\(\int e^{n \text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^{3/2} \, dx\) [766]

Optimal result

Integrand size = 24, antiderivative size = 430 \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1-a x)^{\frac {5-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)}}{2 \left (1-a^2 x^2\right )^{3/2}}-\frac {a (4+n) \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2 (1-a x)^{\frac {5-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)}}{2 \left (1-a^2 x^2\right )^{3/2}}-\frac {3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 (1-a x)^{\frac {5-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)}}{(3-n) \left (1-a^2 x^2\right )^{3/2}}-\frac {a^2 \left (3-n^2\right ) \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-3+n),\frac {1}{2} (-1+n),\frac {1+a x}{1-a x}\right )}{(3-n) \left (1-a^2 x^2\right )^{3/2}}+\frac {2^{\frac {1}{2} (-1+n)} a^2 n \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 (1-a x)^{\frac {5-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {3-n}{2},\frac {5-n}{2},\frac {7-n}{2},\frac {1}{2} (1-a x)\right )}{(3-n) (5-n) \left (1-a^2 x^2\right )^{3/2}} \] Output:

-1/2*(c-c/a^2/x^2)^(3/2)*x*(-a*x+1)^(5/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)/(-a^2 
*x^2+1)^(3/2)-1/2*a*(4+n)*(c-c/a^2/x^2)^(3/2)*x^2*(-a*x+1)^(5/2-1/2*n)*(a* 
x+1)^(-3/2+1/2*n)/(-a^2*x^2+1)^(3/2)-3*a^2*(c-c/a^2/x^2)^(3/2)*x^3*(-a*x+1 
)^(5/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)/(3-n)/(-a^2*x^2+1)^(3/2)-a^2*(-n^2+3)*( 
c-c/a^2/x^2)^(3/2)*x^3*(-a*x+1)^(3/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*hypergeom 
([1, -3/2+1/2*n],[-1/2+1/2*n],(a*x+1)/(-a*x+1))/(3-n)/(-a^2*x^2+1)^(3/2)+2 
^(-1/2+1/2*n)*a^2*n*(c-c/a^2/x^2)^(3/2)*x^3*(-a*x+1)^(5/2-1/2*n)*hypergeom 
([5/2-1/2*n, 3/2-1/2*n],[7/2-1/2*n],-1/2*a*x+1/2)/(3-n)/(5-n)/(-a^2*x^2+1) 
^(3/2)
 

Mathematica [A] (verified)

Time = 2.14 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.44 \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\frac {c e^{n \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \text {csch}\left (\frac {1}{2} \text {arctanh}(a x)\right ) \text {sech}\left (\frac {1}{2} \text {arctanh}(a x)\right ) \left (8 a e^{\text {arctanh}(a x)} n x \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-e^{2 \text {arctanh}(a x)}\right )-4 a e^{\text {arctanh}(a x)} \left (-3+n^2\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \text {arctanh}(a x)}\right )-(1+n) \left (a x (n+a x)+\left (1-a^2 x^2\right ) \cosh (2 \text {arctanh}(a x))\right ) \text {csch}\left (\frac {1}{2} \text {arctanh}(a x)\right ) \text {sech}\left (\frac {1}{2} \text {arctanh}(a x)\right )\right )}{8 (1+n) \left (-1+a^2 x^2\right )} \] Input:

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

Output:

(c*E^(n*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)]*x*Csch[ArcTanh[a*x]/2]*Sech[Ar 
cTanh[a*x]/2]*(8*a*E^ArcTanh[a*x]*n*x*Hypergeometric2F1[1, (1 + n)/2, (3 + 
 n)/2, -E^(2*ArcTanh[a*x])] - 4*a*E^ArcTanh[a*x]*(-3 + n^2)*x*Hypergeometr 
ic2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcTanh[a*x])] - (1 + n)*(a*x*(n + a*x 
) + (1 - a^2*x^2)*Cosh[2*ArcTanh[a*x]])*Csch[ArcTanh[a*x]/2]*Sech[ArcTanh[ 
a*x]/2]))/(8*(1 + n)*(-1 + a^2*x^2))
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.74, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6710, 6700, 139, 88, 79, 2116, 25, 27, 168, 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.

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

\(\Big \downarrow \) 6710

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

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 139

\(\displaystyle \frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (a^3 \int (1-a x)^{\frac {3-n}{2}} (a x+1)^{\frac {n-5}{2}} (a x+4)dx+\int \frac {(1-a x)^{\frac {3-n}{2}} (a x+1)^{\frac {n-5}{2}} \left (6 a^2 x^2+4 a x+1\right )}{x^3}dx\right )}{\left (1-a^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 88

\(\displaystyle \frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (a^3 \left (-\frac {n \int (1-a x)^{\frac {3-n}{2}} (a x+1)^{\frac {n-3}{2}}dx}{3-n}-\frac {3 (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {5-n}{2}}}{a (3-n)}\right )+\int \frac {(1-a x)^{\frac {3-n}{2}} (a x+1)^{\frac {n-5}{2}} \left (6 a^2 x^2+4 a x+1\right )}{x^3}dx\right )}{\left (1-a^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\int \frac {(1-a x)^{\frac {3-n}{2}} (a x+1)^{\frac {n-5}{2}} \left (6 a^2 x^2+4 a x+1\right )}{x^3}dx+a^3 \left (\frac {2^{\frac {n-1}{2}} n (1-a x)^{\frac {5-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {3-n}{2},\frac {5-n}{2},\frac {7-n}{2},\frac {1}{2} (1-a x)\right )}{a (3-n) (5-n)}-\frac {3 (1-a x)^{\frac {5-n}{2}} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}\right )\right )}{\left (1-a^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2116

\(\displaystyle \frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (-\frac {1}{2} \int -\frac {a (1-a x)^{\frac {3-n}{2}} (a x+1)^{\frac {n-5}{2}} (n+13 a x+4)}{x^2}dx+a^3 \left (\frac {2^{\frac {n-1}{2}} n (1-a x)^{\frac {5-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {3-n}{2},\frac {5-n}{2},\frac {7-n}{2},\frac {1}{2} (1-a x)\right )}{a (3-n) (5-n)}-\frac {3 (1-a x)^{\frac {5-n}{2}} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}\right )-\frac {(a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {5-n}{2}}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\frac {1}{2} \int \frac {a (1-a x)^{\frac {3-n}{2}} (a x+1)^{\frac {n-5}{2}} (n+13 a x+4)}{x^2}dx+a^3 \left (\frac {2^{\frac {n-1}{2}} n (1-a x)^{\frac {5-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {3-n}{2},\frac {5-n}{2},\frac {7-n}{2},\frac {1}{2} (1-a x)\right )}{a (3-n) (5-n)}-\frac {3 (1-a x)^{\frac {5-n}{2}} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}\right )-\frac {(a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {5-n}{2}}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\frac {1}{2} a \int \frac {(1-a x)^{\frac {3-n}{2}} (a x+1)^{\frac {n-5}{2}} (n+13 a x+4)}{x^2}dx+a^3 \left (\frac {2^{\frac {n-1}{2}} n (1-a x)^{\frac {5-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {3-n}{2},\frac {5-n}{2},\frac {7-n}{2},\frac {1}{2} (1-a x)\right )}{a (3-n) (5-n)}-\frac {3 (1-a x)^{\frac {5-n}{2}} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}\right )-\frac {(a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {5-n}{2}}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\frac {1}{2} a \left (-\int \frac {a \left (3-n^2\right ) (1-a x)^{\frac {3-n}{2}} (a x+1)^{\frac {n-5}{2}}}{x}dx-\frac {(n+4) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {5-n}{2}}}{x}\right )+a^3 \left (\frac {2^{\frac {n-1}{2}} n (1-a x)^{\frac {5-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {3-n}{2},\frac {5-n}{2},\frac {7-n}{2},\frac {1}{2} (1-a x)\right )}{a (3-n) (5-n)}-\frac {3 (1-a x)^{\frac {5-n}{2}} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}\right )-\frac {(a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {5-n}{2}}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\frac {1}{2} a \left (-a \left (3-n^2\right ) \int \frac {(1-a x)^{\frac {3-n}{2}} (a x+1)^{\frac {n-5}{2}}}{x}dx-\frac {(n+4) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {5-n}{2}}}{x}\right )+a^3 \left (\frac {2^{\frac {n-1}{2}} n (1-a x)^{\frac {5-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {3-n}{2},\frac {5-n}{2},\frac {7-n}{2},\frac {1}{2} (1-a x)\right )}{a (3-n) (5-n)}-\frac {3 (1-a x)^{\frac {5-n}{2}} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}\right )-\frac {(a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {5-n}{2}}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 141

\(\displaystyle \frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (a^3 \left (\frac {2^{\frac {n-1}{2}} n (1-a x)^{\frac {5-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {3-n}{2},\frac {5-n}{2},\frac {7-n}{2},\frac {1}{2} (1-a x)\right )}{a (3-n) (5-n)}-\frac {3 (1-a x)^{\frac {5-n}{2}} (a x+1)^{\frac {n-3}{2}}}{a (3-n)}\right )+\frac {1}{2} a \left (-\frac {2 a \left (3-n^2\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {3-n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {n-3}{2},\frac {n-1}{2},\frac {a x+1}{1-a x}\right )}{3-n}-\frac {(n+4) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {5-n}{2}}}{x}\right )-\frac {(a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {5-n}{2}}}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\)

Input:

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

Output:

((c - c/(a^2*x^2))^(3/2)*x^3*(-1/2*((1 - a*x)^((5 - n)/2)*(1 + a*x)^((-3 + 
 n)/2))/x^2 + (a*(-(((4 + n)*(1 - a*x)^((5 - n)/2)*(1 + a*x)^((-3 + n)/2)) 
/x) - (2*a*(3 - n^2)*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Hypergeo 
metric2F1[1, (-3 + n)/2, (-1 + n)/2, (1 + a*x)/(1 - a*x)])/(3 - n)))/2 + a 
^3*((-3*(1 - a*x)^((5 - n)/2)*(1 + a*x)^((-3 + n)/2))/(a*(3 - n)) + (2^((- 
1 + n)/2)*n*(1 - a*x)^((5 - n)/2)*Hypergeometric2F1[(3 - n)/2, (5 - n)/2, 
(7 - n)/2, (1 - a*x)/2])/(a*(3 - n)*(5 - n)))))/(1 - a^2*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 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 1]
 

rule 139

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

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 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S 
imp[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)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 2116

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_ 
.)*(x_))^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[Px, a + b*x, x], 
 R = PolynomialRemainder[Px, a + b*x, x]}, Simp[b*R*(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] + Si 
mp[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*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f*R*(m + 1 
) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x] 
, x], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && PolyQ[Px, x] && ILtQ[m, 
-1]
 

rule 6700

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

rule 6710

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(exp(n*arctanh(a*x))*(c-c/a^2/x^2)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(exp(n*arctanh(a*x))*(c-c/a^2/x^2)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

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

integrate(exp(n*arctanh(a*x))*(c-c/a^2/x^2)^(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. \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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