\(\int \frac {x^2}{(1-a^2 x^2)^2 \text {arctanh}(a x)^2} \, dx\) [288]

Optimal result

Integrand size = 22, antiderivative size = 38 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2} \, dx=-\frac {x^2}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {\text {Shi}(2 \text {arctanh}(a x))}{a^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2} \, dx=\frac {x^2}{a \left (-1+a^2 x^2\right ) \text {arctanh}(a x)}+\frac {\text {Shi}(2 \text {arctanh}(a x))}{a^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6568, 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.

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

\(\Big \downarrow \) 6568

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

\(\Big \downarrow \) 6596

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

Input:

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

Output:

-(x^2/(a*(1 - a^2*x^2)*ArcTanh[a*x])) + SinhIntegral[2*ArcTanh[a*x]]/a^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 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 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 = 1.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {\frac {1}{2 \,\operatorname {arctanh}\left (a x \right )}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Shi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{a^{3}}\)	\(36\)
default	\(\frac {\frac {1}{2 \,\operatorname {arctanh}\left (a x \right )}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Shi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{a^{3}}\)	\(36\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (36) = 72\).

Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.92 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2} \, dx=\frac {4 \, a^{2} x^{2} + {\left ({\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - {\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{2 \, {\left (a^{5} x^{2} - a^{3}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

2*x^2/((a^3*x^2 - a)*log(a*x + 1) - (a^3*x^2 - a)*log(-a*x + 1)) - 4*integ 
rate(-x/((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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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