\(\int \frac {x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) [375]

Optimal result

Integrand size = 22, antiderivative size = 120 \[ \int \frac {x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {4 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a^2}-\frac {2 i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a^2}+\frac {2 i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a^2} \] Output:

-4*arctan((-a*x+1)^(1/2)/(a*x+1)^(1/2))*arctanh(a*x)/a^2-(-a^2*x^2+1)^(1/2 
)*arctanh(a*x)^2/a^2-2*I*polylog(2,-I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a^2+2* 
I*polylog(2,I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a^2
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.87 \[ \int \frac {x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\text {arctanh}(a x) \left (\sqrt {1-a^2 x^2} \text {arctanh}(a x)+2 i \left (\log \left (1-i e^{-\text {arctanh}(a x)}\right )-\log \left (1+i e^{-\text {arctanh}(a x)}\right )\right )\right )+2 i \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-2 i \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )}{a^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6556, 6512}

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.

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 6512

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

Input:

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

Output:

-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/a^2) + (2*((-2*ArcTan[Sqrt[1 - a*x]/S 
qrt[1 + a*x]]*ArcTanh[a*x])/a - (I*PolyLog[2, ((-I)*Sqrt[1 - a*x])/Sqrt[1 
+ a*x]])/a + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a))/a
 

Defintions of rubi rules used

rule 6512

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol 
] :> Simp[-2*(a + b*ArcTanh[c*x])*(ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*S 
qrt[d])), x] + (-Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 - c*x]/Sqrt[1 + c*x])]/( 
c*Sqrt[d])), x] + Simp[I*b*(PolyLog[2, I*(Sqrt[1 - c*x]/Sqrt[1 + c*x])]/(c* 
Sqrt[d])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 
0]
 

rule 6556

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

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.24

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right )^{2}}{a^{2}}-\frac {2 i \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2}}+\frac {2 i \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2}}-\frac {2 i \operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2}}+\frac {2 i \operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2}}\)	\(149\)

Input:

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

Output:

-(-a^2*x^2+1)^(1/2)*arctanh(a*x)^2/a^2-2*I/a^2*arctanh(a*x)*ln(1+I*(a*x+1) 
/(-a^2*x^2+1)^(1/2))+2*I/a^2*arctanh(a*x)*ln(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2 
))-2*I/a^2*dilog(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))+2*I/a^2*dilog(1-I*(a*x+1) 
/(-a^2*x^2+1)^(1/2))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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