\(\int \frac {x \text {arctanh}(a x)^3}{(1-a^2 x^2)^{3/2}} \, dx\) [406]

Optimal result

Integrand size = 22, antiderivative size = 94 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {6 x}{a \sqrt {1-a^2 x^2}}+\frac {6 \text {arctanh}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 x \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+\frac {\text {arctanh}(a x)^3}{a^2 \sqrt {1-a^2 x^2}} \] Output:

-6*x/a/(-a^2*x^2+1)^(1/2)+6*arctanh(a*x)/a^2/(-a^2*x^2+1)^(1/2)-3*x*arctan 
h(a*x)^2/a/(-a^2*x^2+1)^(1/2)+arctanh(a*x)^3/a^2/(-a^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.48 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {-6 a x+6 \text {arctanh}(a x)-3 a x \text {arctanh}(a x)^2+\text {arctanh}(a x)^3}{a^2 \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(-6*a*x + 6*ArcTanh[a*x] - 3*a*x*ArcTanh[a*x]^2 + ArcTanh[a*x]^3)/(a^2*Sqr 
t[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6556, 6524, 208}

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.

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 6524

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

\(\Big \downarrow \) 208

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 6524

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[(-b)*p*((a + b*ArcTanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) 
), x] + (Simp[x*((a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2])), x] + Simp[b^2 
*p*(p - 1)   Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; 
 FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
 

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 = 0.52 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.60

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (-3 \operatorname {arctanh}\left (a x \right )^{2} a x +\operatorname {arctanh}\left (a x \right )^{3}-6 a x +6 \,\operatorname {arctanh}\left (a x \right )\right )}{a^{2} \left (a^{2} x^{2}-1\right )}\)	\(56\)
orering	\(-\frac {4 \left (a x -1\right ) \left (a x +1\right ) \left (6 a^{6} x^{6}+a^{4} x^{4}+a^{2} x^{2}+2\right ) \operatorname {arctanh}\left (a x \right )^{3}}{x^{2} a^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {2 \left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (18 a^{4} x^{4}+7 a^{2} x^{2}+4\right ) \left (\frac {\operatorname {arctanh}\left (a x \right )^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 x \operatorname {arctanh}\left (a x \right )^{2} a}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {3 x^{2} \operatorname {arctanh}\left (a x \right )^{3} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{a^{4} x^{2}}-\frac {4 \left (a x +1\right )^{3} \left (a x -1\right )^{3} \left (3 a^{2} x^{2}+1\right ) \left (\frac {6 \operatorname {arctanh}\left (a x \right )^{2} a}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {9 \operatorname {arctanh}\left (a x \right )^{3} a^{2} x}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {6 x \,\operatorname {arctanh}\left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}+\frac {24 x^{2} \operatorname {arctanh}\left (a x \right )^{2} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}+\frac {15 x^{3} \operatorname {arctanh}\left (a x \right )^{3} a^{4}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}\right )}{x \,a^{4}}-\frac {\left (a x +1\right )^{4} \left (a x -1\right )^{4} \left (\frac {18 \,\operatorname {arctanh}\left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}+\frac {105 \operatorname {arctanh}\left (a x \right )^{2} a^{3} x}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}+\frac {90 \operatorname {arctanh}\left (a x \right )^{3} a^{4} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}+\frac {9 \operatorname {arctanh}\left (a x \right )^{3} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {6 x \,a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}+\frac {90 x^{2} \operatorname {arctanh}\left (a x \right ) a^{4}}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}+\frac {213 x^{3} \operatorname {arctanh}\left (a x \right )^{2} a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}+\frac {105 x^{4} \operatorname {arctanh}\left (a x \right )^{3} a^{6}}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}\right )}{a^{4}}\)	\(530\)

Input:

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

Output:

-1/a^2*(-a^2*x^2+1)^(1/2)*(-3*arctanh(a*x)^2*a*x+arctanh(a*x)^3-6*a*x+6*ar 
ctanh(a*x))/(a^2*x^2-1)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.97 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} {\left (6 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 48 \, a x - 24 \, \log \left (-\frac {a x + 1}{a x - 1}\right )\right )}}{8 \, {\left (a^{4} x^{2} - a^{2}\right )}} \] Input:

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

Output:

1/8*sqrt(-a^2*x^2 + 1)*(6*a*x*log(-(a*x + 1)/(a*x - 1))^2 - log(-(a*x + 1) 
/(a*x - 1))^3 + 48*a*x - 24*log(-(a*x + 1)/(a*x - 1)))/(a^4*x^2 - a^2)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.94 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {3 \, x \operatorname {artanh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} a} + \frac {\operatorname {artanh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {6 \, {\left (\frac {x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {\operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} a}\right )}}{a} \] Input:

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

Output:

-3*x*arctanh(a*x)^2/(sqrt(-a^2*x^2 + 1)*a) + arctanh(a*x)^3/(sqrt(-a^2*x^2 
 + 1)*a^2) - 6*(x/sqrt(-a^2*x^2 + 1) - arctanh(a*x)/(sqrt(-a^2*x^2 + 1)*a) 
)/a
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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