\(\int \frac {x^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) [367]

Optimal result

Integrand size = 22, antiderivative size = 146 \[ \int \frac {x^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2}}{2 a^3}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 a^2}-\frac {\arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)}{a^3}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a^3}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a^3} \] Output:

-1/2*(-a^2*x^2+1)^(1/2)/a^3-1/2*x*(-a^2*x^2+1)^(1/2)*arctanh(a*x)/a^2-arct 
an((-a*x+1)^(1/2)/(a*x+1)^(1/2))*arctanh(a*x)/a^3-1/2*I*polylog(2,-I*(-a*x 
+1)^(1/2)/(a*x+1)^(1/2))/a^3+1/2*I*polylog(2,I*(-a*x+1)^(1/2)/(a*x+1)^(1/2 
))/a^3
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2}+a x \sqrt {1-a^2 x^2} \text {arctanh}(a x)+i \text {arctanh}(a x) \log \left (1-i e^{-\text {arctanh}(a x)}\right )-i \text {arctanh}(a x) \log \left (1+i e^{-\text {arctanh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )}{2 a^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6578, 241, 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.

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

\(\Big \downarrow \) 6578

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6512

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

Input:

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

Output:

-1/2*Sqrt[1 - a^2*x^2]/a^3 - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(2*a^2) + 
((-2*ArcTan[Sqrt[1 - a*x]/Sqrt[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])/Sq 
rt[1 + a*x]])/a)/(2*a^2)
 

Defintions of rubi rules used

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

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 6578

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

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.04

method	result	size
default	\(-\frac {\left (a x \,\operatorname {arctanh}\left (a x \right )+1\right ) \sqrt {-a^{2} x^{2}+1}}{2 a^{3}}-\frac {i \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{3}}+\frac {i \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{3}}-\frac {i \operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{3}}+\frac {i \operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{3}}\)	\(152\)

Input:

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

Output:

-1/2*(a*x*arctanh(a*x)+1)*(-a^2*x^2+1)^(1/2)/a^3-1/2*I/a^3*arctanh(a*x)*ln 
(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I/a^3*arctanh(a*x)*ln(1-I*(a*x+1)/(-a 
^2*x^2+1)^(1/2))-1/2*I/a^3*dilog(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I/a^3 
*dilog(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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