\(\int e^{-3 \text {arctanh}(a x)} x^3 \, dx\) [66]

Optimal result

Integrand size = 12, antiderivative size = 121 \[ \int e^{-3 \text {arctanh}(a x)} x^3 \, dx=\frac {4 (1-a x)}{a^4 \sqrt {1-a^2 x^2}}+\frac {7 \sqrt {1-a^2 x^2}}{a^4}-\frac {19 x \sqrt {1-a^2 x^2}}{8 a^3}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a}-\frac {\left (1-a^2 x^2\right )^{3/2}}{a^4}+\frac {51 \arcsin (a x)}{8 a^4} \] Output:

4*(-a*x+1)/a^4/(-a^2*x^2+1)^(1/2)+7*(-a^2*x^2+1)^(1/2)/a^4-19/8*x*(-a^2*x^ 
2+1)^(1/2)/a^3-1/4*x^3*(-a^2*x^2+1)^(1/2)/a-(-a^2*x^2+1)^(3/2)/a^4+51/8*ar 
csin(a*x)/a^4
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.58 \[ \int e^{-3 \text {arctanh}(a x)} x^3 \, dx=\sqrt {1-a^2 x^2} \left (\frac {6}{a^4}-\frac {19 x}{8 a^3}+\frac {x^2}{a^2}-\frac {x^3}{4 a}+\frac {4}{a^4 (1+a x)}\right )+\frac {51 \arcsin (a x)}{8 a^4} \] Input:

Integrate[x^3/E^(3*ArcTanh[a*x]),x]
 

Output:

Sqrt[1 - a^2*x^2]*(6/a^4 - (19*x)/(8*a^3) + x^2/a^2 - x^3/(4*a) + 4/(a^4*( 
1 + a*x))) + (51*ArcSin[a*x])/(8*a^4)
 

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.23, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {6674, 2164, 2027, 2164, 27, 563, 25, 2346, 25, 2346, 27, 2346, 27, 455, 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.

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

\(\Big \downarrow \) 6674

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

\(\Big \downarrow \) 2164

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

\(\Big \downarrow \) 2027

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

\(\Big \downarrow \) 2164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 563

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2346

\(\displaystyle \frac {-\frac {\int -\frac {-12 x^3 a^5+19 x^2 a^4-16 x a^3+16 a^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^3}+\frac {4 \sqrt {1-a^2 x^2}}{a^4 (a x+1)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {-12 x^3 a^5+19 x^2 a^4-16 x a^3+16 a^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^3}+\frac {4 \sqrt {1-a^2 x^2}}{a^4 (a x+1)}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {4 a^3 x^2 \sqrt {1-a^2 x^2}-\frac {\int -\frac {3 \left (19 x^2 a^6-24 x a^5+16 a^4\right )}{\sqrt {1-a^2 x^2}}dx}{3 a^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^3}+\frac {4 \sqrt {1-a^2 x^2}}{a^4 (a x+1)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {19 x^2 a^6-24 x a^5+16 a^4}{\sqrt {1-a^2 x^2}}dx}{a^2}+4 a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^3}+\frac {4 \sqrt {1-a^2 x^2}}{a^4 (a x+1)}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {3 a^6 (17-16 a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {19}{2} a^4 x \sqrt {1-a^2 x^2}}{a^2}+4 a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^3}+\frac {4 \sqrt {1-a^2 x^2}}{a^4 (a x+1)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {3}{2} a^4 \int \frac {17-16 a x}{\sqrt {1-a^2 x^2}}dx-\frac {19}{2} a^4 x \sqrt {1-a^2 x^2}}{a^2}+4 a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^3}+\frac {4 \sqrt {1-a^2 x^2}}{a^4 (a x+1)}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {\frac {\frac {3}{2} a^4 \left (17 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {16 \sqrt {1-a^2 x^2}}{a}\right )-\frac {19}{2} a^4 x \sqrt {1-a^2 x^2}}{a^2}+4 a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^3}+\frac {4 \sqrt {1-a^2 x^2}}{a^4 (a x+1)}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {4 \sqrt {1-a^2 x^2}}{a^4 (a x+1)}+\frac {\frac {\frac {\frac {3}{2} a^4 \left (\frac {16 \sqrt {1-a^2 x^2}}{a}+\frac {17 \arcsin (a x)}{a}\right )-\frac {19}{2} a^4 x \sqrt {1-a^2 x^2}}{a^2}+4 a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^3}\)

Input:

Int[x^3/E^(3*ArcTanh[a*x]),x]
 

Output:

(4*Sqrt[1 - a^2*x^2])/(a^4*(1 + a*x)) + (-1/4*(a^2*x^3*Sqrt[1 - a^2*x^2]) 
+ (4*a^3*x^2*Sqrt[1 - a^2*x^2] + ((-19*a^4*x*Sqrt[1 - a^2*x^2])/2 + (3*a^4 
*((16*Sqrt[1 - a^2*x^2])/a + (17*ArcSin[a*x])/a))/2)/a^2)/(4*a^2))/a^3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 455

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

rule 563

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[(-(-c)^(m - n - 2))*d^(2*n - m + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)* 
b^(n + 2)*(c + d*x))), x] - Simp[d^(2*n - m + 2)/b^(n + 1)   Int[(1/Sqrt[a 
+ b*x^2])*ExpandToSum[(2^(-n - 1)*(-c)^(m - n - 1) - d^m*x^m*(-c + d*x)^(-n 
 - 1))/(c + d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2 
, 0] && IGtQ[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
 

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

rule 2164

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] 
:> Simp[d*e   Int[(d + e*x)^(m - 1)*PolynomialQuotient[Pq, a*e + b*d*x, x]* 
(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] 
 && EqQ[b*d^2 + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + b*d*x, x], 0 
]
 

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

rule 6674

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x 
)^m*((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/2)*Sqrt[1 - a^2*x^2])), x] / 
; FreeQ[{a, c, m}, x] && IntegerQ[(n - 1)/2]
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.95

method	result	size
risch	\(\frac {\left (2 a^{3} x^{3}-8 a^{2} x^{2}+19 a x -48\right ) \left (a^{2} x^{2}-1\right )}{8 a^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {51 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{3} \sqrt {a^{2}}}+\frac {4 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a^{5} \left (x +\frac {1}{a}\right )}\)	\(115\)
default	\(\frac {\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}}{a^{3}}-\frac {3 \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )}{a^{4}}-\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-2 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{6}}+\frac {\frac {3 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+9 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )}{a^{5}}\)	\(521\)

Input:

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

Output:

1/8*(2*a^3*x^3-8*a^2*x^2+19*a*x-48)*(a^2*x^2-1)/a^4/(-a^2*x^2+1)^(1/2)+51/ 
8/a^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+4/a^5/(x+1/a)*( 
-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.76 \[ \int e^{-3 \text {arctanh}(a x)} x^3 \, dx=\frac {80 \, a x - 102 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (2 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 11 \, a^{2} x^{2} - 29 \, a x - 80\right )} \sqrt {-a^{2} x^{2} + 1} + 80}{8 \, {\left (a^{5} x + a^{4}\right )}} \] Input:

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

Output:

1/8*(80*a*x - 102*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^ 
4*x^4 - 6*a^3*x^3 + 11*a^2*x^2 - 29*a*x - 80)*sqrt(-a^2*x^2 + 1) + 80)/(a^ 
5*x + a^4)
 

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.78 \[ \int e^{-3 \text {arctanh}(a x)} x^3 \, dx=-\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{6} x^{2} + 2 \, a^{5} x + a^{4}} + \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, {\left (a^{5} x + a^{4}\right )}} + \frac {6 \, \sqrt {-a^{2} x^{2} + 1}}{a^{5} x + a^{4}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, a^{3}} - \frac {3 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} x}{2 \, a^{3}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{8 \, a^{3}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4}} + \frac {3 i \, \arcsin \left (a x + 2\right )}{2 \, a^{4}} + \frac {63 \, \arcsin \left (a x\right )}{8 \, a^{4}} - \frac {3 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3}}{a^{4}} + \frac {9 \, \sqrt {-a^{2} x^{2} + 1}}{2 \, a^{4}} \] Input:

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

Output:

-(-a^2*x^2 + 1)^(3/2)/(a^6*x^2 + 2*a^5*x + a^4) + 3/2*(-a^2*x^2 + 1)^(3/2) 
/(a^5*x + a^4) + 6*sqrt(-a^2*x^2 + 1)/(a^5*x + a^4) + 1/4*(-a^2*x^2 + 1)^( 
3/2)*x/a^3 - 3/2*sqrt(a^2*x^2 + 4*a*x + 3)*x/a^3 + 3/8*sqrt(-a^2*x^2 + 1)* 
x/a^3 - (-a^2*x^2 + 1)^(3/2)/a^4 + 3/2*I*arcsin(a*x + 2)/a^4 + 63/8*arcsin 
(a*x)/a^4 - 3*sqrt(a^2*x^2 + 4*a*x + 3)/a^4 + 9/2*sqrt(-a^2*x^2 + 1)/a^4
 

Giac [F(-2)]

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

integrate(x^3/(a*x+1)^3*(-a^2*x^2+1)^(3/2),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 [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.27 \[ \int e^{-3 \text {arctanh}(a x)} x^3 \, dx=\frac {51\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a^3\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2}{{\left (-a^2\right )}^{3/2}}-\frac {4}{a^2\,\sqrt {-a^2}}-\frac {19\,x\,\sqrt {-a^2}}{8\,a^3}+\frac {a^2\,x^2}{{\left (-a^2\right )}^{3/2}}+\frac {x^3\,{\left (-a^2\right )}^{3/2}}{4\,a^3}\right )}{\sqrt {-a^2}}-\frac {4\,\sqrt {1-a^2\,x^2}}{a^3\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \] Input:

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

Output:

(51*asinh(x*(-a^2)^(1/2)))/(8*a^3*(-a^2)^(1/2)) + ((1 - a^2*x^2)^(1/2)*(2/ 
(-a^2)^(3/2) - 4/(a^2*(-a^2)^(1/2)) - (19*x*(-a^2)^(1/2))/(8*a^3) + (a^2*x 
^2)/(-a^2)^(3/2) + (x^3*(-a^2)^(3/2))/(4*a^3)))/(-a^2)^(1/2) - (4*(1 - a^2 
*x^2)^(1/2))/(a^3*(x*(-a^2)^(1/2) + (-a^2)^(1/2)/a)*(-a^2)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.47 \[ \int e^{-3 \text {arctanh}(a x)} x^3 \, dx=\frac {51 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-51 \mathit {asin} \left (a x \right ) a x -51 \mathit {asin} \left (a x \right )+2 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-6 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+11 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-29 \sqrt {-a^{2} x^{2}+1}\, a x -102 \sqrt {-a^{2} x^{2}+1}+2 a^{5} x^{5}-8 a^{4} x^{4}+17 a^{3} x^{3}-40 a^{2} x^{2}-29 a x +102}{8 a^{4} \left (\sqrt {-a^{2} x^{2}+1}-a x -1\right )} \] Input:

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

Output:

(51*sqrt( - a**2*x**2 + 1)*asin(a*x) - 51*asin(a*x)*a*x - 51*asin(a*x) + 2 
*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 6*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 1 
1*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 29*sqrt( - a**2*x**2 + 1)*a*x - 102*s 
qrt( - a**2*x**2 + 1) + 2*a**5*x**5 - 8*a**4*x**4 + 17*a**3*x**3 - 40*a**2 
*x**2 - 29*a*x + 102)/(8*a**4*(sqrt( - a**2*x**2 + 1) - a*x - 1))