\(\int x^2 (a+b \text {arctanh}(\frac {c}{x^2}))^2 \, dx\) [177]

Optimal result

Integrand size = 16, antiderivative size = 1172 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2 \, dx =\text {Too large to display} \] Output:

1/12*x^3*(2*a-b*ln(1-c/x^2))^2-1/3*I*b^2*c^(3/2)*polylog(2,I*x/c^(1/2))+1/ 
6*I*b^2*c^(3/2)*polylog(2,1+(-1+I)*(c^(1/2)+x)/(c^(1/2)-I*x))+1/6*I*b^2*c^ 
(3/2)*polylog(2,1-(1+I)*(c^(1/2)-x)/(c^(1/2)-I*x))+1/3*I*b^2*c^(3/2)*polyl 
og(2,-1+2*c^(1/2)/(c^(1/2)-I*x))+1/3*I*b^2*c^(3/2)*polylog(2,-I*x/c^(1/2)) 
+1/3*I*b^2*c^(3/2)*arctan(x/c^(1/2))^2+1/3*b^2*c^(3/2)*arctanh(x/c^(1/2))* 
ln(2*c^(1/2)*((-c)^(1/2)+x)/((-c)^(1/2)+c^(1/2))/(c^(1/2)+x))+1/3*b^2*c^(3 
/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))+2/3*b^2*c^(3/2)*arctanh(x/c^(1/2))*ln(2-2*c^(1/2)/(c^(1/2)+x))- 
2/3*b^2*c^(3/2)*arctanh(x/c^(1/2))*ln(2*c^(1/2)/(c^(1/2)+x))-1/3*b^2*c^(3/ 
2)*arctan(x/c^(1/2))*ln((1+I)*(c^(1/2)-x)/(c^(1/2)-I*x))-1/3*b^2*c^(3/2)*a 
rctan(x/c^(1/2))*ln((1-I)*(c^(1/2)+x)/(c^(1/2)-I*x))-2/3*b^2*c^(3/2)*arcta 
n(x/c^(1/2))*ln(2-2*c^(1/2)/(c^(1/2)-I*x))+2/3*b^2*c^(3/2)*arctan(x/c^(1/2 
))*ln(2*c^(1/2)/(c^(1/2)-I*x))-2/3*a*b*c^(3/2)*arctan(x/c^(1/2))-1/3*b^2*c 
^(3/2)*arctanh(x/c^(1/2))*ln(1+c/x^2)-1/3*b^2*c^(3/2)*arctan(x/c^(1/2))*ln 
(1+c/x^2)-1/3*b*c^(3/2)*arctanh(x/c^(1/2))*(2*a-b*ln(1-c/x^2))+1/3*b^2*c^( 
3/2)*arctan(x/c^(1/2))*ln(1-c/x^2)-1/3*I*b^2*c^(3/2)*polylog(2,1-2*c^(1/2) 
/(c^(1/2)-I*x))-1/6*b^2*c^(3/2)*polylog(2,1-2*c^(1/2)*((-c)^(1/2)+x)/((-c) 
^(1/2)+c^(1/2))/(c^(1/2)+x))-1/6*b^2*c^(3/2)*polylog(2,1-2*c^(1/2)*((-c)^( 
1/2)-x)/((-c)^(1/2)-c^(1/2))/(c^(1/2)+x))+1/3*b^2*c^(3/2)*polylog(2,1-2*c^ 
(1/2)/(c^(1/2)+x))-1/3*b^2*c^(3/2)*polylog(2,-1+2*c^(1/2)/(c^(1/2)+x))+...
 

Mathematica [F]

\[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2 \, dx=\int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2 \, dx \] 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 = 2.63 (sec) , antiderivative size = 1172, normalized size of antiderivative = 1.00, 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.

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

\(\Big \downarrow \) 6460

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

\(\Big \downarrow \) 6457

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(4*a*b*c*x)/3 - (2*a*b*c^(3/2)*ArcTan[x/Sqrt[c]])/3 + (4*b^2*c^(3/2)*ArcTa 
n[x/Sqrt[c]])/3 + (I/3)*b^2*c^(3/2)*ArcTan[x/Sqrt[c]]^2 - (4*b^2*c^(3/2)*A 
rcTanh[x/Sqrt[c]])/3 + (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]^2)/3 - (2*b^2*c^(3/ 
2)*ArcTan[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] - I*x)])/3 - (2*b^2*c*x* 
Log[1 - c/x^2])/3 + (b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[1 - c/x^2])/3 - (b* 
c^(3/2)*ArcTanh[x/Sqrt[c]]*(2*a - b*Log[1 - c/x^2]))/3 + (x^3*(2*a - b*Log 
[1 - c/x^2])^2)/12 + (2*b^2*c*x*Log[1 + c/x^2])/3 + (a*b*x^3*Log[1 + c/x^2 
])/3 - (b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[1 + c/x^2])/3 - (b^2*c^(3/2)*Arc 
Tanh[x/Sqrt[c]]*Log[1 + c/x^2])/3 - (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 + (2*b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log 
[(2*Sqrt[c])/(Sqrt[c] - I*x)])/3 - (b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[((1 
+ I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)])/3 - (2*b^2*c^(3/2)*ArcTanh[x/Sqrt[c] 
]*Log[(2*Sqrt[c])/(Sqrt[c] + x)])/3 + (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[ 
(2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))])/3 + (b^2 
*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sq 
rt[c])*(Sqrt[c] + x))])/3 - (b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[((1 - I)*(S 
qrt[c] + x))/(Sqrt[c] - I*x)])/3 + (2*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[2 
 - (2*Sqrt[c])/(Sqrt[c] + x)])/3 - (I/3)*b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqr 
t[c])/(Sqrt[c] - I*x)] + (I/3)*b^2*c^(3/2)*PolyLog[2, -1 + (2*Sqrt[c])/(Sq 
rt[c] - I*x)] + (I/6)*b^2*c^(3/2)*PolyLog[2, 1 - ((1 + I)*(Sqrt[c] - x)...
 

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]

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

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]

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

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

\[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )^2 \, dx=\int { {\left (b \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + a\right )}^{2} x^{2} \,d x } \] 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) - (2*sqrt(c)*arctan(x/sqrt(c)) - s 
qrt(c)*log((x - sqrt(c))/(x + sqrt(c))) - 4*x)*c)*a*b + 1/12*(x^3*log(x^2 
- c)^2 - 3*integrate(-1/3*(3*(x^4 - c*x^2)*log(x^2 + c)^2 - 2*(2*x^4 + 3*( 
x^4 - c*x^2)*log(x^2 + c))*log(x^2 - c))/(x^2 - c), x))*b^2
 

Giac [F]

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

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

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

Output:

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