\(\int \frac {\text {arctanh}(a x)^3}{(1-a^2 x^2)^{7/2}} \, dx\) [477]

Optimal result

Integrand size = 21, antiderivative size = 289 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx=-\frac {6}{625 a \left (1-a^2 x^2\right )^{5/2}}-\frac {272}{3375 a \left (1-a^2 x^2\right )^{3/2}}-\frac {4144}{1125 a \sqrt {1-a^2 x^2}}+\frac {6 x \text {arctanh}(a x)}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac {272 x \text {arctanh}(a x)}{1125 \left (1-a^2 x^2\right )^{3/2}}+\frac {4144 x \text {arctanh}(a x)}{1125 \sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac {4 \text {arctanh}(a x)^2}{15 a \left (1-a^2 x^2\right )^{3/2}}-\frac {8 \text {arctanh}(a x)^2}{5 a \sqrt {1-a^2 x^2}}+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x \text {arctanh}(a x)^3}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x \text {arctanh}(a x)^3}{15 \sqrt {1-a^2 x^2}} \] Output:

-6/625/a/(-a^2*x^2+1)^(5/2)-272/3375/a/(-a^2*x^2+1)^(3/2)-4144/1125/a/(-a^ 
2*x^2+1)^(1/2)+6/125*x*arctanh(a*x)/(-a^2*x^2+1)^(5/2)+272/1125*x*arctanh( 
a*x)/(-a^2*x^2+1)^(3/2)+4144/1125*x*arctanh(a*x)/(-a^2*x^2+1)^(1/2)-3/25*a 
rctanh(a*x)^2/a/(-a^2*x^2+1)^(5/2)-4/15*arctanh(a*x)^2/a/(-a^2*x^2+1)^(3/2 
)-8/5*arctanh(a*x)^2/a/(-a^2*x^2+1)^(1/2)+1/5*x*arctanh(a*x)^3/(-a^2*x^2+1 
)^(5/2)+4/15*x*arctanh(a*x)^3/(-a^2*x^2+1)^(3/2)+8/15*x*arctanh(a*x)^3/(-a 
^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.41 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx=\frac {-63682+125680 a^2 x^2-62160 a^4 x^4+30 a x \left (2235-4280 a^2 x^2+2072 a^4 x^4\right ) \text {arctanh}(a x)-225 \left (149-260 a^2 x^2+120 a^4 x^4\right ) \text {arctanh}(a x)^2+1125 a x \left (15-20 a^2 x^2+8 a^4 x^4\right ) \text {arctanh}(a x)^3}{16875 a \left (1-a^2 x^2\right )^{5/2}} \] Input:

Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2)^(7/2),x]
 

Output:

(-63682 + 125680*a^2*x^2 - 62160*a^4*x^4 + 30*a*x*(2235 - 4280*a^2*x^2 + 2 
072*a^4*x^4)*ArcTanh[a*x] - 225*(149 - 260*a^2*x^2 + 120*a^4*x^4)*ArcTanh[ 
a*x]^2 + 1125*a*x*(15 - 20*a^2*x^2 + 8*a^4*x^4)*ArcTanh[a*x]^3)/(16875*a*( 
1 - a^2*x^2)^(5/2))
 

Rubi [A] (verified)

Time = 1.89 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.52, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6526, 6522, 6522, 6520, 6526, 6522, 6520, 6524, 6520}

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.

\(\displaystyle \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 6526

\(\displaystyle \frac {6}{25} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {4}{5} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 6522

\(\displaystyle \frac {6}{25} \left (\frac {4}{5} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {4}{5} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 6522

\(\displaystyle \frac {6}{25} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {4}{5} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 6520

\(\displaystyle \frac {4}{5} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {6}{25} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )\)

\(\Big \downarrow \) 6526

\(\displaystyle \frac {4}{5} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {2}{3} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {6}{25} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )\)

\(\Big \downarrow \) 6522

\(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {2}{3} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {6}{25} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )\)

\(\Big \downarrow \) 6520

\(\displaystyle \frac {4}{5} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )\right )+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {6}{25} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )\)

\(\Big \downarrow \) 6524

\(\displaystyle \frac {4}{5} \left (\frac {2}{3} \left (6 \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}\right )+\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )\right )+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {6}{25} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )\)

\(\Big \downarrow \) 6520

\(\displaystyle \frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {6}{25} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\right )\)

Input:

Int[ArcTanh[a*x]^3/(1 - a^2*x^2)^(7/2),x]
 

Output:

(-3*ArcTanh[a*x]^2)/(25*a*(1 - a^2*x^2)^(5/2)) + (x*ArcTanh[a*x]^3)/(5*(1 
- a^2*x^2)^(5/2)) + (6*(-1/25*1/(a*(1 - a^2*x^2)^(5/2)) + (x*ArcTanh[a*x]) 
/(5*(1 - a^2*x^2)^(5/2)) + (4*(-1/9*1/(a*(1 - a^2*x^2)^(3/2)) + (x*ArcTanh 
[a*x])/(3*(1 - a^2*x^2)^(3/2)) + (2*(-(1/(a*Sqrt[1 - a^2*x^2])) + (x*ArcTa 
nh[a*x])/Sqrt[1 - a^2*x^2]))/3))/5))/25 + (4*(-1/3*ArcTanh[a*x]^2/(a*(1 - 
a^2*x^2)^(3/2)) + (x*ArcTanh[a*x]^3)/(3*(1 - a^2*x^2)^(3/2)) + (2*(-1/9*1/ 
(a*(1 - a^2*x^2)^(3/2)) + (x*ArcTanh[a*x])/(3*(1 - a^2*x^2)^(3/2)) + (2*(- 
(1/(a*Sqrt[1 - a^2*x^2])) + (x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2]))/3))/3 + ( 
2*((-3*ArcTanh[a*x]^2)/(a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^3)/Sqrt[1 - 
 a^2*x^2] + 6*(-(1/(a*Sqrt[1 - a^2*x^2])) + (x*ArcTanh[a*x])/Sqrt[1 - a^2* 
x^2])))/3))/5
 

Defintions of rubi rules used

rule 6520

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symb 
ol] :> Simp[-b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTanh[c*x])/(d* 
Sqrt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 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 6524

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[(-b)*p*((a + b*ArcTanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) 
), x] + (Simp[x*((a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2])), x] + Simp[b^2 
*p*(p - 1)   Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; 
 FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
 

rule 6526

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

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.53

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (9000 \operatorname {arctanh}\left (a x \right )^{3} a^{5} x^{5}+62160 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}-27000 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}-22500 \operatorname {arctanh}\left (a x \right )^{3} a^{3} x^{3}-62160 a^{4} x^{4}-128400 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )+58500 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}+16875 \operatorname {arctanh}\left (a x \right )^{3} a x +125680 a^{2} x^{2}+67050 a x \,\operatorname {arctanh}\left (a x \right )-33525 \operatorname {arctanh}\left (a x \right )^{2}-63682\right )}{16875 a \left (a^{2} x^{2}-1\right )^{3}}\)	\(153\)
orering	\(\frac {\left (-\frac {232064}{675} a^{8} x^{9}+\frac {769024}{675} a^{6} x^{7}-\frac {68286904}{50625} a^{4} x^{5}+\frac {6618136}{10125} a^{2} x^{3}-\frac {5075776}{50625} x \right ) \operatorname {arctanh}\left (a x \right )^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}-\frac {2 \left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (3605280 a^{6} x^{6}-7644240 a^{4} x^{4}+4400438 a^{2} x^{2}-349409\right ) \left (\frac {3 \operatorname {arctanh}\left (a x \right )^{2} a}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}+\frac {7 \operatorname {arctanh}\left (a x \right )^{3} a^{2} x}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}\right )}{50625 a^{2}}-\frac {16 x \left (a x +1\right )^{3} \left (a x -1\right )^{3} \left (18130 a^{4} x^{4}-36685 a^{2} x^{2}+18609\right ) \left (\frac {6 \,\operatorname {arctanh}\left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {11}{2}}}+\frac {48 \operatorname {arctanh}\left (a x \right )^{2} a^{3} x}{\left (-a^{2} x^{2}+1\right )^{\frac {11}{2}}}+\frac {63 \operatorname {arctanh}\left (a x \right )^{3} a^{4} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {11}{2}}}+\frac {7 \operatorname {arctanh}\left (a x \right )^{3} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}\right )}{16875 a^{2}}-\frac {\left (31080 a^{4} x^{4}-62840 a^{2} x^{2}+31841\right ) \left (a x -1\right )^{4} \left (a x +1\right )^{4} \left (\frac {6 a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {13}{2}}}+\frac {162 \,\operatorname {arctanh}\left (a x \right ) a^{4} x}{\left (-a^{2} x^{2}+1\right )^{\frac {13}{2}}}+\frac {717 \operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {13}{2}}}+\frac {69 \operatorname {arctanh}\left (a x \right )^{2} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {11}{2}}}+\frac {693 \operatorname {arctanh}\left (a x \right )^{3} a^{6} x^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {13}{2}}}+\frac {189 \operatorname {arctanh}\left (a x \right )^{3} a^{4} x}{\left (-a^{2} x^{2}+1\right )^{\frac {11}{2}}}\right )}{50625 a^{4}}\)	\(457\)

Input:

int(arctanh(a*x)^3/(-a^2*x^2+1)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/16875/a*(-a^2*x^2+1)^(1/2)*(9000*arctanh(a*x)^3*a^5*x^5+62160*arctanh(a 
*x)*a^5*x^5-27000*a^4*x^4*arctanh(a*x)^2-22500*arctanh(a*x)^3*a^3*x^3-6216 
0*a^4*x^4-128400*a^3*x^3*arctanh(a*x)+58500*a^2*x^2*arctanh(a*x)^2+16875*a 
rctanh(a*x)^3*a*x+125680*a^2*x^2+67050*a*x*arctanh(a*x)-33525*arctanh(a*x) 
^2-63682)/(a^2*x^2-1)^3
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.61 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx=\frac {{\left (497280 \, a^{4} x^{4} - 1005440 \, a^{2} x^{2} - 1125 \, {\left (8 \, a^{5} x^{5} - 20 \, a^{3} x^{3} + 15 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 450 \, {\left (120 \, a^{4} x^{4} - 260 \, a^{2} x^{2} + 149\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 120 \, {\left (2072 \, a^{5} x^{5} - 4280 \, a^{3} x^{3} + 2235 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 509456\right )} \sqrt {-a^{2} x^{2} + 1}}{135000 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )}} \] Input:

integrate(arctanh(a*x)^3/(-a^2*x^2+1)^(7/2),x, algorithm="fricas")
 

Output:

1/135000*(497280*a^4*x^4 - 1005440*a^2*x^2 - 1125*(8*a^5*x^5 - 20*a^3*x^3 
+ 15*a*x)*log(-(a*x + 1)/(a*x - 1))^3 + 450*(120*a^4*x^4 - 260*a^2*x^2 + 1 
49)*log(-(a*x + 1)/(a*x - 1))^2 - 120*(2072*a^5*x^5 - 4280*a^3*x^3 + 2235* 
a*x)*log(-(a*x + 1)/(a*x - 1)) + 509456)*sqrt(-a^2*x^2 + 1)/(a^7*x^6 - 3*a 
^5*x^4 + 3*a^3*x^2 - a)
 

Sympy [F]

\[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \] Input:

integrate(atanh(a*x)**3/(-a**2*x**2+1)**(7/2),x)
 

Output:

Integral(atanh(a*x)**3/(-(a*x - 1)*(a*x + 1))**(7/2), x)
 

Maxima [F]

\[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(arctanh(a*x)^3/(-a^2*x^2+1)^(7/2),x, algorithm="maxima")
 

Output:

integrate(arctanh(a*x)^3/(-a^2*x^2 + 1)^(7/2), x)
 

Giac [F]

\[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(arctanh(a*x)^3/(-a^2*x^2+1)^(7/2),x, algorithm="giac")
 

Output:

integrate(arctanh(a*x)^3/(-a^2*x^2 + 1)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{{\left (1-a^2\,x^2\right )}^{7/2}} \,d x \] Input:

int(atanh(a*x)^3/(1 - a^2*x^2)^(7/2),x)
 

Output:

int(atanh(a*x)^3/(1 - a^2*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx=-\left (\int \frac {\mathit {atanh} \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}-3 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+3 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}d x \right ) \] Input:

int(atanh(a*x)^3/(-a^2*x^2+1)^(7/2),x)
                                                                                    
                                                                                    
 

Output:

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