\(\int (a+b \text {arctanh}(c x^2))^2 \, dx\) [73]

Optimal result

Integrand size = 12, antiderivative size = 958 \[ \int \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx =\text {Too large to display} \] Output:

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

Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 566, normalized size of antiderivative = 0.59 \[ \int \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\frac {1}{2} x \left (2 a^2+4 a b \text {arctanh}\left (c x^2\right )+\frac {4 a b \left (\arctan \left (\sqrt {c x^2}\right )-\text {arctanh}\left (\sqrt {c x^2}\right )\right )}{\sqrt {c x^2}}+\frac {b^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 )+2 \sqrt {c x^2} \text {arctanh}\left (c x^2\right )^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 )}{\sqrt {c x^2}}\right ) \] Input:

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

Output:

(x*(2*a^2 + 4*a*b*ArcTanh[c*x^2] + (4*a*b*(ArcTan[Sqrt[c*x^2]] - ArcTanh[S 
qrt[c*x^2]]))/Sqrt[c*x^2] + (b^2*((-2*I)*ArcTan[Sqrt[c*x^2]]^2 + 4*ArcTan[ 
Sqrt[c*x^2]]*ArcTanh[c*x^2] + 2*Sqrt[c*x^2]*ArcTanh[c*x^2]^2 + 2*ArcTan[Sq 
rt[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*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)*Arc 
Tan[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])] - P 
olyLog[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
 

Rubi [A] (verified)

Time = 2.66 (sec) , antiderivative size = 958, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6438, 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.

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

\(\Big \downarrow \) 6438

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

a^2*x + (2*a*b*ArcTan[Sqrt[c]*x])/Sqrt[c] + (I*b^2*ArcTan[Sqrt[c]*x]^2)/Sq 
rt[c] - (2*a*b*ArcTanh[Sqrt[c]*x])/Sqrt[c] - (b^2*ArcTanh[Sqrt[c]*x]^2)/Sq 
rt[c] + (2*b^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/Sqrt[c] - (2*b^2 
*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (b^2*ArcTan[Sqrt[c] 
*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (2*b^2*Arc 
Tan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/Sqrt[c] - (2*b^2*ArcTanh[Sqrt[c]* 
x]*Log[2/(1 + Sqrt[c]*x)])/Sqrt[c] + (b^2*ArcTanh[Sqrt[c]*x]*Log[(-2*Sqrt[ 
c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/Sqrt[c] + (b 
^2*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c 
])*(1 + Sqrt[c]*x))])/Sqrt[c] + (b^2*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + S 
qrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] - a*b*x*Log[1 - c*x^2] - (b^2*ArcTa 
n[Sqrt[c]*x]*Log[1 - c*x^2])/Sqrt[c] + (b^2*ArcTanh[Sqrt[c]*x]*Log[1 - c*x 
^2])/Sqrt[c] + (b^2*x*Log[1 - c*x^2]^2)/4 + a*b*x*Log[1 + c*x^2] + (b^2*Ar 
cTan[Sqrt[c]*x]*Log[1 + c*x^2])/Sqrt[c] - (b^2*ArcTanh[Sqrt[c]*x]*Log[1 + 
c*x^2])/Sqrt[c] - (b^2*x*Log[1 - c*x^2]*Log[1 + c*x^2])/2 + (b^2*x*Log[1 + 
 c*x^2]^2)/4 + (b^2*PolyLog[2, 1 - 2/(1 - 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] + (I*b^2*PolyLog[2, 1 - 
 2/(1 + I*Sqrt[c]*x)])/Sqrt[c] + (b^2*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])/S 
qrt[c] - (b^2*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - ...
 

Defintions of rubi rules used

rule 2009

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

rule 6438

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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