\(\int x^2 (a+b \text {arctanh}(c x^2))^2 \, dx\) [72]

Optimal result

Integrand size = 16, antiderivative size = 1129 \[ \int x^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx =\text {Too large to display} \] Output:

-2/9*a*b*x^3-1/6*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^(3/2)-1/6*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^(3/2)+4/3*b^2*arctan(c^(1/2)*x)/c 
^(3/2)-1/3*b^2*arctanh(c^(1/2)*x)^2/c^(3/2)-4/3*b^2*arctanh(c^(1/2)*x)/c^( 
3/2)+1/3*b^2*polylog(2,1-2/(1-c^(1/2)*x))/c^(3/2)+1/3*b^2*polylog(2,1-2/(1 
+c^(1/2)*x))/c^(3/2)+1/9*b*x^3*(2*a-b*ln(-c*x^2+1))+1/9*b^2*x^3*ln(-c*x^2+ 
1)+1/12*b^2*x^3*ln(c*x^2+1)^2+2/3*b^2*x*ln(c*x^2+1)/c+1/3*a*b*x^3*ln(c*x^2 
+1)-1/6*b^2*x^3*ln(-c*x^2+1)*ln(c*x^2+1)-2/3*b^2*x*ln(-c*x^2+1)/c+1/12*x^3 
*(2*a-b*ln(-c*x^2+1))^2+4/3*a*b*x/c+1/6*I*b^2*polylog(2,1+(-1+I)*(1+c^(1/2 
)*x)/(1-I*c^(1/2)*x))/c^(3/2)+1/6*I*b^2*polylog(2,1-(1+I)*(1-c^(1/2)*x)/(1 
-I*c^(1/2)*x))/c^(3/2)+1/3*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^(3/2)+1/3*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^(3 
/2)-2/3*b^2*arctanh(c^(1/2)*x)*ln(2/(1+c^(1/2)*x))/c^(3/2)+2/3*b^2*arctanh 
(c^(1/2)*x)*ln(2/(1-c^(1/2)*x))/c^(3/2)-1/3*b^2*arctan(c^(1/2)*x)*ln((1+I) 
*(1-c^(1/2)*x)/(1-I*c^(1/2)*x))/c^(3/2)-1/3*b^2*arctan(c^(1/2)*x)*ln((1-I) 
*(1+c^(1/2)*x)/(1-I*c^(1/2)*x))/c^(3/2)-2/3*b^2*arctan(c^(1/2)*x)*ln(2/(1+ 
I*c^(1/2)*x))/c^(3/2)+2/3*b^2*arctan(c^(1/2)*x)*ln(2/(1-I*c^(1/2)*x))/c^(3 
/2)-2/3*a*b*arctan(c^(1/2)*x)/c^(3/2)-1/3*b^2*arctanh(c^(1/2)*x)*ln(c*x^2+ 
1)/c^(3/2)-1/3*b^2*arctan(c^(1/2)*x)*ln(c*x^2+1)/c^(3/2)-1/3*b*arctanh(...
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 2.50 (sec) , antiderivative size = 1129, 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.

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

\(\Big \downarrow \) 6456

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{12} \left (2 a-b \log \left (1-c x^2\right )\right )^2 x^3+\frac {1}{12} b^2 \log ^2\left (c x^2+1\right ) x^3-\frac {2}{9} a b x^3+\frac {1}{9} b^2 \log \left (1-c x^2\right ) x^3+\frac {1}{9} b \left (2 a-b \log \left (1-c x^2\right )\right ) x^3+\frac {1}{3} a b \log \left (c x^2+1\right ) x^3-\frac {1}{6} b^2 \log \left (1-c x^2\right ) \log \left (c x^2+1\right ) x^3-\frac {2 b^2 \log \left (1-c x^2\right ) x}{3 c}+\frac {2 b^2 \log \left (c x^2+1\right ) x}{3 c}+\frac {4 a b x}{3 c}-\frac {i b^2 \arctan \left (\sqrt {c} x\right )^2}{3 c^{3/2}}-\frac {b^2 \text {arctanh}\left (\sqrt {c} x\right )^2}{3 c^{3/2}}+\frac {4 b^2 \arctan \left (\sqrt {c} x\right )}{3 c^{3/2}}-\frac {2 a b \arctan \left (\sqrt {c} x\right )}{3 c^{3/2}}-\frac {4 b^2 \text {arctanh}\left (\sqrt {c} x\right )}{3 c^{3/2}}+\frac {2 b^2 \text {arctanh}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{3 c^{3/2}}+\frac {2 b^2 \arctan \left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{3 c^{3/2}}-\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 )}{3 c^{3/2}}-\frac {2 b^2 \arctan \left (\sqrt {c} x\right ) \log \left (\frac {2}{i \sqrt {c} x+1}\right )}{3 c^{3/2}}-\frac {2 b^2 \text {arctanh}\left (\sqrt {c} x\right ) \log \left (\frac {2}{\sqrt {c} x+1}\right )}{3 c^{3/2}}+\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 )}{3 c^{3/2}}+\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 )}{3 c^{3/2}}-\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 )}{3 c^{3/2}}+\frac {b^2 \arctan \left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{3 c^{3/2}}-\frac {b \text {arctanh}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{3 c^{3/2}}-\frac {b^2 \arctan \left (\sqrt {c} x\right ) \log \left (c x^2+1\right )}{3 c^{3/2}}-\frac {b^2 \text {arctanh}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right )}{3 c^{3/2}}+\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt {c} x}\right )}{3 c^{3/2}}-\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i \sqrt {c} x}\right )}{3 c^{3/2}}+\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{6 c^{3/2}}-\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i \sqrt {c} x+1}\right )}{3 c^{3/2}}+\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt {c} x+1}\right )}{3 c^{3/2}}-\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 )}{6 c^{3/2}}-\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 )}{6 c^{3/2}}+\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right )}{6 c^{3/2}}\)

Input:

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

Output:

(4*a*b*x)/(3*c) - (2*a*b*x^3)/9 - (2*a*b*ArcTan[Sqrt[c]*x])/(3*c^(3/2)) + 
(4*b^2*ArcTan[Sqrt[c]*x])/(3*c^(3/2)) - ((I/3)*b^2*ArcTan[Sqrt[c]*x]^2)/c^ 
(3/2) - (4*b^2*ArcTanh[Sqrt[c]*x])/(3*c^(3/2)) - (b^2*ArcTanh[Sqrt[c]*x]^2 
)/(3*c^(3/2)) + (2*b^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/(3*c^(3/ 
2)) + (2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/(3*c^(3/2)) - (b^ 
2*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(3*c 
^(3/2)) - (2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/(3*c^(3/2)) - 
 (2*b^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)])/(3*c^(3/2)) + (b^2*ArcT 
anh[Sqrt[c]*x]*Log[(-2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 
+ Sqrt[c]*x))])/(3*c^(3/2)) + (b^2*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + 
Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(3*c^(3/2)) - (b^2*A 
rcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(3*c^(3 
/2)) - (2*b^2*x*Log[1 - c*x^2])/(3*c) + (b^2*x^3*Log[1 - c*x^2])/9 + (b^2* 
ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/(3*c^(3/2)) + (b*x^3*(2*a - b*Log[1 - c* 
x^2]))/9 - (b*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]))/(3*c^(3/2)) + ( 
x^3*(2*a - b*Log[1 - c*x^2])^2)/12 + (2*b^2*x*Log[1 + c*x^2])/(3*c) + (a*b 
*x^3*Log[1 + c*x^2])/3 - (b^2*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2])/(3*c^(3/2) 
) - (b^2*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/(3*c^(3/2)) - (b^2*x^3*Log[1 - 
 c*x^2]*Log[1 + c*x^2])/6 + (b^2*x^3*Log[1 + c*x^2]^2)/12 + (b^2*PolyLog[2 
, 1 - 2/(1 - Sqrt[c]*x)])/(3*c^(3/2)) - ((I/3)*b^2*PolyLog[2, 1 - 2/(1 ...
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/3*a^2*x^3 + 1/3*(2*x^3*arctanh(c*x^2) + c*(4*x/c^2 - 2*arctan(sqrt(c)*x) 
/c^(5/2) + log((c*x - sqrt(c))/(c*x + sqrt(c)))/c^(5/2)))*a*b + 1/12*(x^3* 
log(-c*x^2 + 1)^2 - 3*integrate(-1/3*(3*(c*x^4 - x^2)*log(c*x^2 + 1)^2 - 2 
*(2*c*x^4 + 3*(c*x^4 - x^2)*log(c*x^2 + 1))*log(-c*x^2 + 1))/(c*x^2 - 1), 
x))*b^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\frac {-2 \sqrt {c}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}}\right ) a b +4 \sqrt {c}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}}\right ) b^{2}+\mathit {atanh} \left (c \,x^{2}\right )^{2} b^{2} c^{2} x^{3}+2 \sqrt {c}\, \mathit {atanh} \left (c \,x^{2}\right ) a b +4 \sqrt {c}\, \mathit {atanh} \left (c \,x^{2}\right ) b^{2}+2 \mathit {atanh} \left (c \,x^{2}\right ) a b \,c^{2} x^{3}+4 \mathit {atanh} \left (c \,x^{2}\right ) b^{2} c x +2 \sqrt {c}\, \mathrm {log}\left (\sqrt {c}\, x -1\right ) a b +4 \sqrt {c}\, \mathrm {log}\left (\sqrt {c}\, x -1\right ) b^{2}-\sqrt {c}\, \mathrm {log}\left (c \,x^{2}+1\right ) a b -2 \sqrt {c}\, \mathrm {log}\left (c \,x^{2}+1\right ) b^{2}+4 \left (\int \frac {\mathit {atanh} \left (c \,x^{2}\right )}{c^{2} x^{4}-1}d x \right ) b^{2} c +a^{2} c^{2} x^{3}+4 a b c x}{3 c^{2}} \] Input:

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

Output:

( - 2*sqrt(c)*atan((c*x)/sqrt(c))*a*b + 4*sqrt(c)*atan((c*x)/sqrt(c))*b**2 
 + atanh(c*x**2)**2*b**2*c**2*x**3 + 2*sqrt(c)*atanh(c*x**2)*a*b + 4*sqrt( 
c)*atanh(c*x**2)*b**2 + 2*atanh(c*x**2)*a*b*c**2*x**3 + 4*atanh(c*x**2)*b* 
*2*c*x + 2*sqrt(c)*log(sqrt(c)*x - 1)*a*b + 4*sqrt(c)*log(sqrt(c)*x - 1)*b 
**2 - sqrt(c)*log(c*x**2 + 1)*a*b - 2*sqrt(c)*log(c*x**2 + 1)*b**2 + 4*int 
(atanh(c*x**2)/(c**2*x**4 - 1),x)*b**2*c + a**2*c**2*x**3 + 4*a*b*c*x)/(3* 
c**2)