\(\int \frac {x^3 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) [373]

Optimal result

Integrand size = 24, antiderivative size = 205 \[ \int \frac {x^3 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2}}{3 a^4}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^3}-\frac {10 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{3 a^2}-\frac {5 i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{3 a^4}+\frac {5 i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{3 a^4} \] Output:

-1/3*(-a^2*x^2+1)^(1/2)/a^4-1/3*x*(-a^2*x^2+1)^(1/2)*arctanh(a*x)/a^3-10/3 
*arctan((-a*x+1)^(1/2)/(a*x+1)^(1/2))*arctanh(a*x)/a^4-2/3*(-a^2*x^2+1)^(1 
/2)*arctanh(a*x)^2/a^4-1/3*x^2*(-a^2*x^2+1)^(1/2)*arctanh(a*x)^2/a^2-5/3*I 
*polylog(2,-I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a^4+5/3*I*polylog(2,I*(-a*x+1) 
^(1/2)/(a*x+1)^(1/2))/a^4
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.78 \[ \int \frac {x^3 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (-1-a x \text {arctanh}(a x)-3 \text {arctanh}(a x)^2+\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2-\frac {5 i \text {arctanh}(a x) \left (\log \left (1-i e^{-\text {arctanh}(a x)}\right )-\log \left (1+i e^{-\text {arctanh}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}-\frac {5 i \left (\operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{3 a^4} \] Input:

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

Output:

(Sqrt[1 - a^2*x^2]*(-1 - a*x*ArcTanh[a*x] - 3*ArcTanh[a*x]^2 + (1 - a^2*x^ 
2)*ArcTanh[a*x]^2 - ((5*I)*ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 
 + I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2] - ((5*I)*(PolyLog[2, (-I)/E^ArcTa 
nh[a*x]] - PolyLog[2, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(3*a^4)
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.57, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6578, 6556, 6512, 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.

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

\(\Big \downarrow \) 6578

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 6512

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

\(\Big \downarrow \) 6578

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6512

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

Input:

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

Output:

-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/a^2 + (2*(-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])/Sqrt[1 + a*x]])/a)/(2* 
a^2)))/(3*a) + (2*(-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/a^2) + (2*((-2*Arc 
Tan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a - (I*PolyLog[2, ((-I)*Sqr 
t[1 - a*x])/Sqrt[1 + a*x]])/a + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a 
*x]])/a))/a))/(3*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 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]
 

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 = 0.50 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {\left (a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}+a x \,\operatorname {arctanh}\left (a x \right )+2 \operatorname {arctanh}\left (a x \right )^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}-\frac {5 i \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{3 a^{4}}+\frac {5 i \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{3 a^{4}}-\frac {5 i \operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{3 a^{4}}+\frac {5 i \operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{3 a^{4}}\)	\(173\)

Input:

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

Output:

-1/3*(a^2*x^2*arctanh(a*x)^2+a*x*arctanh(a*x)+2*arctanh(a*x)^2+1)*(-a^2*x^ 
2+1)^(1/2)/a^4-5/3*I/a^4*arctanh(a*x)*ln(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))+5 
/3*I/a^4*arctanh(a*x)*ln(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))-5/3*I/a^4*dilog(1 
+I*(a*x+1)/(-a^2*x^2+1)^(1/2))+5/3*I/a^4*dilog(1-I*(a*x+1)/(-a^2*x^2+1)^(1 
/2))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [F]

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

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

Output:

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