\(\int \frac {e^{n \coth ^{-1}(a x)} x^3}{(c-a^2 c x^2)^{3/2}} \, dx\) [725]

Optimal result

Integrand size = 27, antiderivative size = 356 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {(2+n) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (2+2 n+n^2\right ) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{a \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^4}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {2 n \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1+n),\frac {1+n}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a (1-n) \left (c-a^2 c x^2\right )^{3/2}} \] Output:

-(2+n)*(1-1/a^2/x^2)^(3/2)*(1-1/a/x)^(-1/2-1/2*n)*(1+1/a/x)^(-1/2+1/2*n)*x 
^3/a/(1+n)/(-a^2*c*x^2+c)^(3/2)+(n^2+2*n+2)*(1-1/a^2/x^2)^(3/2)*(1-1/a/x)^ 
(1/2-1/2*n)*(1+1/a/x)^(-1/2+1/2*n)*x^3/a/(-n^2+1)/(-a^2*c*x^2+c)^(3/2)+(1- 
1/a^2/x^2)^(3/2)*(1-1/a/x)^(-1/2-1/2*n)*(1+1/a/x)^(-1/2+1/2*n)*x^4/(-a^2*c 
*x^2+c)^(3/2)-2*n*(1-1/a^2/x^2)^(3/2)*(1-1/a/x)^(1/2-1/2*n)*(1+1/a/x)^(-1/ 
2+1/2*n)*x^3*hypergeom([1, -1/2+1/2*n],[1/2+1/2*n],(a+1/x)/(a-1/x))/a/(1-n 
)/(-a^2*c*x^2+c)^(3/2)
 

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.37 \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\frac {c e^{n \coth ^{-1}(a x)} (-1+a n x)}{-1+n^2}-\frac {c \left (-1+a^2 x^2\right ) \left (e^{n \coth ^{-1}(a x)} (1+n)+\frac {2 e^{(1+n) \coth ^{-1}(a x)} n \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \coth ^{-1}(a x)}\right )}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\right )}{1+n}}{a^4 c^2 \sqrt {c-a^2 c x^2}} \] Input:

Integrate[(E^(n*ArcCoth[a*x])*x^3)/(c - a^2*c*x^2)^(3/2),x]
 

Output:

((c*E^(n*ArcCoth[a*x])*(-1 + a*n*x))/(-1 + n^2) - (c*(-1 + a^2*x^2)*(E^(n* 
ArcCoth[a*x])*(1 + n) + (2*E^((1 + n)*ArcCoth[a*x])*n*Hypergeometric2F1[1, 
 (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])])/(a*Sqrt[1 - 1/(a^2*x^2)]*x)))/ 
(1 + n))/(a^4*c^2*Sqrt[c - a^2*c*x^2])
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.77, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6746, 6748, 144, 25, 27, 172, 25, 27, 172, 27, 141}

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.

\(\displaystyle \int \frac {x^3 e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 6746

\(\displaystyle \frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}dx}{\left (c-a^2 c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 6748

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

\(\Big \downarrow \) 144

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 172

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 172

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 141

\(\displaystyle -\frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {\frac {\frac {2 a n \left (1-n^2\right ) \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {n-1}{2},\frac {n+1}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{(1-n)^2}-\frac {a \left (n^2+2 n+2\right ) \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}}}{1-n}}{n+1}+\frac {a (n+2) \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n+1}}{a^2}-x \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\)

Input:

Int[(E^(n*ArcCoth[a*x])*x^3)/(c - a^2*c*x^2)^(3/2),x]
 

Output:

-(((1 - 1/(a^2*x^2))^(3/2)*x^3*(-((1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x)) 
^((-1 + n)/2)*x) + ((a*(2 + n)*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^(( 
-1 + n)/2))/(1 + n) + (-((a*(2 + 2*n + n^2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 
 1/(a*x))^((-1 + n)/2))/(1 - n)) + (2*a*n*(1 - n^2)*(1 - 1/(a*x))^((1 - n) 
/2)*(1 + 1/(a*x))^((-1 + n)/2)*Hypergeometric2F1[1, (-1 + n)/2, (1 + n)/2, 
 (a + x^(-1))/(a - x^(-1))])/(1 - n)^2)/(1 + n))/a^2))/(c - a^2*c*x^2)^(3/ 
2))
 

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 141

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( 
n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f 
))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, 
p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !Su 
mSimplerQ[p, 1]) &&  !ILtQ[m, 0]
 

rule 144

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[b*(a + b*x)^(m + 1)*( 
c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] 
+ Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) 
- b*d*f*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | 
| ( !SumSimplerQ[n, 1] &&  !SumSimplerQ[p, 1]))] /; FreeQ[{a, b, c, d, e, f 
, m, n, p}, x] && NeQ[m, -1]
 

rule 172

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[ 
(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1) 
*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f 
)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g 
 - a*h)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | 
| ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1 
])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
 

rule 6746

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[(c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p)   Int[u*x^(2*p)*(1 - 
 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && 
EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2] &&  !IntegerQ[p]
 

rule 6748

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> 
 Simp[-c^p   Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/a)^(p + n/2)/x^2), x], x 
, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !IntegerQ[ 
n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegersQ[2*p, p + n/2]
 

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} x^{3}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]

Input:

int(exp(n*arccoth(a*x))*x^3/(-a^2*c*x^2+c)^(3/2),x)
 

Output:

int(exp(n*arccoth(a*x))*x^3/(-a^2*c*x^2+c)^(3/2),x)
 

Fricas [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(exp(n*arccoth(a*x))*x^3/(-a^2*c*x^2+c)^(3/2),x, algorithm="frica 
s")
 

Output:

integral(sqrt(-a^2*c*x^2 + c)*x^3*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^4*c^2*x 
^4 - 2*a^2*c^2*x^2 + c^2), x)
 

Sympy [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^{3} e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(exp(n*acoth(a*x))*x**3/(-a**2*c*x**2+c)**(3/2),x)
 

Output:

Integral(x**3*exp(n*acoth(a*x))/(-c*(a*x - 1)*(a*x + 1))**(3/2), x)
 

Maxima [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(exp(n*arccoth(a*x))*x^3/(-a^2*c*x^2+c)^(3/2),x, algorithm="maxim 
a")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(exp(n*arccoth(a*x))*x^3/(-a^2*c*x^2+c)^(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 [F(-1)]

Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^3\,{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \] Input:

int((x^3*exp(n*acoth(a*x)))/(c - a^2*c*x^2)^(3/2),x)
 

Output:

int((x^3*exp(n*acoth(a*x)))/(c - a^2*c*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {\int \frac {e^{\mathit {acoth} \left (a x \right ) n} x^{3}}{\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}d x}{\sqrt {c}\, c} \] Input:

int(exp(n*acoth(a*x))*x^3/(-a^2*c*x^2+c)^(3/2),x)
 

Output:

( - int((e**(acoth(a*x)*n)*x**3)/(sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( 
 - a**2*x**2 + 1)),x))/(sqrt(c)*c)