2)+2*a* 
b*arctan(x/c^(1/2))*c^(1/2)-2*a*b*arctanh(x/c^(1/2))*c^(1/2)-1/2*b^2*ln(1- 
c/x^2)*ln(-x+c^(1/2))*c^(1/2)-2*b^2*arctan(x/c^(1/2))*ln(2*c^(1/2)/(-I*x+c 
^(1/2)))*c^(1/2)-2*b^2*arctanh(x/c^(1/2))*ln(2*c^(1/2)/(x+c^(1/2)))*c^(...
 

3.2.78.2 Mathematica [A] (verified)

Time = 2.41 (sec) , antiderivative size = 565, normalized size of antiderivative = 0.36 \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2 \, dx=a^2 x-2 a b \sqrt {\frac {c}{x^2}} x \left (\arctan \left (\sqrt {\frac {c}{x^2}}\right )+\text {arctanh}\left (\sqrt {\frac {c}{x^2}}\right )\right )+2 a b x \text {arctanh}\left (\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {\frac {c}{x^2}} x \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 )-\frac {2 \text {arctanh}\left (\frac {c}{x^2}\right )^2}{\sqrt {\frac {c}{x^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 ) \]

input

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

output

a^2*x - 2*a*b*Sqrt[c/x^2]*x*(ArcTan[Sqrt[c/x^2]] + ArcTanh[Sqrt[c/x^2]]) + 
 2*a*b*x*ArcTanh[c/x^2] - (b^2*Sqrt[c/x^2]*x*((-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*ArcTan 
h[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*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] + PolyL 
og[2, (-1/2 - I/2)*(-1 + Sqrt[c/x^2])] + PolyLog[2, (-1/2 + I/2)*(-1 + Sqr 
t[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
 

3.2.78.3 Rubi [A] (verified)

Time = 1.75 (sec) , antiderivative size = 1050, normalized size of antiderivative = 0.68, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6440, 6439, 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.

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

\(\Big \downarrow \) 6440

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

\(\Big \downarrow \) 6439

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

\(\Big \downarrow \) 2009

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

input

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

output

a^2*x + 2*a*b*Sqrt[c]*ArcTan[x/Sqrt[c]] - I*b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]^ 
2 - 2*a*b*Sqrt[c]*ArcTanh[x/Sqrt[c]] + b^2*Sqrt[c]*ArcTanh[x/Sqrt[c]]^2 + 
2*b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] - I*x)] - a*b 
*x*Log[1 - c/x^2] - b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[1 - c/x^2] + b^2*Sqr 
t[c]*ArcTanh[x/Sqrt[c]]*Log[1 - c/x^2] + (b^2*x*Log[1 - c/x^2]^2)/4 + a*b* 
x*Log[1 + c/x^2] + b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[1 + c/x^2] - b^2*Sqrt 
[c]*ArcTanh[x/Sqrt[c]]*Log[1 + c/x^2] - (b^2*x*Log[1 - c/x^2]*Log[1 + c/x^ 
2])/2 + (b^2*x*Log[1 + c/x^2]^2)/4 - 2*b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[( 
2*Sqrt[c])/(Sqrt[c] - I*x)] + b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[((1 + I)*( 
Sqrt[c] - x))/(Sqrt[c] - I*x)] - 2*b^2*Sqrt[c]*ArcTanh[x/Sqrt[c]]*Log[(2*S 
qrt[c])/(Sqrt[c] + x)] + b^2*Sqrt[c]*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sq 
rt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))] + b^2*Sqrt[c]*ArcTanh[x 
/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + 
x))] + b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[((1 - I)*(Sqrt[c] + x))/(Sqrt[c] 
- I*x)] + 2*b^2*Sqrt[c]*ArcTanh[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] + 
x)] + I*b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] - I*x)] - I*b^2*Sq 
rt[c]*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] - I*x)] - (I/2)*b^2*Sqrt[c]*Pol 
yLog[2, 1 - ((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)] + b^2*Sqrt[c]*PolyLog 
[2, -(x/Sqrt[c])] - I*b^2*Sqrt[c]*PolyLog[2, ((-I)*x)/Sqrt[c]] + I*b^2*Sqr 
t[c]*PolyLog[2, (I*x)/Sqrt[c]] - b^2*Sqrt[c]*PolyLog[2, x/Sqrt[c]] + b^...
 

3.2.78.3.1 Defintions of rubi rules used

rule 2009

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

rule 6439

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_)]*(b_.))^(p_), x_Symbol] :> Int[ExpandI 
ntegrand[(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]
 

rule 6440

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

3.2.78.4 Maple [F]

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

input

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

output

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

3.2.78.5 Fricas [F]

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

input

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

output

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

3.2.78.6 Sympy [F]

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

input

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

output

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

3.2.78.7 Maxima [F]

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

input

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

output

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

3.2.78.8 Giac [F]

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

input

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

output

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

3.2.78.9 Mupad [F(-1)]

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

input

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

output

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