\(\int \frac {\text {arctanh}(a x)}{x^2 (1-a^2 x^2)^3} \, dx\) [307]

Optimal result

Integrand size = 20, antiderivative size = 123 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^3} \, dx=-\frac {a}{16 \left (1-a^2 x^2\right )^2}-\frac {7 a}{16 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)}{x}+\frac {a^2 x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \text {arctanh}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {15}{16} a \text {arctanh}(a x)^2+a \log (x)-\frac {1}{2} a \log \left (1-a^2 x^2\right ) \] Output:

-1/16*a/(-a^2*x^2+1)^2-7*a/(-16*a^2*x^2+16)-arctanh(a*x)/x+1/4*a^2*x*arcta 
nh(a*x)/(-a^2*x^2+1)^2+7*a^2*x*arctanh(a*x)/(-8*a^2*x^2+8)+15/16*a*arctanh 
(a*x)^2+a*ln(x)-1/2*a*ln(-a^2*x^2+1)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.76 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^3} \, dx=\frac {1}{16} \left (-\frac {2 \left (8-25 a^2 x^2+15 a^4 x^4\right ) \text {arctanh}(a x)}{x \left (-1+a^2 x^2\right )^2}+15 a \text {arctanh}(a x)^2+a \left (\frac {-8+7 a^2 x^2}{\left (-1+a^2 x^2\right )^2}+16 \log (x)-8 \log \left (1-a^2 x^2\right )\right )\right ) \] Input:

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

Output:

((-2*(8 - 25*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x])/(x*(-1 + a^2*x^2)^2) + 15 
*a*ArcTanh[a*x]^2 + a*((-8 + 7*a^2*x^2)/(-1 + a^2*x^2)^2 + 16*Log[x] - 8*L 
og[1 - a^2*x^2]))/16
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.67, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6592, 6522, 6518, 241, 6592, 6518, 241, 6544, 6452, 243, 47, 14, 16, 6510}

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.

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6522

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

\(\Big \downarrow \) 6518

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6518

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 6510

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 6452

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x 
], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 
] && IntegerQ[m])) && NeQ[m, -1]
 

rule 6510

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

rule 6544

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

rule 6592

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[1/d   Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh 
[c*x])^p, x], x] - Simp[e/d   Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh[c* 
x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Integers 
Q[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.61

method	result	size
parallelrisch	\(\frac {15 \operatorname {arctanh}\left (a x \right )^{2} a x -30 \operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}+15 \operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}-16 a x \,\operatorname {arctanh}\left (a x \right )-16 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}+32 \ln \left (a x -1\right ) x^{3} a^{3}+8 a^{5} x^{5}-9 a^{3} x^{3}+16 \ln \left (x \right ) a^{5} x^{5}+16 a \ln \left (x \right ) x -32 \ln \left (x \right ) a^{3} x^{3}+32 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )-30 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )+50 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-16 \,\operatorname {arctanh}\left (a x \right )-16 \ln \left (a x -1\right ) x^{5} a^{5}-16 \ln \left (a x -1\right ) a x}{16 \left (a^{2} x^{2}-1\right )^{2} x}\)	\(198\)
derivativedivides	\(a \left (\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {7 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )}-\frac {15 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{16}-\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {7 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )}+\frac {15 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{16}-\frac {\operatorname {arctanh}\left (a x \right )}{a x}-\frac {15 \ln \left (a x -1\right )^{2}}{64}+\frac {15 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{32}+\frac {15 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {15 \ln \left (a x +1\right )^{2}}{64}-\frac {1}{64 \left (a x -1\right )^{2}}+\frac {15}{64 \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{2}-\frac {1}{64 \left (a x +1\right )^{2}}-\frac {15}{64 \left (a x +1\right )}-\frac {\ln \left (a x +1\right )}{2}+\ln \left (a x \right )\right )\)	\(208\)
default	\(a \left (\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {7 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )}-\frac {15 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{16}-\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {7 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )}+\frac {15 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{16}-\frac {\operatorname {arctanh}\left (a x \right )}{a x}-\frac {15 \ln \left (a x -1\right )^{2}}{64}+\frac {15 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{32}+\frac {15 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{32}-\frac {15 \ln \left (a x +1\right )^{2}}{64}-\frac {1}{64 \left (a x -1\right )^{2}}+\frac {15}{64 \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{2}-\frac {1}{64 \left (a x +1\right )^{2}}-\frac {15}{64 \left (a x +1\right )}-\frac {\ln \left (a x +1\right )}{2}+\ln \left (a x \right )\right )\)	\(208\)
parts	\(-\frac {\operatorname {arctanh}\left (a x \right )}{x}-\frac {\operatorname {arctanh}\left (a x \right ) a}{16 \left (a x +1\right )^{2}}-\frac {7 \,\operatorname {arctanh}\left (a x \right ) a}{16 \left (a x +1\right )}+\frac {15 a \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{16}+\frac {\operatorname {arctanh}\left (a x \right ) a}{16 \left (a x -1\right )^{2}}-\frac {7 \,\operatorname {arctanh}\left (a x \right ) a}{16 \left (a x -1\right )}-\frac {15 a \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{16}-\frac {a \left (\frac {15 \ln \left (a x -1\right )^{2}}{4}-\frac {15 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {15 \ln \left (a x +1\right )^{2}}{4}-\frac {15 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-16 \ln \left (x \right )+\frac {1}{4 \left (a x +1\right )^{2}}+\frac {15}{4 \left (a x +1\right )}+8 \ln \left (a x +1\right )+\frac {1}{4 \left (a x -1\right )^{2}}-\frac {15}{4 \left (a x -1\right )}+8 \ln \left (a x -1\right )\right )}{16}\)	\(213\)
risch	\(\frac {15 a \ln \left (a x +1\right )^{2}}{64}-\frac {\left (15 x^{5} \ln \left (-a x +1\right ) a^{5}+30 a^{4} x^{4}-30 a^{3} x^{3} \ln \left (-a x +1\right )-50 a^{2} x^{2}+15 a x \ln \left (-a x +1\right )+16\right ) \ln \left (a x +1\right )}{32 x \left (a^{2} x^{2}-1\right )^{2}}+\frac {15 a^{5} x^{5} \ln \left (-a x +1\right )^{2}+64 \ln \left (x \right ) a^{5} x^{5}-32 \ln \left (a^{2} x^{2}-1\right ) a^{5} x^{5}+60 x^{4} \ln \left (-a x +1\right ) a^{4}-30 a^{3} x^{3} \ln \left (-a x +1\right )^{2}-128 \ln \left (x \right ) a^{3} x^{3}+64 \ln \left (a^{2} x^{2}-1\right ) a^{3} x^{3}+28 a^{3} x^{3}-100 x^{2} \ln \left (-a x +1\right ) a^{2}+15 a \ln \left (-a x +1\right )^{2} x +64 a \ln \left (x \right ) x -32 a \ln \left (a^{2} x^{2}-1\right ) x -32 a x +32 \ln \left (-a x +1\right )}{64 x \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\)	\(299\)

Input:

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

Output:

1/16*(15*arctanh(a*x)^2*a*x-30*arctanh(a*x)^2*a^3*x^3+15*arctanh(a*x)^2*a^ 
5*x^5-16*a*x*arctanh(a*x)-16*arctanh(a*x)*a^5*x^5+32*ln(a*x-1)*x^3*a^3+8*a 
^5*x^5-9*a^3*x^3+16*ln(x)*a^5*x^5+16*a*ln(x)*x-32*ln(x)*a^3*x^3+32*a^3*x^3 
*arctanh(a*x)-30*a^4*x^4*arctanh(a*x)+50*a^2*x^2*arctanh(a*x)-16*arctanh(a 
*x)-16*ln(a*x-1)*x^5*a^5-16*ln(a*x-1)*a*x)/(a^2*x^2-1)^2/x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.31 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^3} \, dx=\frac {28 \, a^{3} x^{3} + 15 \, {\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 32 \, a x - 32 \, {\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \log \left (a^{2} x^{2} - 1\right ) + 64 \, {\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \log \left (x\right ) - 4 \, {\left (15 \, a^{4} x^{4} - 25 \, a^{2} x^{2} + 8\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{64 \, {\left (a^{4} x^{5} - 2 \, a^{2} x^{3} + x\right )}} \] Input:

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

Output:

1/64*(28*a^3*x^3 + 15*(a^5*x^5 - 2*a^3*x^3 + a*x)*log(-(a*x + 1)/(a*x - 1) 
)^2 - 32*a*x - 32*(a^5*x^5 - 2*a^3*x^3 + a*x)*log(a^2*x^2 - 1) + 64*(a^5*x 
^5 - 2*a^3*x^3 + a*x)*log(x) - 4*(15*a^4*x^4 - 25*a^2*x^2 + 8)*log(-(a*x + 
 1)/(a*x - 1)))/(a^4*x^5 - 2*a^2*x^3 + x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (107) = 214\).

Time = 1.61 (sec) , antiderivative size = 549, normalized size of antiderivative = 4.46 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^3} \, dx=\begin {cases} \frac {16 a^{5} x^{5} \log {\left (x \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} - \frac {16 a^{5} x^{5} \log {\left (x - \frac {1}{a} \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} + \frac {15 a^{5} x^{5} \operatorname {atanh}^{2}{\left (a x \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} - \frac {16 a^{5} x^{5} \operatorname {atanh}{\left (a x \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} - \frac {30 a^{4} x^{4} \operatorname {atanh}{\left (a x \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} - \frac {32 a^{3} x^{3} \log {\left (x \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} + \frac {32 a^{3} x^{3} \log {\left (x - \frac {1}{a} \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} - \frac {30 a^{3} x^{3} \operatorname {atanh}^{2}{\left (a x \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} + \frac {32 a^{3} x^{3} \operatorname {atanh}{\left (a x \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} + \frac {7 a^{3} x^{3}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} + \frac {50 a^{2} x^{2} \operatorname {atanh}{\left (a x \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} + \frac {16 a x \log {\left (x \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} - \frac {16 a x \log {\left (x - \frac {1}{a} \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} + \frac {15 a x \operatorname {atanh}^{2}{\left (a x \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} - \frac {16 a x \operatorname {atanh}{\left (a x \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} - \frac {8 a x}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} - \frac {16 \operatorname {atanh}{\left (a x \right )}}{16 a^{4} x^{5} - 32 a^{2} x^{3} + 16 x} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((16*a**5*x**5*log(x)/(16*a**4*x**5 - 32*a**2*x**3 + 16*x) - 16*a 
**5*x**5*log(x - 1/a)/(16*a**4*x**5 - 32*a**2*x**3 + 16*x) + 15*a**5*x**5* 
atanh(a*x)**2/(16*a**4*x**5 - 32*a**2*x**3 + 16*x) - 16*a**5*x**5*atanh(a* 
x)/(16*a**4*x**5 - 32*a**2*x**3 + 16*x) - 30*a**4*x**4*atanh(a*x)/(16*a**4 
*x**5 - 32*a**2*x**3 + 16*x) - 32*a**3*x**3*log(x)/(16*a**4*x**5 - 32*a**2 
*x**3 + 16*x) + 32*a**3*x**3*log(x - 1/a)/(16*a**4*x**5 - 32*a**2*x**3 + 1 
6*x) - 30*a**3*x**3*atanh(a*x)**2/(16*a**4*x**5 - 32*a**2*x**3 + 16*x) + 3 
2*a**3*x**3*atanh(a*x)/(16*a**4*x**5 - 32*a**2*x**3 + 16*x) + 7*a**3*x**3/ 
(16*a**4*x**5 - 32*a**2*x**3 + 16*x) + 50*a**2*x**2*atanh(a*x)/(16*a**4*x* 
*5 - 32*a**2*x**3 + 16*x) + 16*a*x*log(x)/(16*a**4*x**5 - 32*a**2*x**3 + 1 
6*x) - 16*a*x*log(x - 1/a)/(16*a**4*x**5 - 32*a**2*x**3 + 16*x) + 15*a*x*a 
tanh(a*x)**2/(16*a**4*x**5 - 32*a**2*x**3 + 16*x) - 16*a*x*atanh(a*x)/(16* 
a**4*x**5 - 32*a**2*x**3 + 16*x) - 8*a*x/(16*a**4*x**5 - 32*a**2*x**3 + 16 
*x) - 16*atanh(a*x)/(16*a**4*x**5 - 32*a**2*x**3 + 16*x), Ne(a, 0)), (0, T 
rue))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.66 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^3} \, dx=\frac {1}{64} \, a {\left (\frac {28 \, a^{2} x^{2} - 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 30 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 32}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - 32 \, \log \left (a x + 1\right ) - 32 \, \log \left (a x - 1\right ) + 64 \, \log \left (x\right )\right )} + \frac {1}{16} \, {\left (15 \, a \log \left (a x + 1\right ) - 15 \, a \log \left (a x - 1\right ) - \frac {2 \, {\left (15 \, a^{4} x^{4} - 25 \, a^{2} x^{2} + 8\right )}}{a^{4} x^{5} - 2 \, a^{2} x^{3} + x}\right )} \operatorname {artanh}\left (a x\right ) \] Input:

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

Output:

1/64*a*((28*a^2*x^2 - 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 + 30*(a^ 
4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(a*x - 1) - 15*(a^4*x^4 - 2*a^2*x^2 
 + 1)*log(a*x - 1)^2 - 32)/(a^4*x^4 - 2*a^2*x^2 + 1) - 32*log(a*x + 1) - 3 
2*log(a*x - 1) + 64*log(x)) + 1/16*(15*a*log(a*x + 1) - 15*a*log(a*x - 1) 
- 2*(15*a^4*x^4 - 25*a^2*x^2 + 8)/(a^4*x^5 - 2*a^2*x^3 + x))*arctanh(a*x)
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 4.11 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.49 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^3} \, dx=\frac {15\,a\,{\ln \left (a\,x+1\right )}^2}{64}-\frac {4\,a-\frac {7\,a^3\,x^2}{2}}{8\,a^4\,x^4-16\,a^2\,x^2+8}+\frac {15\,a\,{\ln \left (1-a\,x\right )}^2}{64}-\frac {a\,\ln \left (a^2\,x^2-1\right )}{2}+a\,\ln \left (x\right )+\ln \left (1-a\,x\right )\,\left (\frac {\frac {15\,a^4\,x^4}{8}-\frac {25\,a^2\,x^2}{8}+1}{2\,a^4\,x^5-4\,a^2\,x^3+2\,x}-\frac {15\,a\,\ln \left (a\,x+1\right )}{32}\right )-\frac {\ln \left (a\,x+1\right )\,\left (\frac {1}{2\,a}-\frac {25\,a\,x^2}{16}+\frac {15\,a^3\,x^4}{16}\right )}{\frac {x}{a}-2\,a\,x^3+a^3\,x^5} \] Input:

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

Output:

(15*a*log(a*x + 1)^2)/64 - (4*a - (7*a^3*x^2)/2)/(8*a^4*x^4 - 16*a^2*x^2 + 
 8) + (15*a*log(1 - a*x)^2)/64 - (a*log(a^2*x^2 - 1))/2 + a*log(x) + log(1 
 - a*x)*(((15*a^4*x^4)/8 - (25*a^2*x^2)/8 + 1)/(2*x - 4*a^2*x^3 + 2*a^4*x^ 
5) - (15*a*log(a*x + 1))/32) - (log(a*x + 1)*(1/(2*a) - (25*a*x^2)/16 + (1 
5*a^3*x^4)/16))/(x/a - 2*a*x^3 + a^3*x^5)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.73 \[ \int \frac {\text {arctanh}(a x)}{x^2 \left (1-a^2 x^2\right )^3} \, dx=\frac {30 \mathit {atanh} \left (a x \right )^{2} a^{5} x^{5}-60 \mathit {atanh} \left (a x \right )^{2} a^{3} x^{3}+30 \mathit {atanh} \left (a x \right )^{2} a x -32 \mathit {atanh} \left (a x \right ) a^{5} x^{5}-60 \mathit {atanh} \left (a x \right ) a^{4} x^{4}+64 \mathit {atanh} \left (a x \right ) a^{3} x^{3}+100 \mathit {atanh} \left (a x \right ) a^{2} x^{2}-32 \mathit {atanh} \left (a x \right ) a x -32 \mathit {atanh} \left (a x \right )-32 \,\mathrm {log}\left (a^{2} x -a \right ) a^{5} x^{5}+64 \,\mathrm {log}\left (a^{2} x -a \right ) a^{3} x^{3}-32 \,\mathrm {log}\left (a^{2} x -a \right ) a x +32 \,\mathrm {log}\left (x \right ) a^{5} x^{5}-64 \,\mathrm {log}\left (x \right ) a^{3} x^{3}+32 \,\mathrm {log}\left (x \right ) a x +7 a^{5} x^{5}-9 a x}{32 x \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )} \] Input:

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

Output:

(30*atanh(a*x)**2*a**5*x**5 - 60*atanh(a*x)**2*a**3*x**3 + 30*atanh(a*x)** 
2*a*x - 32*atanh(a*x)*a**5*x**5 - 60*atanh(a*x)*a**4*x**4 + 64*atanh(a*x)* 
a**3*x**3 + 100*atanh(a*x)*a**2*x**2 - 32*atanh(a*x)*a*x - 32*atanh(a*x) - 
 32*log(a**2*x - a)*a**5*x**5 + 64*log(a**2*x - a)*a**3*x**3 - 32*log(a**2 
*x - a)*a*x + 32*log(x)*a**5*x**5 - 64*log(x)*a**3*x**3 + 32*log(x)*a*x + 
7*a**5*x**5 - 9*a*x)/(32*x*(a**4*x**4 - 2*a**2*x**2 + 1))