\(\int \frac {x^2}{(1-a^2 x^2)^3 \text {arctanh}(a x)^2} \, dx\) [332]

Optimal result

Integrand size = 22, antiderivative size = 41 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\frac {x^2}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}+\frac {\text {Shi}(4 \text {arctanh}(a x))}{2 a^3} \] Output:

-x^2/a/(-a^2*x^2+1)^2/arctanh(a*x)+1/2*Shi(4*arctanh(a*x))/a^3
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.37 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\frac {-2 a^2 x^2+\left (-1+a^2 x^2\right )^2 \text {arctanh}(a x) \text {Shi}(4 \text {arctanh}(a x))}{2 a^3 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)} \] Input:

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

Output:

(-2*a^2*x^2 + (-1 + a^2*x^2)^2*ArcTanh[a*x]*SinhIntegral[4*ArcTanh[a*x]])/ 
(2*a^3*(-1 + a^2*x^2)^2*ArcTanh[a*x])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(97\) vs. \(2(41)=82\).

Time = 0.81 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.37, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6590, 6528, 6596, 5971, 27, 2009, 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.

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

\(\Big \downarrow \) 6590

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

\(\Big \downarrow \) 6528

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

\(\Big \downarrow \) 6596

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {\int -\frac {i \sin (2 i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}}{a^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}-\frac {i \int \frac {\sin (2 i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}}{a^2}\)

\(\Big \downarrow \) 3779

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

Input:

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

Output:

-((-(1/(a*(1 - a^2*x^2)*ArcTanh[a*x])) + SinhIntegral[2*ArcTanh[a*x]]/a)/a 
^2) + (-(1/(a*(1 - a^2*x^2)^2*ArcTanh[a*x])) + (4*(SinhIntegral[2*ArcTanh[ 
a*x]]/4 + SinhIntegral[4*ArcTanh[a*x]]/8))/a)/a^2
 

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 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 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 6590

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[1/e   Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*A 
rcTanh[c*x])^p, x], x] - Simp[d/e   Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcT 
anh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && In 
tegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[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 = 1.42 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{8 \,\operatorname {arctanh}\left (a x \right )}+\frac {\operatorname {Shi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{2}+\frac {1}{8 \,\operatorname {arctanh}\left (a x \right )}}{a^{3}}\)	\(38\)
default	\(\frac {-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{8 \,\operatorname {arctanh}\left (a x \right )}+\frac {\operatorname {Shi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{2}+\frac {1}{8 \,\operatorname {arctanh}\left (a x \right )}}{a^{3}}\)	\(38\)

Input:

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

Output:

1/a^3*(-1/8/arctanh(a*x)*cosh(4*arctanh(a*x))+1/2*Shi(4*arctanh(a*x))+1/8/ 
arctanh(a*x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (38) = 76\).

Time = 0.08 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.00 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\frac {8 \, a^{2} x^{2} - {\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{4 \, {\left (a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )} \] Input:

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

Output:

-1/4*(8*a^2*x^2 - ((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^2 
*x^2 - 2*a*x + 1)/(a^2*x^2 + 2*a*x + 1)))*log(-(a*x + 1)/(a*x - 1)))/((a^7 
*x^4 - 2*a^5*x^2 + a^3)*log(-(a*x + 1)/(a*x - 1)))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-2*x^2/((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)) + integrate(-4*(a^2*x^3 + x)/((a^7*x^6 - 3*a^5*x^4 + 3*a^3 
*x^2 - a)*log(a*x + 1) - (a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)*log(-a*x + 
1)), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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