\(\int \frac {\sqrt {\text {arctanh}(a x)}}{(1-a^2 x^2)^4} \, dx\) [348]

Optimal result

Integrand size = 21, antiderivative size = 252 \[ \int \frac {\sqrt {\text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^4} \, dx=\frac {5 \text {arctanh}(a x)^{3/2}}{24 a}+\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arctanh}(a x)}\right )}{512 a}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arctanh}(a x)}\right )}{256 a}+\frac {\sqrt {\frac {\pi }{6}} \text {erf}\left (\sqrt {6} \sqrt {\text {arctanh}(a x)}\right )}{768 a}-\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arctanh}(a x)}\right )}{512 a}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arctanh}(a x)}\right )}{256 a}-\frac {\sqrt {\frac {\pi }{6}} \text {erfi}\left (\sqrt {6} \sqrt {\text {arctanh}(a x)}\right )}{768 a}+\frac {15 \sqrt {\text {arctanh}(a x)} \sinh (2 \text {arctanh}(a x))}{64 a}+\frac {3 \sqrt {\text {arctanh}(a x)} \sinh (4 \text {arctanh}(a x))}{64 a}+\frac {\sqrt {\text {arctanh}(a x)} \sinh (6 \text {arctanh}(a x))}{192 a} \] Output:

5/24*arctanh(a*x)^(3/2)/a+3/512*Pi^(1/2)*erf(2*arctanh(a*x)^(1/2))/a+15/51 
2*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*arctanh(a*x)^(1/2))/a+1/4608*6^(1/2)*Pi^(1/ 
2)*erf(6^(1/2)*arctanh(a*x)^(1/2))/a-3/512*Pi^(1/2)*erfi(2*arctanh(a*x)^(1 
/2))/a-15/512*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*arctanh(a*x)^(1/2))/a-1/4608*6 
^(1/2)*Pi^(1/2)*erfi(6^(1/2)*arctanh(a*x)^(1/2))/a+15/64*arctanh(a*x)^(1/2 
)*sinh(2*arctanh(a*x))/a+3/64*arctanh(a*x)^(1/2)*sinh(4*arctanh(a*x))/a+1/ 
192*arctanh(a*x)^(1/2)*sinh(6*arctanh(a*x))/a
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {\text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^4} \, dx=\frac {-\frac {3168 x \sqrt {\text {arctanh}(a x)}}{\left (-1+a^2 x^2\right )^3}+\frac {3840 a^2 x^3 \sqrt {\text {arctanh}(a x)}}{\left (-1+a^2 x^2\right )^3}-\frac {1440 a^4 x^5 \sqrt {\text {arctanh}(a x)}}{\left (-1+a^2 x^2\right )^3}+\frac {960 \text {arctanh}(a x)^{3/2}}{a}+\frac {\sqrt {6} \sqrt {\text {arctanh}(a x)} \Gamma \left (\frac {1}{2},-6 \text {arctanh}(a x)\right )}{a \sqrt {-\text {arctanh}(a x)}}+\frac {27 \sqrt {\text {arctanh}(a x)} \Gamma \left (\frac {1}{2},-4 \text {arctanh}(a x)\right )}{a \sqrt {-\text {arctanh}(a x)}}+\frac {135 \sqrt {2} \sqrt {\text {arctanh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arctanh}(a x)\right )}{a \sqrt {-\text {arctanh}(a x)}}-\frac {135 \sqrt {2} \Gamma \left (\frac {1}{2},2 \text {arctanh}(a x)\right )}{a}-\frac {27 \Gamma \left (\frac {1}{2},4 \text {arctanh}(a x)\right )}{a}-\frac {\sqrt {6} \Gamma \left (\frac {1}{2},6 \text {arctanh}(a x)\right )}{a}}{4608} \] Input:

Integrate[Sqrt[ArcTanh[a*x]]/(1 - a^2*x^2)^4,x]
 

Output:

((-3168*x*Sqrt[ArcTanh[a*x]])/(-1 + a^2*x^2)^3 + (3840*a^2*x^3*Sqrt[ArcTan 
h[a*x]])/(-1 + a^2*x^2)^3 - (1440*a^4*x^5*Sqrt[ArcTanh[a*x]])/(-1 + a^2*x^ 
2)^3 + (960*ArcTanh[a*x]^(3/2))/a + (Sqrt[6]*Sqrt[ArcTanh[a*x]]*Gamma[1/2, 
 -6*ArcTanh[a*x]])/(a*Sqrt[-ArcTanh[a*x]]) + (27*Sqrt[ArcTanh[a*x]]*Gamma[ 
1/2, -4*ArcTanh[a*x]])/(a*Sqrt[-ArcTanh[a*x]]) + (135*Sqrt[2]*Sqrt[ArcTanh 
[a*x]]*Gamma[1/2, -2*ArcTanh[a*x]])/(a*Sqrt[-ArcTanh[a*x]]) - (135*Sqrt[2] 
*Gamma[1/2, 2*ArcTanh[a*x]])/a - (27*Gamma[1/2, 4*ArcTanh[a*x]])/a - (Sqrt 
[6]*Gamma[1/2, 6*ArcTanh[a*x]])/a)/4608
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 226, 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.

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

\(\Big \downarrow \) 6530

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sqrt {\text {arctanh}(a x)} \sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^6d\text {arctanh}(a x)}{a}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {\int \left (\frac {15}{32} \sqrt {\text {arctanh}(a x)} \cosh (2 \text {arctanh}(a x))+\frac {3}{16} \sqrt {\text {arctanh}(a x)} \cosh (4 \text {arctanh}(a x))+\frac {1}{32} \sqrt {\text {arctanh}(a x)} \cosh (6 \text {arctanh}(a x))+\frac {5}{16} \sqrt {\text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {3}{512} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arctanh}(a x)}\right )+\frac {15}{256} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arctanh}(a x)}\right )+\frac {1}{768} \sqrt {\frac {\pi }{6}} \text {erf}\left (\sqrt {6} \sqrt {\text {arctanh}(a x)}\right )-\frac {3}{512} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arctanh}(a x)}\right )-\frac {15}{256} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arctanh}(a x)}\right )-\frac {1}{768} \sqrt {\frac {\pi }{6}} \text {erfi}\left (\sqrt {6} \sqrt {\text {arctanh}(a x)}\right )+\frac {5}{24} \text {arctanh}(a x)^{3/2}+\frac {15}{64} \sqrt {\text {arctanh}(a x)} \sinh (2 \text {arctanh}(a x))+\frac {3}{64} \sqrt {\text {arctanh}(a x)} \sinh (4 \text {arctanh}(a x))+\frac {1}{192} \sqrt {\text {arctanh}(a x)} \sinh (6 \text {arctanh}(a x))}{a}\)

Input:

Int[Sqrt[ArcTanh[a*x]]/(1 - a^2*x^2)^4,x]
 

Output:

((5*ArcTanh[a*x]^(3/2))/24 + (3*Sqrt[Pi]*Erf[2*Sqrt[ArcTanh[a*x]]])/512 + 
(15*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcTanh[a*x]]])/256 + (Sqrt[Pi/6]*Erf[Sqrt 
[6]*Sqrt[ArcTanh[a*x]]])/768 - (3*Sqrt[Pi]*Erfi[2*Sqrt[ArcTanh[a*x]]])/512 
 - (15*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcTanh[a*x]]])/256 - (Sqrt[Pi/6]*Erfi 
[Sqrt[6]*Sqrt[ArcTanh[a*x]]])/768 + (15*Sqrt[ArcTanh[a*x]]*Sinh[2*ArcTanh[ 
a*x]])/64 + (3*Sqrt[ArcTanh[a*x]]*Sinh[4*ArcTanh[a*x]])/64 + (Sqrt[ArcTanh 
[a*x]]*Sinh[6*ArcTanh[a*x]])/192)/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]

\[\int \frac {\sqrt {\operatorname {arctanh}\left (a x \right )}}{\left (-a^{2} x^{2}+1\right )^{4}}d x\]

Input:

int(arctanh(a*x)^(1/2)/(-a^2*x^2+1)^4,x)
 

Output:

int(arctanh(a*x)^(1/2)/(-a^2*x^2+1)^4,x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {\text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(arctanh(a*x)^(1/2)/(-a^2*x^2+1)^4,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(arctanh(a*x)^(1/2)/(-a^2*x^2+1)^4,x, algorithm="maxima")
 

Output:

integrate(sqrt(arctanh(a*x))/(a^2*x^2 - 1)^4, x)
 

Giac [F]

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

integrate(arctanh(a*x)^(1/2)/(-a^2*x^2+1)^4,x, algorithm="giac")
 

Output:

integrate(sqrt(arctanh(a*x))/(a^2*x^2 - 1)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {\text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^4} \, dx=\int \frac {\sqrt {\mathrm {atanh}\left (a\,x\right )}}{{\left (a^2\,x^2-1\right )}^4} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {\text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^4} \, dx=\int \frac {\sqrt {\mathit {atanh} \left (a x \right )}}{a^{8} x^{8}-4 a^{6} x^{6}+6 a^{4} x^{4}-4 a^{2} x^{2}+1}d x \] Input:

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

Output:

int(sqrt(atanh(a*x))/(a**8*x**8 - 4*a**6*x**6 + 6*a**4*x**4 - 4*a**2*x**2 
+ 1),x)