\(\int \frac {1}{(1-a^2 x^2)^{3/2} \text {arctanh}(a x)^2} \, dx\) [419]

Optimal result

Integrand size = 21, antiderivative size = 35 \[ \int \frac {1}{\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)^2} \, dx=-\frac {1}{a \sqrt {1-a^2 x^2} \text {arctanh}(a x)}+\frac {\text {Shi}(\text {arctanh}(a x))}{a} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)^2} \, dx=\frac {-\frac {1}{\sqrt {1-a^2 x^2} \text {arctanh}(a x)}+\text {Shi}(\text {arctanh}(a x))}{a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6528, 6596, 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.

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

\(\Big \downarrow \) 6528

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

\(\Big \downarrow \) 6596

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

Input:

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

Output:

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

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 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 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 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 = 0.55 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.77

method	result	size
default	\(\frac {\operatorname {arctanh}\left (a x \right ) \operatorname {Shi}\left (\operatorname {arctanh}\left (a x \right )\right ) a^{2} x^{2}-\operatorname {Shi}\left (\operatorname {arctanh}\left (a x \right )\right ) \operatorname {arctanh}\left (a x \right )+\sqrt {-a^{2} x^{2}+1}}{a \,\operatorname {arctanh}\left (a x \right ) \left (a^{2} x^{2}-1\right )}\)	\(62\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(-a^2*x^2 + 1)/((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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