\(\int x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx\) [427]

Optimal result

Integrand size = 22, antiderivative size = 136 \[ \int x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx=\frac {x \sqrt {1-a^2 x^2}}{24 a^3}+\frac {x^3 \sqrt {1-a^2 x^2}}{20 a}+\frac {11 \arcsin (a x)}{120 a^4}-\frac {2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{15 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x) \] Output:

1/24*x*(-a^2*x^2+1)^(1/2)/a^3+1/20*x^3*(-a^2*x^2+1)^(1/2)/a+11/120*arcsin( 
a*x)/a^4-2/15*(-a^2*x^2+1)^(1/2)*arctanh(a*x)/a^4-1/15*x^2*(-a^2*x^2+1)^(1 
/2)*arctanh(a*x)/a^2+1/5*x^4*(-a^2*x^2+1)^(1/2)*arctanh(a*x)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.58 \[ \int x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx=\frac {a x \sqrt {1-a^2 x^2} \left (5+6 a^2 x^2\right )+11 \arcsin (a x)+8 \sqrt {1-a^2 x^2} \left (-2-a^2 x^2+3 a^4 x^4\right ) \text {arctanh}(a x)}{120 a^4} \] Input:

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

Output:

(a*x*Sqrt[1 - a^2*x^2]*(5 + 6*a^2*x^2) + 11*ArcSin[a*x] + 8*Sqrt[1 - a^2*x 
^2]*(-2 - a^2*x^2 + 3*a^4*x^4)*ArcTanh[a*x])/(120*a^4)
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.54, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6572, 262, 262, 223, 6578, 262, 223, 6556, 223}

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.

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

\(\Big \downarrow \) 6572

\(\displaystyle \frac {1}{5} \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{5} a \int \frac {x^4}{\sqrt {1-a^2 x^2}}dx+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{5} \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{5} a \left (\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{5} \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{5} a \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{5} \int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\)

\(\Big \downarrow \) 6578

\(\displaystyle \frac {1}{5} \left (\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int \frac {x^2}{\sqrt {1-a^2 x^2}}dx}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{5} \left (\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{5} \left (\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\)

\(\Big \downarrow \) 6556

\(\displaystyle \frac {1}{5} \left (\frac {2 \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)+\frac {1}{5} \left (\frac {2 \left (\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}\right )-\frac {1}{5} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )\)

Input:

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

Output:

-1/5*(a*(-1/4*(x^3*Sqrt[1 - a^2*x^2])/a^2 + (3*(-1/2*(x*Sqrt[1 - a^2*x^2]) 
/a^2 + ArcSin[a*x]/(2*a^3)))/(4*a^2))) + (x^4*Sqrt[1 - a^2*x^2]*ArcTanh[a* 
x])/5 + ((-1/2*(x*Sqrt[1 - a^2*x^2])/a^2 + ArcSin[a*x]/(2*a^3))/(3*a) - (x 
^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(3*a^2) + (2*(ArcSin[a*x]/a^2 - (Sqrt[1 
 - a^2*x^2]*ArcTanh[a*x])/a^2))/(3*a^2))/5
 

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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

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 6572

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

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 [C] (verified)

Result contains complex when optimal does not.

Time = 1.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (24 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )+6 a^{3} x^{3}-8 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )+5 a x -16 \,\operatorname {arctanh}\left (a x \right )\right )}{120 a^{4}}+\frac {11 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}+i\right )}{120 a^{4}}-\frac {11 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-i\right )}{120 a^{4}}\)	\(120\)

Input:

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

Output:

1/120/a^4*(-(a*x-1)*(a*x+1))^(1/2)*(24*a^4*x^4*arctanh(a*x)+6*a^3*x^3-8*a^ 
2*x^2*arctanh(a*x)+5*a*x-16*arctanh(a*x))+11/120*I*ln((a*x+1)/(-a^2*x^2+1) 
^(1/2)+I)/a^4-11/120*I*ln((a*x+1)/(-a^2*x^2+1)^(1/2)-I)/a^4
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.67 \[ \int x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx=\frac {{\left (6 \, a^{3} x^{3} + 5 \, a x + 4 \, {\left (3 \, a^{4} x^{4} - a^{2} x^{2} - 2\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1} - 22 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{120 \, a^{4}} \] Input:

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

Output:

1/120*((6*a^3*x^3 + 5*a*x + 4*(3*a^4*x^4 - a^2*x^2 - 2)*log(-(a*x + 1)/(a* 
x - 1)))*sqrt(-a^2*x^2 + 1) - 22*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a 
^4
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.94 \[ \int x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x) \, dx=-\frac {1}{120} \, a {\left (\frac {3 \, {\left (\frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )}}{a^{2}} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} x + \frac {\arcsin \left (a x\right )}{a}\right )}}{a^{4}}\right )} - \frac {1}{15} \, {\left (\frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{a^{2}} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4}}\right )} \operatorname {artanh}\left (a x\right ) \] Input:

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

Output:

-1/120*a*(3*(2*(-a^2*x^2 + 1)^(3/2)*x/a^2 - sqrt(-a^2*x^2 + 1)*x/a^2 - arc 
sin(a*x)/a^3)/a^2 - 8*(sqrt(-a^2*x^2 + 1)*x + arcsin(a*x)/a)/a^4) - 1/15*( 
3*(-a^2*x^2 + 1)^(3/2)*x^2/a^2 + 2*(-a^2*x^2 + 1)^(3/2)/a^4)*arctanh(a*x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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