### 3.756 $$\int \frac{e^{n \coth ^{-1}(a x)} x^4}{(c-a^2 c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=463 $-\frac{2 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{n-1}{2},\frac{n+1}{2},\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\right )}{(1-n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (n^2+6 n+15\right ) x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{(1-n) (n+1) (n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (-n^3-2 n^2+7 n+18\right ) x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}}}{\left (n^4-10 n^2+9\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{(n+6) x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{(n+1) (n+3) \left (c-a^2 c x^2\right )^{5/2}}$

[Out]

-(((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))) - ((6 + n)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^((-3 + n)/2)*x^5)/((1 + n
)*(3 + n)*(c - a^2*c*x^2)^(5/2)) + ((15 + 6*n + n^2)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/
(a*x))^((-3 + n)/2)*x^5)/((1 - n)*(1 + n)*(3 + n)*(c - a^2*c*x^2)^(5/2)) - ((18 + 7*n - 2*n^2 - n^3)*(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)) - (2*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2)*x^5*Hypergeometric2F1[
1, (-1 + n)/2, (1 + n)/2, (a + x^(-1))/(a - x^(-1))])/((1 - n)*(c - a^2*c*x^2)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.534858, antiderivative size = 467, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 6, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {6192, 6195, 129, 155, 12, 131} $\frac{2 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}} \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{(3-n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (n^2+6 n+15\right ) x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{(1-n) (n+1) (n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (-n^3-2 n^2+7 n+18\right ) x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}}}{\left (n^4-10 n^2+9\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{(n+6) x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{(n+1) (n+3) \left (c-a^2 c x^2\right )^{5/2}}$

Warning: Unable to verify antiderivative.

[In]

Int[(E^(n*ArcCoth[a*x])*x^4)/(c - a^2*c*x^2)^(5/2),x]

[Out]

-(((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))) - ((6 + n)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^((-3 + n)/2)*x^5)/((1 + n
)*(3 + n)*(c - a^2*c*x^2)^(5/2)) + ((15 + 6*n + n^2)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/
(a*x))^((-3 + n)/2)*x^5)/((1 - n)*(1 + n)*(3 + n)*(c - a^2*c*x^2)^(5/2)) - ((18 + 7*n - 2*n^2 - n^3)*(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)) + (2*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((-3 + n)/2)*x^5*Hypergeometric2F1[
1, (3 - n)/2, (5 - n)/2, (a - x^(-1))/(a + x^(-1))])/((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 129

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

Rule 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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 + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 131

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

Rubi steps

\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)} x^4}{\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} \, 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 \frac{\left (1-\frac{x}{a}\right )^{-\frac{5}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\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{\left (a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{3+n}{a}-\frac{3 x}{a^2}\right ) \left (1-\frac{x}{a}\right )^{-\frac{3}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\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{(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (a^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \left (\frac{(1+n) (3+n)}{a^2}+\frac{2 (6+n) x}{a^3}\right )}{x} \, dx,x,\frac{1}{x}\right )}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\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{(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (15+6 n+n^2\right ) \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}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (a^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \left (\frac{(1-n) (1+n) (3+n)}{a^3}-\frac{\left (15+6 n+n^2\right ) x}{a^4}\right )}{x} \, dx,x,\frac{1}{x}\right )}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\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{(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (15+6 n+n^2\right ) \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}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (18+7 n-2 n^2-n^3\right ) \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{\left (a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \frac{\left (9-10 n^2+n^4\right ) \left (1-\frac{x}{a}\right )^{\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}}}{a^4 x} \, dx,x,\frac{1}{x}\right )}{(1-n) (3-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\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{(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (15+6 n+n^2\right ) \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}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (18+7 n-2 n^2-n^3\right ) \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{\left (\left (9-10 n^2+n^4\right ) \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{(1-n) (3-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\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{(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (15+6 n+n^2\right ) \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}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (18+7 n-2 n^2-n^3\right ) \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{2 \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 \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{(3-n) \left (c-a^2 c x^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 1.9755, size = 201, normalized size = 0.43 $\frac{\left (a^2 x^2-1\right ) \left (-\frac{8 a x \sqrt{1-\frac{1}{a^2 x^2}} e^{(n+1) \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},e^{2 \coth ^{-1}(a x)}\right )}{n+1}+\frac{e^{n \coth ^{-1}(a x)} \left (-3 a \left (n^2-1\right ) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (3 \coth ^{-1}(a x)\right )+3 a n^2 x+2 \left (n^2-1\right ) n \cosh \left (2 \coth ^{-1}(a x)\right )-27 a x-2 n^3+26 n\right )}{n^4-10 n^2+9}+\frac{8 (n-a x) e^{n \coth ^{-1}(a x)}}{n^2-1}\right )}{4 a^5 c \left (c-a^2 c x^2\right )^{3/2}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^(n*ArcCoth[a*x])*x^4)/(c - a^2*c*x^2)^(5/2),x]

[Out]

((-1 + a^2*x^2)*((8*E^(n*ArcCoth[a*x])*(n - a*x))/(-1 + n^2) + (E^(n*ArcCoth[a*x])*(26*n - 2*n^3 - 27*a*x + 3*
a*n^2*x + 2*n*(-1 + n^2)*Cosh[2*ArcCoth[a*x]] - 3*a*(-1 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]]))/
(9 - 10*n^2 + n^4) - (8*a*E^((1 + n)*ArcCoth[a*x])*Sqrt[1 - 1/(a^2*x^2)]*x*Hypergeometric2F1[1, (1 + n)/2, (3
+ n)/2, E^(2*ArcCoth[a*x])])/(1 + n)))/(4*a^5*c*(c - a^2*c*x^2)^(3/2))

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Maple [F]  time = 0.303, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}{x}^{4} \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^4/(-a^2*c*x^2+c)^(5/2),x)

[Out]

int(exp(n*arccoth(a*x))*x^4/(-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{x^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))*x^4/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(x^4*((a*x - 1)/(a*x + 1))^(1/2*n)/(-a^2*c*x^2 + c)^(5/2), 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} x^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))*x^4/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*c*x^2 + c)*x^4*((a*x - 1)/(a*x + 1))^(1/2*n)/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2
- c^3), 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**4/(-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{x^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))*x^4/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate(x^4*((a*x - 1)/(a*x + 1))^(1/2*n)/(-a^2*c*x^2 + c)^(5/2), x)