\(\int \frac {\text {arctanh}(a x)^2}{(1-a^2 x^2)^{5/2}} \, dx\) [472]

Optimal result

Integrand size = 21, antiderivative size = 139 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\frac {2 x}{27 \left (1-a^2 x^2\right )^{3/2}}+\frac {40 x}{27 \sqrt {1-a^2 x^2}}-\frac {2 \text {arctanh}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}}-\frac {4 \text {arctanh}(a x)}{3 a \sqrt {1-a^2 x^2}}+\frac {x \text {arctanh}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \text {arctanh}(a x)^2}{3 \sqrt {1-a^2 x^2}} \] Output:

2/27*x/(-a^2*x^2+1)^(3/2)+40/27*x/(-a^2*x^2+1)^(1/2)-2/9*arctanh(a*x)/a/(- 
a^2*x^2+1)^(3/2)-4/3*arctanh(a*x)/a/(-a^2*x^2+1)^(1/2)+1/3*x*arctanh(a*x)^ 
2/(-a^2*x^2+1)^(3/2)+2/3*x*arctanh(a*x)^2/(-a^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.50 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\frac {42 a x-40 a^3 x^3+6 \left (-7+6 a^2 x^2\right ) \text {arctanh}(a x)-9 a x \left (-3+2 a^2 x^2\right ) \text {arctanh}(a x)^2}{27 a \left (1-a^2 x^2\right )^{3/2}} \] Input:

Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^(5/2),x]
 

Output:

(42*a*x - 40*a^3*x^3 + 6*(-7 + 6*a^2*x^2)*ArcTanh[a*x] - 9*a*x*(-3 + 2*a^2 
*x^2)*ArcTanh[a*x]^2)/(27*a*(1 - a^2*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6526, 209, 208, 6524, 208}

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.

\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6526

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

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 6524

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

\(\Big \downarrow \) 208

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

Input:

Int[ArcTanh[a*x]^2/(1 - a^2*x^2)^(5/2),x]
 

Output:

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

Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 209

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
 

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 = 0.45 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.60

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (18 \operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}+40 a^{3} x^{3}-36 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-27 \operatorname {arctanh}\left (a x \right )^{2} a x -42 a x +42 \,\operatorname {arctanh}\left (a x \right )\right )}{27 a \left (a^{2} x^{2}-1\right )^{2}}\)	\(84\)
orering	\(\frac {\left (\frac {200}{9} a^{6} x^{7}-\frac {1222}{27} a^{4} x^{5}+\frac {595}{27} a^{2} x^{3}+x \right ) \operatorname {arctanh}\left (a x \right )^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {\left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (80 a^{4} x^{4}-78 a^{2} x^{2}-7\right ) \left (\frac {2 \,\operatorname {arctanh}\left (a x \right ) a}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}+\frac {5 \operatorname {arctanh}\left (a x \right )^{2} a^{2} x}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}\right )}{9 a^{2}}+\frac {x \left (20 a^{2} x^{2}-21\right ) \left (a x +1\right )^{3} \left (a x -1\right )^{3} \left (\frac {2 a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}+\frac {24 \,\operatorname {arctanh}\left (a x \right ) a^{3} x}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}+\frac {35 \operatorname {arctanh}\left (a x \right )^{2} a^{4} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}+\frac {5 \operatorname {arctanh}\left (a x \right )^{2} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}\right )}{27 a^{2}}\)	\(247\)

Input:

int(arctanh(a*x)^2/(-a^2*x^2+1)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/27/a*(-a^2*x^2+1)^(1/2)*(18*arctanh(a*x)^2*a^3*x^3+40*a^3*x^3-36*a^2*x^ 
2*arctanh(a*x)-27*arctanh(a*x)^2*a*x-42*a*x+42*arctanh(a*x))/(a^2*x^2-1)^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.76 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx=-\frac {{\left (160 \, a^{3} x^{3} + 9 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 168 \, a x - 12 \, {\left (6 \, a^{2} x^{2} - 7\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{108 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )}} \] Input:

integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(5/2),x, algorithm="fricas")
 

Output:

-1/108*(160*a^3*x^3 + 9*(2*a^3*x^3 - 3*a*x)*log(-(a*x + 1)/(a*x - 1))^2 - 
168*a*x - 12*(6*a^2*x^2 - 7)*log(-(a*x + 1)/(a*x - 1)))*sqrt(-a^2*x^2 + 1) 
/(a^5*x^4 - 2*a^3*x^2 + a)
 

Sympy [F]

\[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(atanh(a*x)**2/(-a**2*x**2+1)**(5/2),x)
 

Output:

Integral(atanh(a*x)**2/(-(a*x - 1)*(a*x + 1))**(5/2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (115) = 230\).

Time = 0.16 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.19 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {1}{27} \, a {\left (\frac {\frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} a^{2} x + \sqrt {-a^{2} x^{2} + 1} a}}{a} + \frac {\frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} a^{2} x - \sqrt {-a^{2} x^{2} + 1} a}}{a} - \frac {18 \, \sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} x + a\right )} a} - \frac {18 \, \sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} x - a\right )} a} - \frac {18 \, \log \left (a x + 1\right )}{\sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {18 \, \log \left (-a x + 1\right )}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {3 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}} + \frac {3 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}}\right )} \] Input:

integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(5/2),x, algorithm="maxima")
 

Output:

1/3*(2*x/sqrt(-a^2*x^2 + 1) + x/(-a^2*x^2 + 1)^(3/2))*arctanh(a*x)^2 + 1/2 
7*a*((2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*a^2*x + sqrt(-a^2*x^2 
 + 1)*a))/a + (2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*a^2*x - sqrt 
(-a^2*x^2 + 1)*a))/a - 18*sqrt(-a^2*x^2 + 1)/((a^2*x + a)*a) - 18*sqrt(-a^ 
2*x^2 + 1)/((a^2*x - a)*a) - 18*log(a*x + 1)/(sqrt(-a^2*x^2 + 1)*a^2) + 18 
*log(-a*x + 1)/(sqrt(-a^2*x^2 + 1)*a^2) - 3*log(a*x + 1)/((-a^2*x^2 + 1)^( 
3/2)*a^2) + 3*log(-a*x + 1)/((-a^2*x^2 + 1)^(3/2)*a^2))
 

Giac [F]

\[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(5/2),x, algorithm="giac")
 

Output:

integrate(arctanh(a*x)^2/(-a^2*x^2 + 1)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{{\left (1-a^2\,x^2\right )}^{5/2}} \,d x \] Input:

int(atanh(a*x)^2/(1 - a^2*x^2)^(5/2),x)
 

Output:

int(atanh(a*x)^2/(1 - a^2*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\int \frac {\mathit {atanh} \left (a x \right )^{2}}{\sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+\sqrt {-a^{2} x^{2}+1}}d x \] Input:

int(atanh(a*x)^2/(-a^2*x^2+1)^(5/2),x)
 

Output:

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