\(\int \frac {1}{(1-a^2 x^2)^3 \text {arctanh}(a x)^3} \, dx\) [340]

Optimal result

Integrand size = 19, antiderivative size = 69 \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}-\frac {2 x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}+\frac {\text {Chi}(2 \text {arctanh}(a x))}{a}+\frac {\text {Chi}(4 \text {arctanh}(a x))}{a} \] Output:

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\frac {-1-4 a x \text {arctanh}(a x)+2 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^2 \text {Chi}(2 \text {arctanh}(a x))+2 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^2 \text {Chi}(4 \text {arctanh}(a x))}{2 a \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^2} \] Input:

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

Output:

(-1 - 4*a*x*ArcTanh[a*x] + 2*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2*CoshIntegral[ 
2*ArcTanh[a*x]] + 2*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2*CoshIntegral[4*ArcTanh 
[a*x]])/(2*a*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2)
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.68, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {6528, 6594, 6530, 3042, 3793, 2009, 6596, 5971, 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.

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

\(\Big \downarrow \) 6528

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

\(\Big \downarrow \) 6594

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

\(\Big \downarrow \) 6530

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}+2 a \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^4}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )\)

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 a \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\)

\(\Big \downarrow \) 6596

\(\displaystyle 2 a \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\)

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 a \left (\frac {3 \left (\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))-\frac {1}{8} \log (\text {arctanh}(a x))\right )}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\)

Input:

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

Output:

-1/2*1/(a*(1 - a^2*x^2)^2*ArcTanh[a*x]^2) + 2*a*(-(x/(a*(1 - a^2*x^2)^2*Ar 
cTanh[a*x])) + (3*(CoshIntegral[4*ArcTanh[a*x]]/8 - Log[ArcTanh[a*x]]/8))/ 
a^2 + (CoshIntegral[2*ArcTanh[a*x]]/2 + CoshIntegral[4*ArcTanh[a*x]]/8 + ( 
3*Log[ArcTanh[a*x]])/8)/a^2)
 

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 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 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])
 

rule 6594

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

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 = 0.64 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.28

method	result	size
derivativedivides	\(\frac {-\frac {3}{16 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{16 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{a}\)	\(88\)
default	\(\frac {-\frac {3}{16 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{16 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{a}\)	\(88\)

Input:

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

Output:

1/a*(-3/16/arctanh(a*x)^2-1/4/arctanh(a*x)^2*cosh(2*arctanh(a*x))-1/2/arct 
anh(a*x)*sinh(2*arctanh(a*x))+Chi(2*arctanh(a*x))-1/16/arctanh(a*x)^2*cosh 
(4*arctanh(a*x))-1/4*sinh(4*arctanh(a*x))/arctanh(a*x)+Chi(4*arctanh(a*x)) 
)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (65) = 130\).

Time = 0.08 (sec) , antiderivative size = 241, normalized size of antiderivative = 3.49 \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\frac {8 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) - {\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4}{2 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \] Input:

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

Output:

-1/2*(8*a*x*log(-(a*x + 1)/(a*x - 1)) - ((a^4*x^4 - 2*a^2*x^2 + 1)*log_int 
egral((a^2*x^2 + 2*a*x + 1)/(a^2*x^2 - 2*a*x + 1)) + (a^4*x^4 - 2*a^2*x^2 
+ 1)*log_integral((a^2*x^2 - 2*a*x + 1)/(a^2*x^2 + 2*a*x + 1)) + (a^4*x^4 
- 2*a^2*x^2 + 1)*log_integral(-(a*x + 1)/(a*x - 1)) + (a^4*x^4 - 2*a^2*x^2 
 + 1)*log_integral(-(a*x - 1)/(a*x + 1)))*log(-(a*x + 1)/(a*x - 1))^2 + 4) 
/((a^5*x^4 - 2*a^3*x^2 + a)*log(-(a*x + 1)/(a*x - 1))^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-2*(2*a*x*log(a*x + 1) - 2*a*x*log(-a*x + 1) + 1)/((a^5*x^4 - 2*a^3*x^2 + 
a)*log(a*x + 1)^2 - 2*(a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1)*log(-a*x + 1) 
 + (a^5*x^4 - 2*a^3*x^2 + a)*log(-a*x + 1)^2) + integrate(-4*(3*a^2*x^2 + 
1)/((a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1) - (a^6*x^6 - 3*a^4* 
x^4 + 3*a^2*x^2 - 1)*log(-a*x + 1)), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\frac {-4 \mathit {atanh} \left (a x \right )^{2} \left (\int \frac {x}{\mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{2}}d x \right ) a^{6} x^{4}+8 \mathit {atanh} \left (a x \right )^{2} \left (\int \frac {x}{\mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{2}}d x \right ) a^{4} x^{2}-4 \mathit {atanh} \left (a x \right )^{2} \left (\int \frac {x}{\mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{2}}d x \right ) a^{2}-1}{2 \mathit {atanh} \left (a x \right )^{2} a \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 4*atanh(a*x)**2*int(x/(atanh(a*x)**2*a**6*x**6 - 3*atanh(a*x)**2*a**4* 
x**4 + 3*atanh(a*x)**2*a**2*x**2 - atanh(a*x)**2),x)*a**6*x**4 + 8*atanh(a 
*x)**2*int(x/(atanh(a*x)**2*a**6*x**6 - 3*atanh(a*x)**2*a**4*x**4 + 3*atan 
h(a*x)**2*a**2*x**2 - atanh(a*x)**2),x)*a**4*x**2 - 4*atanh(a*x)**2*int(x/ 
(atanh(a*x)**2*a**6*x**6 - 3*atanh(a*x)**2*a**4*x**4 + 3*atanh(a*x)**2*a** 
2*x**2 - atanh(a*x)**2),x)*a**2 - 1)/(2*atanh(a*x)**2*a*(a**4*x**4 - 2*a** 
2*x**2 + 1))