\(\int \frac {\text {arctanh}(a x)}{(1-a^2 x^2)^3} \, dx\) [305]

Optimal result

Integrand size = 17, antiderivative size = 94 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {1}{16 a \left (1-a^2 x^2\right )^2}-\frac {3}{16 a \left (1-a^2 x^2\right )}+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arctanh}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {3 \text {arctanh}(a x)^2}{16 a} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.69 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=\frac {-4+3 a^2 x^2+\left (10 a x-6 a^3 x^3\right ) \text {arctanh}(a x)+3 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^2}{16 a \left (-1+a^2 x^2\right )^2} \] Input:

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

Output:

(-4 + 3*a^2*x^2 + (10*a*x - 6*a^3*x^3)*ArcTanh[a*x] + 3*(-1 + a^2*x^2)^2*A 
rcTanh[a*x]^2)/(16*a*(-1 + a^2*x^2)^2)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6522, 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.

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

\(\Big \downarrow \) 6522

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

\(\Big \downarrow \) 6518

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

\(\Big \downarrow \) 241

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

Input:

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

Output:

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

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 6522

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbo 
l] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + 
 e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/( 
2*d*(q + 1))   Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x]) /; Fre 
eQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
 

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.96

method	result	size
parallelrisch	\(-\frac {-3 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}-4 a^{4} x^{4}+6 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )+6 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}+5 a^{2} x^{2}-10 a x \,\operatorname {arctanh}\left (a x \right )-3 \operatorname {arctanh}\left (a x \right )^{2}}{16 \left (a^{2} x^{2}-1\right )^{2} a}\)	\(90\)
derivativedivides	\(\frac {\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{16}-\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{16}-\frac {3 \ln \left (a x -1\right )^{2}}{64}+\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {3 \ln \left (a x +1\right )^{2}}{64}+\frac {3 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {1}{64 \left (a x -1\right )^{2}}+\frac {7}{64 \left (a x -1\right )}-\frac {1}{64 \left (a x +1\right )^{2}}-\frac {7}{64 \left (a x +1\right )}}{a}\)	\(178\)
default	\(\frac {\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{16}-\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{16}-\frac {3 \ln \left (a x -1\right )^{2}}{64}+\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {3 \ln \left (a x +1\right )^{2}}{64}+\frac {3 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {1}{64 \left (a x -1\right )^{2}}+\frac {7}{64 \left (a x -1\right )}-\frac {1}{64 \left (a x +1\right )^{2}}-\frac {7}{64 \left (a x +1\right )}}{a}\)	\(178\)
risch	\(\frac {3 \ln \left (a x +1\right )^{2}}{64 a}-\frac {\left (3 x^{4} \ln \left (-a x +1\right ) a^{4}+6 a^{3} x^{3}-6 x^{2} \ln \left (-a x +1\right ) a^{2}-10 a x +3 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{32 \left (a^{2} x^{2}-1\right )^{2} a}+\frac {3 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+12 a^{3} x^{3} \ln \left (-a x +1\right )-6 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+12 a^{2} x^{2}-20 a x \ln \left (-a x +1\right )+3 \ln \left (-a x +1\right )^{2}-16}{64 a \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )}\)	\(200\)
parts	\(-\frac {\operatorname {arctanh}\left (a x \right )}{16 a \left (a x +1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 a \left (a x +1\right )}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{16 a}+\frac {\operatorname {arctanh}\left (a x \right )}{16 a \left (a x -1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 a \left (a x -1\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{16 a}-\frac {a \left (\frac {\frac {3 \ln \left (a x -1\right )^{2}}{4}-\frac {3 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}}{a^{2}}-\frac {3 \left (-\frac {\ln \left (a x +1\right )^{2}}{4}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}\right )}{a^{2}}+\frac {1}{4 a^{2} \left (a x +1\right )^{2}}+\frac {7}{4 \left (a x +1\right ) a^{2}}+\frac {1}{4 a^{2} \left (a x -1\right )^{2}}-\frac {7}{4 \left (a x -1\right ) a^{2}}\right )}{16}\)	\(238\)

Input:

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

Output:

-1/16*(-3*a^4*x^4*arctanh(a*x)^2-4*a^4*x^4+6*a^3*x^3*arctanh(a*x)+6*a^2*x^ 
2*arctanh(a*x)^2+5*a^2*x^2-10*a*x*arctanh(a*x)-3*arctanh(a*x)^2)/(a^2*x^2- 
1)^2/a
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.03 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=\frac {12 \, a^{2} x^{2} + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 4 \, {\left (3 \, a^{3} x^{3} - 5 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 16}{64 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )}} \] Input:

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

Output:

1/64*(12*a^2*x^2 + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^2 
 - 4*(3*a^3*x^3 - 5*a*x)*log(-(a*x + 1)/(a*x - 1)) - 16)/(a^5*x^4 - 2*a^3* 
x^2 + a)
 

Sympy [F]

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

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

Output:

-Integral(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. 182 vs. \(2 (80) = 160\).

Time = 0.03 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.94 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {1}{16} \, {\left (\frac {2 \, {\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - \frac {3 \, \log \left (a x + 1\right )}{a} + \frac {3 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right ) + \frac {{\left (12 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16\right )} a}{64 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )}} \] Input:

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

Output:

-1/16*(2*(3*a^2*x^3 - 5*x)/(a^4*x^4 - 2*a^2*x^2 + 1) - 3*log(a*x + 1)/a + 
3*log(a*x - 1)/a)*arctanh(a*x) + 1/64*(12*a^2*x^2 - 3*(a^4*x^4 - 2*a^2*x^2 
 + 1)*log(a*x + 1)^2 + 6*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(a*x - 
1) - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 16)*a/(a^6*x^4 - 2*a^4*x 
^2 + a^2)
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 4.00 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.64 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=\frac {\frac {3\,a\,x^2}{2}-\frac {2}{a}}{8\,a^4\,x^4-16\,a^2\,x^2+8}-\ln \left (1-a\,x\right )\,\left (\frac {3\,\ln \left (a\,x+1\right )}{32\,a}+\frac {\frac {5\,x}{8}-\frac {3\,a^2\,x^3}{8}}{2\,a^4\,x^4-4\,a^2\,x^2+2}\right )+\frac {3\,{\ln \left (a\,x+1\right )}^2}{64\,a}+\frac {3\,{\ln \left (1-a\,x\right )}^2}{64\,a}+\frac {\ln \left (a\,x+1\right )\,\left (\frac {5\,x}{16\,a}-\frac {3\,a\,x^3}{16}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.96 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=\frac {6 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}-12 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}+6 \mathit {atanh} \left (a x \right )^{2}-12 \mathit {atanh} \left (a x \right ) a^{3} x^{3}+20 \mathit {atanh} \left (a x \right ) a x +3 a^{4} x^{4}-5}{32 a \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )} \] Input:

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

Output:

(6*atanh(a*x)**2*a**4*x**4 - 12*atanh(a*x)**2*a**2*x**2 + 6*atanh(a*x)**2 
- 12*atanh(a*x)*a**3*x**3 + 20*atanh(a*x)*a*x + 3*a**4*x**4 - 5)/(32*a*(a* 
*4*x**4 - 2*a**2*x**2 + 1))