\(\int \frac {1}{(1-a^2 x^2)^2 \text {arctanh}(a x)^4} \, dx\) [297]

Optimal result

Integrand size = 19, antiderivative size = 97 \[ \int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4} \, dx=-\frac {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}-\frac {x}{3 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {1+a^2 x^2}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {2 \text {Shi}(2 \text {arctanh}(a x))}{3 a} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4} \, dx=\frac {1+a x \text {arctanh}(a x)+\left (1+a^2 x^2\right ) \text {arctanh}(a x)^2+2 \left (-1+a^2 x^2\right ) \text {arctanh}(a x)^3 \text {Shi}(2 \text {arctanh}(a x))}{3 a \left (-1+a^2 x^2\right ) \text {arctanh}(a x)^3} \] Input:

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

Output:

(1 + a*x*ArcTanh[a*x] + (1 + a^2*x^2)*ArcTanh[a*x]^2 + 2*(-1 + a^2*x^2)*Ar 
cTanh[a*x]^3*SinhIntegral[2*ArcTanh[a*x]])/(3*a*(-1 + a^2*x^2)*ArcTanh[a*x 
]^3)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6528, 6558, 6596, 5971, 27, 3042, 26, 3779}

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.

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

\(\Big \downarrow \) 6528

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

\(\Big \downarrow \) 6558

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

\(\Big \downarrow \) 6596

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

Input:

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

Output:

-1/3*1/(a*(1 - a^2*x^2)*ArcTanh[a*x]^3) + (2*a*(-1/2*x/(a*(1 - a^2*x^2)*Ar 
cTanh[a*x]^2) - (1 + a^2*x^2)/(2*a^2*(1 - a^2*x^2)*ArcTanh[a*x]) + SinhInt 
egral[2*ArcTanh[a*x]]/a^2))/3
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6528

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

rule 6558

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

rule 6596

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) 
^2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1)   Subst[Int[(a + b*x)^p*(Sinh[x]^ 
m/Cosh[x]^(m + 2*(q + 1))), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, 
e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (In 
tegerQ[q] || GtQ[d, 0])
 

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {-\frac {1}{6 \operatorname {arctanh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{6 \operatorname {arctanh}\left (a x \right )^{3}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{6 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{3 \,\operatorname {arctanh}\left (a x \right )}+\frac {2 \,\operatorname {Shi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{3}}{a}\)	\(68\)
default	\(\frac {-\frac {1}{6 \operatorname {arctanh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{6 \operatorname {arctanh}\left (a x \right )^{3}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{6 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{3 \,\operatorname {arctanh}\left (a x \right )}+\frac {2 \,\operatorname {Shi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{3}}{a}\)	\(68\)

Input:

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

Output:

1/a*(-1/6/arctanh(a*x)^3-1/6/arctanh(a*x)^3*cosh(2*arctanh(a*x))-1/6/arcta 
nh(a*x)^2*sinh(2*arctanh(a*x))-1/3/arctanh(a*x)*cosh(2*arctanh(a*x))+2/3*S 
hi(2*arctanh(a*x)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.56 \[ \int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4} \, dx=\frac {{\left ({\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - {\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 4 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) + 2 \, {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 8}{3 \, {\left (a^{3} x^{2} - a\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3}} \] Input:

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

Output:

1/3*(((a^2*x^2 - 1)*log_integral(-(a*x + 1)/(a*x - 1)) - (a^2*x^2 - 1)*log 
_integral(-(a*x - 1)/(a*x + 1)))*log(-(a*x + 1)/(a*x - 1))^3 + 4*a*x*log(- 
(a*x + 1)/(a*x - 1)) + 2*(a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^2 + 8)/(( 
a^3*x^2 - a)*log(-(a*x + 1)/(a*x - 1))^3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-8*a*integrate(-1/3*x/((a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1) - (a^4*x^4 - 
 2*a^2*x^2 + 1)*log(-a*x + 1)), x) + 2/3*(2*a*x*log(a*x + 1) + (a^2*x^2 + 
1)*log(a*x + 1)^2 + (a^2*x^2 + 1)*log(-a*x + 1)^2 - 2*(a*x + (a^2*x^2 + 1) 
*log(a*x + 1))*log(-a*x + 1) + 4)/((a^3*x^2 - a)*log(a*x + 1)^3 - 3*(a^3*x 
^2 - a)*log(a*x + 1)^2*log(-a*x + 1) + 3*(a^3*x^2 - a)*log(a*x + 1)*log(-a 
*x + 1)^2 - (a^3*x^2 - a)*log(-a*x + 1)^3)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4} \, dx=\frac {\mathit {atanh} \left (a x \right )^{3} \left (\int \frac {x^{2}}{\mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}-2 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}+\mathit {atanh} \left (a x \right )^{2}}d x \right ) a^{5} x^{2}-\mathit {atanh} \left (a x \right )^{3} \left (\int \frac {x^{2}}{\mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}-2 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}+\mathit {atanh} \left (a x \right )^{2}}d x \right ) a^{3}+2 \mathit {atanh} \left (a x \right )^{3} \left (\int \frac {x}{\mathit {atanh} \left (a x \right ) a^{4} x^{4}-2 \mathit {atanh} \left (a x \right ) a^{2} x^{2}+\mathit {atanh} \left (a x \right )}d x \right ) a^{4} x^{2}-2 \mathit {atanh} \left (a x \right )^{3} \left (\int \frac {x}{\mathit {atanh} \left (a x \right ) a^{4} x^{4}-2 \mathit {atanh} \left (a x \right ) a^{2} x^{2}+\mathit {atanh} \left (a x \right )}d x \right ) a^{2}+\mathit {atanh} \left (a x \right )^{2}+\mathit {atanh} \left (a x \right ) a x +1}{3 \mathit {atanh} \left (a x \right )^{3} a \left (a^{2} x^{2}-1\right )} \] Input:

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

Output:

(atanh(a*x)**3*int(x**2/(atanh(a*x)**2*a**4*x**4 - 2*atanh(a*x)**2*a**2*x* 
*2 + atanh(a*x)**2),x)*a**5*x**2 - atanh(a*x)**3*int(x**2/(atanh(a*x)**2*a 
**4*x**4 - 2*atanh(a*x)**2*a**2*x**2 + atanh(a*x)**2),x)*a**3 + 2*atanh(a* 
x)**3*int(x/(atanh(a*x)*a**4*x**4 - 2*atanh(a*x)*a**2*x**2 + atanh(a*x)),x 
)*a**4*x**2 - 2*atanh(a*x)**3*int(x/(atanh(a*x)*a**4*x**4 - 2*atanh(a*x)*a 
**2*x**2 + atanh(a*x)),x)*a**2 + atanh(a*x)**2 + atanh(a*x)*a*x + 1)/(3*at 
anh(a*x)**3*a*(a**2*x**2 - 1))