\(\int \frac {x^2 \text {arctanh}(a x)}{(1-a^2 x^2)^3} \, dx\) [303]

Optimal result

Integrand size = 20, antiderivative size = 100 \[ \int \frac {x^2 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {1}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac {1}{16 a^3 \left (1-a^2 x^2\right )}+\frac {x \text {arctanh}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \text {arctanh}(a x)}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)^2}{16 a^3} \] Output:

-1/16/a^3/(-a^2*x^2+1)^2+1/16/a^3/(-a^2*x^2+1)+1/4*x*arctanh(a*x)/a^2/(-a^ 
2*x^2+1)^2-1/8*x*arctanh(a*x)/a^2/(-a^2*x^2+1)-1/16*arctanh(a*x)^2/a^3
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.61 \[ \int \frac {x^2 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {a^2 x^2-2 \left (a x+a^3 x^3\right ) \text {arctanh}(a x)+\left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^2}{16 a^3 \left (-1+a^2 x^2\right )^2} \] Input:

Integrate[(x^2*ArcTanh[a*x])/(1 - a^2*x^2)^3,x]
 

Output:

-1/16*(a^2*x^2 - 2*(a*x + a^3*x^3)*ArcTanh[a*x] + (-1 + a^2*x^2)^2*ArcTanh 
[a*x]^2)/(a^3*(-1 + a^2*x^2)^2)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6560, 6518, 241}

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.

\(\displaystyle \int \frac {x^2 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx\)

\(\Big \downarrow \) 6560

\(\displaystyle -\frac {\int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{4 a^2}+\frac {x \text {arctanh}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a^3 \left (1-a^2 x^2\right )^2}\)

\(\Big \downarrow \) 6518

\(\displaystyle -\frac {-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{4 a^2}+\frac {x \text {arctanh}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a^3 \left (1-a^2 x^2\right )^2}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {x \text {arctanh}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{4 a^2}-\frac {1}{16 a^3 \left (1-a^2 x^2\right )^2}\)

Input:

Int[(x^2*ArcTanh[a*x])/(1 - a^2*x^2)^3,x]
 

Output:

-1/16*1/(a^3*(1 - a^2*x^2)^2) + (x*ArcTanh[a*x])/(4*a^2*(1 - a^2*x^2)^2) - 
 (-1/4*1/(a*(1 - a^2*x^2)) + (x*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + ArcTanh[ 
a*x]^2/(4*a))/(4*a^2)
 

Defintions of rubi rules used

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 6518

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Sy 
mbol] :> Simp[x*((a + b*ArcTanh[c*x])^p/(2*d*(d + e*x^2))), x] + (Simp[(a + 
 b*ArcTanh[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] - Simp[b*c*(p/2)   Int[x*( 
(a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
 

rule 6560

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^2*((d_) + (e_.)*(x_)^2)^(q_), 
x_Symbol] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c^3*d*(q + 1)^2)), x] + (-Si 
mp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(2*c^2*d*(q + 1))), x] + Sim 
p[1/(2*c^2*d*(q + 1))   Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x 
]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q 
, -5/2]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.78

method	result	size
parallelrisch	\(-\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}-2 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )-2 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}+a^{2} x^{2}-2 a x \,\operatorname {arctanh}\left (a x \right )+\operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a^{2} x^{2}-1\right )^{2} a^{3}}\)	\(78\)
derivativedivides	\(\frac {\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )}{16 a x -16}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{16}-\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )}{16 a x +16}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{16}+\frac {\ln \left (a x +1\right )^{2}}{64}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{32}+\frac {\ln \left (a x -1\right )^{2}}{64}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {1}{64 \left (a x -1\right )^{2}}-\frac {1}{64 \left (a x -1\right )}-\frac {1}{64 \left (a x +1\right )^{2}}+\frac {1}{64 a x +64}}{a^{3}}\)	\(178\)
default	\(\frac {\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )}{16 a x -16}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{16}-\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )}{16 a x +16}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{16}+\frac {\ln \left (a x +1\right )^{2}}{64}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{32}+\frac {\ln \left (a x -1\right )^{2}}{64}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {1}{64 \left (a x -1\right )^{2}}-\frac {1}{64 \left (a x -1\right )}-\frac {1}{64 \left (a x +1\right )^{2}}+\frac {1}{64 a x +64}}{a^{3}}\)	\(178\)
risch	\(-\frac {\ln \left (a x +1\right )^{2}}{64 a^{3}}+\frac {\left (x^{4} \ln \left (-a x +1\right ) a^{4}+2 a^{3} x^{3}-2 x^{2} \ln \left (-a x +1\right ) a^{2}+2 a x +\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{32 a^{3} \left (a^{2} x^{2}-1\right )^{2}}-\frac {a^{4} x^{4} \ln \left (-a x +1\right )^{2}+4 a^{3} x^{3} \ln \left (-a x +1\right )-2 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+4 a^{2} x^{2}+4 a x \ln \left (-a x +1\right )+\ln \left (-a x +1\right )^{2}}{64 a^{3} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\)	\(193\)
parts	\(-\frac {\operatorname {arctanh}\left (a x \right )}{16 a^{3} \left (a x +1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )}{16 a^{3} \left (a x +1\right )}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{16 a^{3}}+\frac {\operatorname {arctanh}\left (a x \right )}{16 a^{3} \left (a x -1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )}{16 a^{3} \left (a x -1\right )}+\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{16 a^{3}}-\frac {a \left (\frac {-\frac {\ln \left (a x +1\right )^{2}}{4}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}}{a^{4}}-\frac {\frac {\ln \left (a x -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}}{a^{4}}-\frac {2 \left (-\frac {1}{8 a^{2} \left (a x +1\right )^{2}}+\frac {1}{8 \left (a x +1\right ) a^{2}}-\frac {1}{8 a^{2} \left (a x -1\right )^{2}}-\frac {1}{8 \left (a x -1\right ) a^{2}}\right )}{a^{2}}\right )}{16}\)	\(243\)

Input:

int(x^2*arctanh(a*x)/(-a^2*x^2+1)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/16*(a^4*x^4*arctanh(a*x)^2-2*a^3*x^3*arctanh(a*x)-2*a^2*x^2*arctanh(a*x 
)^2+a^2*x^2-2*a*x*arctanh(a*x)+arctanh(a*x)^2)/(a^2*x^2-1)^2/a^3
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.95 \[ \int \frac {x^2 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {4 \, a^{2} x^{2} + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 4 \, {\left (a^{3} x^{3} + a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{64 \, {\left (a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}\right )}} \] Input:

integrate(x^2*arctanh(a*x)/(-a^2*x^2+1)^3,x, algorithm="fricas")
 

Output:

-1/64*(4*a^2*x^2 + (a^4*x^4 - 2*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^2 - 
 4*(a^3*x^3 + a*x)*log(-(a*x + 1)/(a*x - 1)))/(a^7*x^4 - 2*a^5*x^2 + a^3)
 

Sympy [F]

\[ \int \frac {x^2 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=- \int \frac {x^{2} \operatorname {atanh}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \] Input:

integrate(x**2*atanh(a*x)/(-a**2*x**2+1)**3,x)
 

Output:

-Integral(x**2*atanh(a*x)/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (86) = 172\).

Time = 0.03 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.79 \[ \int \frac {x^2 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=\frac {1}{16} \, {\left (\frac {2 \, {\left (a^{2} x^{3} + x\right )}}{a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {artanh}\left (a x\right ) - \frac {{\left (4 \, a^{2} x^{2} - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2}\right )} a}{64 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \] Input:

integrate(x^2*arctanh(a*x)/(-a^2*x^2+1)^3,x, algorithm="maxima")
 

Output:

1/16*(2*(a^2*x^3 + x)/(a^6*x^4 - 2*a^4*x^2 + a^2) - log(a*x + 1)/a^3 + log 
(a*x - 1)/a^3)*arctanh(a*x) - 1/64*(4*a^2*x^2 - (a^4*x^4 - 2*a^2*x^2 + 1)* 
log(a*x + 1)^2 + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(a*x - 1) - ( 
a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2)*a/(a^8*x^4 - 2*a^6*x^2 + a^4)
 

Giac [F]

\[ \int \frac {x^2 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=\int { -\frac {x^{2} \operatorname {artanh}\left (a x\right )}{{\left (a^{2} x^{2} - 1\right )}^{3}} \,d x } \] Input:

integrate(x^2*arctanh(a*x)/(-a^2*x^2+1)^3,x, algorithm="giac")
 

Output:

integrate(-x^2*arctanh(a*x)/(a^2*x^2 - 1)^3, x)
 

Mupad [B] (verification not implemented)

Time = 3.95 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.50 \[ \int \frac {x^2 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=\ln \left (1-a\,x\right )\,\left (\frac {\ln \left (a\,x+1\right )}{32\,a^3}-\frac {\frac {x}{8\,a^2}+\frac {x^3}{8}}{2\,a^4\,x^4-4\,a^2\,x^2+2}\right )-\frac {{\ln \left (a\,x+1\right )}^2}{64\,a^3}-\frac {{\ln \left (1-a\,x\right )}^2}{64\,a^3}-\frac {x^2}{2\,\left (8\,a^5\,x^4-16\,a^3\,x^2+8\,a\right )}+\frac {\ln \left (a\,x+1\right )\,\left (\frac {x}{16\,a^3}+\frac {x^3}{16\,a}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4} \] Input:

int(-(x^2*atanh(a*x))/(a^2*x^2 - 1)^3,x)
 

Output:

log(1 - a*x)*(log(a*x + 1)/(32*a^3) - (x/(8*a^2) + x^3/8)/(2*a^4*x^4 - 4*a 
^2*x^2 + 2)) - log(a*x + 1)^2/(64*a^3) - log(1 - a*x)^2/(64*a^3) - x^2/(2* 
(8*a - 16*a^3*x^2 + 8*a^5*x^4)) + (log(a*x + 1)*(x/(16*a^3) + x^3/(16*a))) 
/(1/a - 2*a*x^2 + a^3*x^4)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=\frac {-2 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+4 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-2 \mathit {atanh} \left (a x \right )^{2}+4 \mathit {atanh} \left (a x \right ) a^{3} x^{3}+4 \mathit {atanh} \left (a x \right ) a x -a^{4} x^{4}-1}{32 a^{3} \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )} \] Input:

int(x^2*atanh(a*x)/(-a^2*x^2+1)^3,x)
 

Output:

( - 2*atanh(a*x)**2*a**4*x**4 + 4*atanh(a*x)**2*a**2*x**2 - 2*atanh(a*x)** 
2 + 4*atanh(a*x)*a**3*x**3 + 4*atanh(a*x)*a*x - a**4*x**4 - 1)/(32*a**3*(a 
**4*x**4 - 2*a**2*x**2 + 1))