### 3.761 $$\int \frac{e^{n \coth ^{-1}(a x)}}{x (c-a^2 c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=944 $\text{result too large to display}$

[Out]

-((a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((-3 + n)/2)*x^5)/((3 + n)*(c - a^2*c*
x^2)^(5/2))) - (3*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^((-3 + n)/2)*x^5)/((3 +
4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((
-3 + n)/2)*x^5)/((3 + n)*(1 - n^2)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 -
n)/2)*(1 + 1/(a*x))^((-3 + n)/2)*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)) + (4*a^5*(1 - 1/(a^2*x^2))^(
5/2)*(1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) + (8*a^5*(1 -
1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)
^(5/2)) - (8*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*x^5)/((3 + n)*(1
- n^2)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((1 +
n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1
/(a*x))^((1 + n)/2)*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (4*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x)
)^((-3 - n)/2)*(1 + 1/(a*x))^((3 + n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) - (2^((5 + n)/2)*a^5*(1 - 1/(a^2
*x^2))^(5/2)*(1 - 1/(a*x))^((-3 - n)/2)*x^5*Hypergeometric2F1[(-3 - n)/2, (-3 - n)/2, (-1 - n)/2, (a - x^(-1))
/(2*a)])/((3 + n)*(c - a^2*c*x^2)^(5/2))

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Rubi [A]  time = 0.632143, antiderivative size = 944, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 6, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {6192, 6195, 128, 45, 37, 69} $-\frac{2^{\frac{n+5}{2}} a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5 \, _2F_1\left (\frac{1}{2} (-n-3),\frac{1}{2} (-n-3);\frac{1}{2} (-n-1);\frac{a-\frac{1}{x}}{2 a}\right ) \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-3}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-1}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n+1}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n+3}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-3}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-1}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n+1}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-3}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{(n+3) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-1}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{(n+3) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{\frac{n-3}{2}} x^5 \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}}}{\left (n^4-10 n^2+9\right ) \left (c-a^2 c x^2\right )^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((-3 + n)/2)*x^5)/((3 + n)*(c - a^2*c*
x^2)^(5/2))) - (3*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^((-3 + n)/2)*x^5)/((3 +
4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((
-3 + n)/2)*x^5)/((3 + n)*(1 - n^2)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 -
n)/2)*(1 + 1/(a*x))^((-3 + n)/2)*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)) + (4*a^5*(1 - 1/(a^2*x^2))^(
5/2)*(1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) + (8*a^5*(1 -
1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)
^(5/2)) - (8*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*x^5)/((3 + n)*(1
- n^2)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((1 +
n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1
/(a*x))^((1 + n)/2)*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (4*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x)
)^((-3 - n)/2)*(1 + 1/(a*x))^((3 + n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) - (2^((5 + n)/2)*a^5*(1 - 1/(a^2
*x^2))^(5/2)*(1 - 1/(a*x))^((-3 - n)/2)*x^5*Hypergeometric2F1[(-3 - n)/2, (-3 - n)/2, (-1 - n)/2, (a - x^(-1))
/(2*a)])/((3 + n)*(c - a^2*c*x^2)^(5/2))

Rule 6192

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(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 6195

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^p, Subst[Int[((
1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2))/x^(m + 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] && IntegerQ[m]

Rule 128

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (IGtQ[m, 0] || (
ILtQ[m, 0] && ILtQ[n, 0]))

Rule 45

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

Rule 37

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

Rule 69

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

Rubi steps

\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{e^{n \coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^6} \, dx}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int x^4 \left (1-\frac{x}{a}\right )^{-\frac{5}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (a^4 \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}}-4 a^4 \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}}+6 a^4 \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{-\frac{1}{2}+\frac{n}{2}}-4 a^4 \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{\frac{1}{2}+\frac{n}{2}}+a^4 \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{\frac{3}{2}+\frac{n}{2}}\right ) \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\left (a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{\frac{3}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (4 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (4 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{\frac{1}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (6 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-5-n)} \left (1+\frac{x}{a}\right )^{-\frac{1}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{2^{\frac{5+n}{2}} a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} x^5 \, _2F_1\left (\frac{1}{2} (-3-n),\frac{1}{2} (-3-n);\frac{1}{2} (-1-n);\frac{a-\frac{1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (3 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (6 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{x}{a}\right )^{-\frac{1}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \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^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{2^{\frac{5+n}{2}} a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} x^5 \, _2F_1\left (\frac{1}{2} (-3-n),\frac{1}{2} (-3-n);\frac{1}{2} (-1-n);\frac{a-\frac{1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (6 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \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^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{2^{\frac{5+n}{2}} a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} x^5 \, _2F_1\left (\frac{1}{2} (-3-n),\frac{1}{2} (-3-n);\frac{1}{2} (-1-n);\frac{a-\frac{1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (6 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1-n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (9-10 n^2+n^4\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \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^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{8 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1+n}{2}} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{2^{\frac{5+n}{2}} a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} x^5 \, _2F_1\left (\frac{1}{2} (-3-n),\frac{1}{2} (-3-n);\frac{1}{2} (-1-n);\frac{a-\frac{1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 1.5572, size = 220, normalized size = 0.23 $\frac{e^{n \coth ^{-1}(a x)} \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (5 a n^3 x-45 a n x-2 n^2+42\right )+6 a \left (n^2-1\right ) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (2 \coth ^{-1}(a x)\right )-n \left (n^2-1\right ) \left (a^2 x^2-1\right ) \cosh \left (3 \coth ^{-1}(a x)\right )\right )-8 \left (n^3-n^2-9 n+9\right ) \left (a^2 x^2-1\right ) e^{(n+1) \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},-e^{2 \coth ^{-1}(a x)}\right )}{4 a c^2 (n-1) (n+1) \left (n^2-9\right ) x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a^2 c x^2}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(E^(n*ArcCoth[a*x])*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(42 - 2*n^2 - 45*a*n*x + 5*a*n^3*x) + 6*a*(-1 + n^2)*Sqrt[1 - 1
/(a^2*x^2)]*x*Cosh[2*ArcCoth[a*x]] - n*(-1 + n^2)*(-1 + a^2*x^2)*Cosh[3*ArcCoth[a*x]]) - 8*E^((1 + n)*ArcCoth[
a*x])*(9 - 9*n - n^2 + n^3)*(-1 + a^2*x^2)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -E^(2*ArcCoth[a*x])])/(4
*a*c^2*(-1 + n)*(1 + n)*(-9 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a^2*c*x^2])

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Maple [F]  time = 0.342, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{x} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} c x^{2} + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{6} c^{3} x^{7} - 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} - c^{3} x}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-a^2*c*x^2 + c)*((a*x - 1)/(a*x + 1))^(1/2*n)/(a^6*c^3*x^7 - 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 - c^
3*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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