\(\int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) [364]

Optimal result

Integrand size = 22, antiderivative size = 139 \[ \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {5 x \sqrt {1-a^2 x^2}}{24 a^5}-\frac {x^3 \sqrt {1-a^2 x^2}}{20 a^3}+\frac {89 \arcsin (a x)}{120 a^6}-\frac {8 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{15 a^6}-\frac {4 x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{15 a^4}-\frac {x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 a^2} \] Output:

-5/24*x*(-a^2*x^2+1)^(1/2)/a^5-1/20*x^3*(-a^2*x^2+1)^(1/2)/a^3+89/120*arcs 
in(a*x)/a^6-8/15*(-a^2*x^2+1)^(1/2)*arctanh(a*x)/a^6-4/15*x^2*(-a^2*x^2+1) 
^(1/2)*arctanh(a*x)/a^4-1/5*x^4*(-a^2*x^2+1)^(1/2)*arctanh(a*x)/a^2
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.57 \[ \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {a x \sqrt {1-a^2 x^2} \left (25+6 a^2 x^2\right )-89 \arcsin (a x)+8 \sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \text {arctanh}(a x)}{120 a^6} \] Input:

Integrate[(x^5*ArcTanh[a*x])/Sqrt[1 - a^2*x^2],x]
 

Output:

-1/120*(a*x*Sqrt[1 - a^2*x^2]*(25 + 6*a^2*x^2) - 89*ArcSin[a*x] + 8*Sqrt[1 
 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcTanh[a*x])/a^6
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.57, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6578, 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.

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

\(\Big \downarrow \) 6578

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 6578

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 223

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

Input:

Int[(x^5*ArcTanh[a*x])/Sqrt[1 - a^2*x^2],x]
 

Output:

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

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 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 = 1.76 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.85

method	result	size
default	\(-\frac {\left (24 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )+6 a^{3} x^{3}+32 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )+25 a x +64 \,\operatorname {arctanh}\left (a x \right )\right ) \sqrt {-a^{2} x^{2}+1}}{120 a^{6}}+\frac {89 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}+i\right )}{120 a^{6}}-\frac {89 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-i\right )}{120 a^{6}}\)	\(118\)

Input:

int(x^5*arctanh(a*x)/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/120*(24*a^4*x^4*arctanh(a*x)+6*a^3*x^3+32*a^2*x^2*arctanh(a*x)+25*a*x+6 
4*arctanh(a*x))*(-a^2*x^2+1)^(1/2)/a^6+89/120*I/a^6*ln((a*x+1)/(-a^2*x^2+1 
)^(1/2)+I)-89/120*I/a^6*ln((a*x+1)/(-a^2*x^2+1)^(1/2)-I)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.65 \[ \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {{\left (6 \, a^{3} x^{3} + 25 \, a x + 4 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1} + 178 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{120 \, a^{6}} \] Input:

integrate(x^5*arctanh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

-1/120*((6*a^3*x^3 + 25*a*x + 4*(3*a^4*x^4 + 4*a^2*x^2 + 8)*log(-(a*x + 1) 
/(a*x - 1)))*sqrt(-a^2*x^2 + 1) + 178*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x 
)))/a^6
 

Sympy [F]

\[ \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{5} \operatorname {atanh}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:

integrate(x**5*atanh(a*x)/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral(x**5*atanh(a*x)/sqrt(-(a*x - 1)*(a*x + 1)), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.17 \[ \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {1}{120} \, a {\left (\frac {3 \, {\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{3}}{a^{2}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{a^{4}} - \frac {3 \, \arcsin \left (a x\right )}{a^{5}}\right )}}{a^{2}} + \frac {16 \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )}}{a^{4}} - \frac {64 \, \arcsin \left (a x\right )}{a^{7}}\right )} - \frac {1}{15} \, {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6}}\right )} \operatorname {artanh}\left (a x\right ) \] Input:

integrate(x^5*arctanh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

-1/120*a*(3*(2*sqrt(-a^2*x^2 + 1)*x^3/a^2 + 3*sqrt(-a^2*x^2 + 1)*x/a^4 - 3 
*arcsin(a*x)/a^5)/a^2 + 16*(sqrt(-a^2*x^2 + 1)*x/a^2 - arcsin(a*x)/a^3)/a^ 
4 - 64*arcsin(a*x)/a^7) - 1/15*(3*sqrt(-a^2*x^2 + 1)*x^4/a^2 + 4*sqrt(-a^2 
*x^2 + 1)*x^2/a^4 + 8*sqrt(-a^2*x^2 + 1)/a^6)*arctanh(a*x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^5*arctanh(a*x)/(-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. \[ \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^5\,\mathrm {atanh}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \] Input:

int((x^5*atanh(a*x))/(1 - a^2*x^2)^(1/2),x)
 

Output:

int((x^5*atanh(a*x))/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^5 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {atanh} \left (a x \right ) x^{5}}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:

int(x^5*atanh(a*x)/(-a^2*x^2+1)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int((atanh(a*x)*x**5)/sqrt( - a**2*x**2 + 1),x)