\(\int \frac {x}{(1-a^2 x^2)^2 \text {arctanh}(a x)^2} \, dx\) [289]

Optimal result

Integrand size = 20, antiderivative size = 36 \[ \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2} \, dx=-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {\text {Chi}(2 \text {arctanh}(a x))}{a^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2} \, dx=\frac {\frac {a x}{\left (-1+a^2 x^2\right ) \text {arctanh}(a x)}+\text {Chi}(2 \text {arctanh}(a x))}{a^2} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(36)=72\).

Time = 0.89 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6594, 6530, 3042, 3793, 2009, 6596, 3042, 25, 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.

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

\(\Big \downarrow \) 6594

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

\(\Big \downarrow \) 6530

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6596

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-(x/(a*(1 - a^2*x^2)*ArcTanh[a*x])) + (CoshIntegral[2*ArcTanh[a*x]]/2 - Lo 
g[ArcTanh[a*x]]/2)/a^2 + (CoshIntegral[2*ArcTanh[a*x]]/2 + Log[ArcTanh[a*x 
]]/2)/a^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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])
 

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 = 1.47 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{a^{2}}\)	\(28\)
default	\(\frac {-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{a^{2}}\)	\(28\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (34) = 68\).

Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.94 \[ \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2} \, dx=\frac {4 \, a x + {\left ({\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + {\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{2 \, {\left (a^{4} x^{2} - a^{2}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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