\(\int \frac {x^2}{(1-a^2 x^2)^2 \text {arctanh}(a x)^3} \, dx\) [293]

Optimal result

Integrand size = 22, antiderivative size = 64 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {x}{a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {\text {Chi}(2 \text {arctanh}(a x))}{a^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\frac {\frac {a x (a x+2 \text {arctanh}(a x))}{\left (-1+a^2 x^2\right ) \text {arctanh}(a x)^2}+2 \text {Chi}(2 \text {arctanh}(a x))}{2 a^3} \] Input:

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

Output:

((a*x*(a*x + 2*ArcTanh[a*x]))/((-1 + a^2*x^2)*ArcTanh[a*x]^2) + 2*CoshInte 
gral[2*ArcTanh[a*x]])/(2*a^3)
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.69, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6568, 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.

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

\(\Big \downarrow \) 6568

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

\(\Big \downarrow \) 6594

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

\(\Big \downarrow \) 6530

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6596

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 6568

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

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 = 1.49 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(\frac {\frac {1}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{a^{3}}\)	\(51\)
default	\(\frac {\frac {1}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{a^{3}}\)	\(51\)

Input:

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

Output:

1/a^3*(1/4/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)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (59) = 118\).

Time = 0.07 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.05 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\frac {4 \, a^{2} x^{2} + 4 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) + {\left ({\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + {\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{2 \, {\left (a^{5} x^{2} - a^{3}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \] Input:

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

Output:

1/2*(4*a^2*x^2 + 4*a*x*log(-(a*x + 1)/(a*x - 1)) + ((a^2*x^2 - 1)*log_inte 
gral(-(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))^2)/((a^5*x^2 - a^3)*log(-(a*x + 1)/(a*x - 1 
))^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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