\(\int \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx\) [471]

Optimal result

Integrand size = 21, antiderivative size = 158 \[ \int \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=-\frac {\arcsin (a x)}{a}+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2+\frac {\arctan \left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2}{a}-\frac {i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )}{a}+\frac {i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )}{a}+\frac {i \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )}{a}-\frac {i \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )}{a} \] Output:

-arcsin(a*x)/a+(-a^2*x^2+1)^(1/2)*arctanh(a*x)/a+1/2*x*(-a^2*x^2+1)^(1/2)* 
arctanh(a*x)^2+arctan((a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)^2/a-I*arcta 
nh(a*x)*polylog(2,-I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a+I*arctanh(a*x)*polylog( 
2,I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a+I*polylog(3,-I*(a*x+1)/(-a^2*x^2+1)^(1/2 
))/a-I*polylog(3,I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.18 \[ \int \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\frac {\sqrt {1-a^2 x^2} \left (2 \text {arctanh}(a x)+a x \text {arctanh}(a x)^2-\frac {i \left (-4 i \arctan \left (\tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )+\text {arctanh}(a x)^2 \log \left (1-i e^{-\text {arctanh}(a x)}\right )-\text {arctanh}(a x)^2 \log \left (1+i e^{-\text {arctanh}(a x)}\right )+2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{-\text {arctanh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{-\text {arctanh}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{2 a} \] Input:

Integrate[Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2,x]
 

Output:

(Sqrt[1 - a^2*x^2]*(2*ArcTanh[a*x] + a*x*ArcTanh[a*x]^2 - (I*((-4*I)*ArcTa 
n[Tanh[ArcTanh[a*x]/2]] + ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - ArcTa 
nh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] + 2*ArcTanh[a*x]*PolyLog[2, (-I)/E^Arc 
Tanh[a*x]] - 2*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] + 2*PolyLog[3, (- 
I)/E^ArcTanh[a*x]] - 2*PolyLog[3, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/ 
(2*a)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6506, 223, 6514, 3042, 4668, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 6506

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 6514

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

\(\displaystyle \frac {-2 i \int \text {arctanh}(a x) \log \left (1-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 i \int \text {arctanh}(a x) \log \left (1+i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a}-\frac {\arcsin (a x)}{a}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {2 i \left (\int \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a}-\frac {\arcsin (a x)}{a}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a}-\frac {\arcsin (a x)}{a}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {1}{2} x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a}-\frac {\arcsin (a x)}{a}+\frac {2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )+2 i \left (\operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )}{2 a}\)

Input:

Int[Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2,x]
 

Output:

-(ArcSin[a*x]/a) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a + (x*Sqrt[1 - a^2*x^ 
2]*ArcTanh[a*x]^2)/2 + (2*ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 + (2*I)*(- 
(ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]]) + PolyLog[3, (-I)*E^ArcTanh 
[a*x]]) - (2*I)*(-(ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]]) + PolyLog[3, 
 I*E^ArcTanh[a*x]]))/(2*a)
 

Defintions of rubi rules used

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 6506

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x 
_Symbol] :> Simp[b*p*(d + e*x^2)^q*((a + b*ArcTanh[c*x])^(p - 1)/(2*c*q*(2* 
q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTanh[c*x])^p/(2*q + 1)), x] 
 + Simp[2*d*(q/(2*q + 1))   Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, 
 x], x] - Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1)))   Int[(d + e*x^2)^(q - 1)* 
(a + b*ArcTanh[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c 
^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 1]
 

rule 6514

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[1/(c*Sqrt[d])   Subst[Int[(a + b*x)^p*Sech[x], x], x, ArcTa 
nh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0 
] && GtQ[d, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int \sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right )^{2}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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