\(\int \frac {\sqrt {\text {arctanh}(a x)}}{(1-a^2 x^2)^3} \, dx\) [320]

Optimal result

Integrand size = 21, antiderivative size = 168 \[ \int \frac {\sqrt {\text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^3} \, dx=\frac {\text {arctanh}(a x)^{3/2}}{4 a}+\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arctanh}(a x)}\right )}{256 a}+\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arctanh}(a x)}\right )}{16 a}-\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arctanh}(a x)}\right )}{256 a}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arctanh}(a x)}\right )}{16 a}+\frac {\sqrt {\text {arctanh}(a x)} \sinh (2 \text {arctanh}(a x))}{4 a}+\frac {\sqrt {\text {arctanh}(a x)} \sinh (4 \text {arctanh}(a x))}{32 a} \] Output:

1/4*arctanh(a*x)^(3/2)/a+1/256*Pi^(1/2)*erf(2*arctanh(a*x)^(1/2))/a+1/32*2 
^(1/2)*Pi^(1/2)*erf(2^(1/2)*arctanh(a*x)^(1/2))/a-1/256*Pi^(1/2)*erfi(2*ar 
ctanh(a*x)^(1/2))/a-1/32*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*arctanh(a*x)^(1/2)) 
/a+1/4*arctanh(a*x)^(1/2)*sinh(2*arctanh(a*x))/a+1/32*arctanh(a*x)^(1/2)*s 
inh(4*arctanh(a*x))/a
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {\text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^3} \, dx=\frac {\frac {32 \sqrt {\text {arctanh}(a x)} \left (5 a x-3 a^3 x^3+2 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)\right )}{\left (-1+a^2 x^2\right )^2}+\frac {\sqrt {\text {arctanh}(a x)} \Gamma \left (\frac {1}{2},-4 \text {arctanh}(a x)\right )}{\sqrt {-\text {arctanh}(a x)}}+\frac {8 \sqrt {2} \sqrt {\text {arctanh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arctanh}(a x)\right )}{\sqrt {-\text {arctanh}(a x)}}-8 \sqrt {2} \Gamma \left (\frac {1}{2},2 \text {arctanh}(a x)\right )-\Gamma \left (\frac {1}{2},4 \text {arctanh}(a x)\right )}{256 a} \] Input:

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

Output:

((32*Sqrt[ArcTanh[a*x]]*(5*a*x - 3*a^3*x^3 + 2*(-1 + a^2*x^2)^2*ArcTanh[a* 
x]))/(-1 + a^2*x^2)^2 + (Sqrt[ArcTanh[a*x]]*Gamma[1/2, -4*ArcTanh[a*x]])/S 
qrt[-ArcTanh[a*x]] + (8*Sqrt[2]*Sqrt[ArcTanh[a*x]]*Gamma[1/2, -2*ArcTanh[a 
*x]])/Sqrt[-ArcTanh[a*x]] - 8*Sqrt[2]*Gamma[1/2, 2*ArcTanh[a*x]] - Gamma[1 
/2, 4*ArcTanh[a*x]])/(256*a)
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6530, 3042, 3793, 2009}

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.

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

\(\Big \downarrow \) 6530

\(\displaystyle \frac {\int \frac {\sqrt {\text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^2}d\text {arctanh}(a x)}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sqrt {\text {arctanh}(a x)} \sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^4d\text {arctanh}(a x)}{a}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {\int \left (\frac {1}{2} \sqrt {\text {arctanh}(a x)} \cosh (2 \text {arctanh}(a x))+\frac {1}{8} \sqrt {\text {arctanh}(a x)} \cosh (4 \text {arctanh}(a x))+\frac {3}{8} \sqrt {\text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {1}{256} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arctanh}(a x)}\right )+\frac {1}{16} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arctanh}(a x)}\right )-\frac {1}{256} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arctanh}(a x)}\right )-\frac {1}{16} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arctanh}(a x)}\right )+\frac {1}{4} \text {arctanh}(a x)^{3/2}+\frac {1}{4} \sqrt {\text {arctanh}(a x)} \sinh (2 \text {arctanh}(a x))+\frac {1}{32} \sqrt {\text {arctanh}(a x)} \sinh (4 \text {arctanh}(a x))}{a}\)

Input:

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

Output:

(ArcTanh[a*x]^(3/2)/4 + (Sqrt[Pi]*Erf[2*Sqrt[ArcTanh[a*x]]])/256 + (Sqrt[P 
i/2]*Erf[Sqrt[2]*Sqrt[ArcTanh[a*x]]])/16 - (Sqrt[Pi]*Erfi[2*Sqrt[ArcTanh[a 
*x]]])/256 - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcTanh[a*x]]])/16 + (Sqrt[ArcT 
anh[a*x]]*Sinh[2*ArcTanh[a*x]])/4 + (Sqrt[ArcTanh[a*x]]*Sinh[4*ArcTanh[a*x 
]])/32)/a
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6530

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x 
_Symbol] :> Simp[d^q/c   Subst[Int[(a + b*x)^p/Cosh[x]^(2*(q + 1)), x], x, 
ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && I 
LtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
 

Maple [F]

\[\int \frac {\sqrt {\operatorname {arctanh}\left (a x \right )}}{\left (-a^{2} x^{2}+1\right )^{3}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {\text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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