\(\int \frac {e^{n \text {arctanh}(a x)}}{x^2 (c-a^2 c x^2)^{5/2}} \, dx\) [1391]

Optimal result

Integrand size = 27, antiderivative size = 507 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {a (4+n) (1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (3+n) \sqrt {c-a^2 c x^2}}-\frac {(1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 x \sqrt {c-a^2 c x^2}}+\frac {a \left (12+6 n+n^2\right ) (1-a x)^{\frac {1}{2} (-1-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt {c-a^2 c x^2}}-\frac {a \left (24+15 n+6 n^2+n^3\right ) (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 (3+n) \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}+\frac {a \left (24+18 n+7 n^2-2 n^3-n^4\right ) (1-a x)^{\frac {3-n}{2}} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2}}{c^2 \left (9-10 n^2+n^4\right ) \sqrt {c-a^2 c x^2}}+\frac {2 a n (1-a x)^{\frac {1-n}{2}} (1+a x)^{\frac {1}{2} (-1+n)} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1+n),\frac {1+n}{2},\frac {1+a x}{1-a x}\right )}{c^2 (1-n) \sqrt {c-a^2 c x^2}} \] Output:

a*(4+n)*(-a*x+1)^(-3/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2+1)^(1/2)/c^2/ 
(3+n)/(-a^2*c*x^2+c)^(1/2)-(-a*x+1)^(-3/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^ 
2*x^2+1)^(1/2)/c^2/x/(-a^2*c*x^2+c)^(1/2)+a*(n^2+6*n+12)*(-a*x+1)^(-1/2-1/ 
2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2+1)^(1/2)/c^2/(1+n)/(3+n)/(-a^2*c*x^2+c 
)^(1/2)-a*(n^3+6*n^2+15*n+24)*(-a*x+1)^(1/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(- 
a^2*x^2+1)^(1/2)/c^2/(3+n)/(-n^2+1)/(-a^2*c*x^2+c)^(1/2)+a*(-n^4-2*n^3+7*n 
^2+18*n+24)*(-a*x+1)^(3/2-1/2*n)*(a*x+1)^(-3/2+1/2*n)*(-a^2*x^2+1)^(1/2)/c 
^2/(n^4-10*n^2+9)/(-a^2*c*x^2+c)^(1/2)+2*a*n*(-a*x+1)^(1/2-1/2*n)*(a*x+1)^ 
(-1/2+1/2*n)*(-a^2*x^2+1)^(1/2)*hypergeom([1, -1/2+1/2*n],[1/2+1/2*n],(a*x 
+1)/(-a*x+1))/c^2/(1-n)/(-a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.40 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.53 \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {(1-a x)^{\frac {1}{2} (-3-n)} (1+a x)^{\frac {1}{2} (-3+n)} \sqrt {1-a^2 x^2} \left (9-36 a^2 x^2+24 a^4 x^4-n^4 (-1+a x)^3 (1+a x)+a n^3 x \left (-4+2 a x+3 a^2 x^2-2 a^3 x^3\right )+a n x \left (34-18 a x-33 a^2 x^2+18 a^3 x^3\right )+n^2 \left (-10+18 a x+6 a^2 x^2-18 a^3 x^3+7 a^4 x^4\right )+2 a n \left (-3-n+3 n^2+n^3\right ) x (-1+a x)^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2}-\frac {n}{2},\frac {5}{2}-\frac {n}{2},\frac {1-a x}{1+a x}\right )\right )}{c^2 (-3+n) (-1+n) (1+n) (3+n) x \sqrt {c-a^2 c x^2}} \] Input:

Integrate[E^(n*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)^(5/2)),x]
 

Output:

-(((1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2]*(9 - 36 
*a^2*x^2 + 24*a^4*x^4 - n^4*(-1 + a*x)^3*(1 + a*x) + a*n^3*x*(-4 + 2*a*x + 
 3*a^2*x^2 - 2*a^3*x^3) + a*n*x*(34 - 18*a*x - 33*a^2*x^2 + 18*a^3*x^3) + 
n^2*(-10 + 18*a*x + 6*a^2*x^2 - 18*a^3*x^3 + 7*a^4*x^4) + 2*a*n*(-3 - n + 
3*n^2 + n^3)*x*(-1 + a*x)^3*Hypergeometric2F1[1, 3/2 - n/2, 5/2 - n/2, (1 
- a*x)/(1 + a*x)]))/(c^2*(-3 + n)*(-1 + n)*(1 + n)*(3 + n)*x*Sqrt[c - a^2* 
c*x^2]))
 

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.69, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {6703, 6700, 144, 25, 27, 172, 25, 27, 172, 25, 27, 172, 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.

\(\displaystyle \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6703

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (1-a^2 x^2\right )^{5/2}}dx}{c^2 \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 6700

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {(1-a x)^{\frac {1}{2} (-n-5)} (a x+1)^{\frac {n-5}{2}}}{x^2}dx}{c^2 \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 144

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 172

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 172

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (a \left (\frac {\frac {\left (n^2+6 n+12\right ) (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-3}{2}}}{n+1}-\frac {\int -\frac {a (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-5}{2}} \left (n (n+1) (n+3)+2 a \left (n^2+6 n+12\right ) x\right )}{x}dx}{a (n+1)}}{n+3}+\frac {(n+4) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )}{c^2 \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (a \left (\frac {\frac {\int \frac {a (1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-5}{2}} \left (n (n+1) (n+3)+2 a \left (n^2+6 n+12\right ) x\right )}{x}dx}{a (n+1)}+\frac {\left (n^2+6 n+12\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {(n+4) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )}{c^2 \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (a \left (\frac {\frac {\int \frac {(1-a x)^{\frac {1}{2} (-n-1)} (a x+1)^{\frac {n-5}{2}} \left (n (n+1) (n+3)+2 a \left (n^2+6 n+12\right ) x\right )}{x}dx}{n+1}+\frac {\left (n^2+6 n+12\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {(n+4) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )}{c^2 \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 172

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (a \left (\frac {\frac {\frac {\int \frac {a (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-5}{2}} \left ((1-n) n (n+1) (n+3)-a \left (n^3+6 n^2+15 n+24\right ) x\right )}{x}dx}{a (1-n)}-\frac {\left (n^3+6 n^2+15 n+24\right ) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {\left (n^2+6 n+12\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {(n+4) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )}{c^2 \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (a \left (\frac {\frac {\frac {\int \frac {(1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-5}{2}} \left ((1-n) n (n+1) (n+3)-a \left (n^3+6 n^2+15 n+24\right ) x\right )}{x}dx}{1-n}-\frac {\left (n^3+6 n^2+15 n+24\right ) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {\left (n^2+6 n+12\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {(n+4) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )}{c^2 \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 172

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (a \left (\frac {\frac {\frac {\frac {\int \frac {a (1-n) (3-n) n (n+1) (n+3) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{x}dx}{a (3-n)}+\frac {\left (-n^4-2 n^3+7 n^2+18 n+24\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {3-n}{2}}}{3-n}}{1-n}-\frac {\left (n^3+6 n^2+15 n+24\right ) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {\left (n^2+6 n+12\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {(n+4) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )}{c^2 \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (a \left (\frac {\frac {\frac {(1-n) n (n+1) (n+3) \int \frac {(1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{x}dx+\frac {\left (-n^4-2 n^3+7 n^2+18 n+24\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {3-n}{2}}}{3-n}}{1-n}-\frac {\left (n^3+6 n^2+15 n+24\right ) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {\left (n^2+6 n+12\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {(n+4) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )}{c^2 \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 141

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (a \left (\frac {\frac {\frac {2 n (n+1) (n+3) (a x+1)^{\frac {n-1}{2}} (1-a x)^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {n-1}{2},\frac {n+1}{2},\frac {a x+1}{1-a x}\right )+\frac {\left (-n^4-2 n^3+7 n^2+18 n+24\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {3-n}{2}}}{3-n}}{1-n}-\frac {\left (n^3+6 n^2+15 n+24\right ) (1-a x)^{\frac {1-n}{2}} (a x+1)^{\frac {n-3}{2}}}{1-n}}{n+1}+\frac {\left (n^2+6 n+12\right ) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-1)}}{n+1}}{n+3}+\frac {(n+4) (a x+1)^{\frac {n-3}{2}} (1-a x)^{\frac {1}{2} (-n-3)}}{n+3}\right )-\frac {(1-a x)^{\frac {1}{2} (-n-3)} (a x+1)^{\frac {n-3}{2}}}{x}\right )}{c^2 \sqrt {c-a^2 c x^2}}\)

Input:

Int[E^(n*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)^(5/2)),x]
 

Output:

(Sqrt[1 - a^2*x^2]*(-(((1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2))/x) + 
 a*(((4 + n)*(1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2))/(3 + n) + (((1 
2 + 6*n + n^2)*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2))/(1 + n) + (- 
(((24 + 15*n + 6*n^2 + n^3)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-3 + n)/2))/ 
(1 - n)) + (((24 + 18*n + 7*n^2 - 2*n^3 - n^4)*(1 - a*x)^((3 - n)/2)*(1 + 
a*x)^((-3 + n)/2))/(3 - n) + 2*n*(1 + n)*(3 + n)*(1 - a*x)^((1 - n)/2)*(1 
+ a*x)^((-1 + n)/2)*Hypergeometric2F1[1, (-1 + n)/2, (1 + n)/2, (1 + a*x)/ 
(1 - a*x)])/(1 - n))/(1 + n))/(3 + n))))/(c^2*Sqrt[c - a^2*c*x^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 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 6703

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

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{x^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]

Input:

int(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c)^(5/2),x)
 

Output:

int(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c)^(5/2),x)
 

Fricas [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c)^(5/2),x, algorithm="frica 
s")
 

Output:

integral(-sqrt(-a^2*c*x^2 + c)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^6*c^3*x^8 
 - 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 - c^3*x^2), x)
 

Sympy [F]

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

integrate(exp(n*atanh(a*x))/x**2/(-a**2*c*x**2+c)**(5/2),x)
 

Output:

Integral(exp(n*atanh(a*x))/(x**2*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)
 

Maxima [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxim 
a")
 

Output:

integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/((-a^2*c*x^2 + c)^(5/2)*x^2), x)
 

Giac [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))/x^2/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac" 
)
 

Output:

integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/((-a^2*c*x^2 + c)^(5/2)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x^2\,{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \] Input:

int(exp(n*atanh(a*x))/(x^2*(c - a^2*c*x^2)^(5/2)),x)
 

Output:

int(exp(n*atanh(a*x))/(x^2*(c - a^2*c*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {e^{n \text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\int \frac {e^{\mathit {atanh} \left (a x \right ) n}}{\sqrt {-a^{2} x^{2}+1}\, a^{4} x^{6}-2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{4}+\sqrt {-a^{2} x^{2}+1}\, x^{2}}d x}{\sqrt {c}\, c^{2}} \] Input:

int(exp(n*atanh(a*x))/x^2/(-a^2*c*x^2+c)^(5/2),x)
 

Output:

int(e**(atanh(a*x)*n)/(sqrt( - a**2*x**2 + 1)*a**4*x**6 - 2*sqrt( - a**2*x 
**2 + 1)*a**2*x**4 + sqrt( - a**2*x**2 + 1)*x**2),x)/(sqrt(c)*c**2)