3.799 \(\int \frac {e^{n \tanh ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{5/2}} \, dx\)

Optimal. Leaf size=1039 \[ -\frac {(a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-3)}}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}+\frac {(n+4) (a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-3)}}{a^3 (n+3) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}-\frac {3 (n+4) (a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-1)}}{a^5 (n+3) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}+\frac {n (a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-1)}}{a^6 (n+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {3 (2-n) (n+4) (a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-1)}}{a^6 \left (9-n^2\right ) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}+\frac {3 (n+4) \left (-n^2+2 n+1\right ) (a x+1)^{\frac {n-1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-1)}}{a^6 (3-n) (n+1) (n+3) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {3 n (a x+1)^{\frac {n-1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-1)}}{a^6 (n+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}+\frac {3 n (a x+1)^{\frac {n+1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-1)}}{a^6 (n+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {2^{\frac {n+3}{2}} n \left (1-a^2 x^2\right )^{5/2} \, _2F_1\left (\frac {1}{2} (-n-1),\frac {1}{2} (-n-1);\frac {1-n}{2};\frac {1}{2} (1-a x)\right ) (1-a x)^{\frac {1}{2} (-n-1)}}{a^6 (n+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {2 n (a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1-n}{2}}}{a^6 \left (1-n^2\right ) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}+\frac {3 n (a x+1)^{\frac {n-1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1-n}{2}}}{a^6 \left (1-n^2\right ) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {3 (n+4) \left (-n^2+2 n+1\right ) (a x+1)^{\frac {n-1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1-n}{2}}}{a^6 \left (n^4-10 n^2+9\right ) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}+\frac {2 n (a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {3-n}{2}}}{a^6 (n+1) \left (n^2-4 n+3\right ) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \]

[Out]

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

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Rubi [A]  time = 0.77, antiderivative size = 1039, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6160, 6150, 100, 159, 128, 45, 37, 69, 94, 90, 79} \[ -\frac {(a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-3)}}{a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}+\frac {(n+4) (a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-3)}}{a^3 (n+3) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}-\frac {3 (n+4) (a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-1)}}{a^5 (n+3) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}+\frac {n (a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-1)}}{a^6 (n+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {3 (2-n) (n+4) (a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-1)}}{a^6 \left (9-n^2\right ) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}+\frac {3 (n+4) \left (-n^2+2 n+1\right ) (a x+1)^{\frac {n-1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-1)}}{a^6 (3-n) (n+1) (n+3) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {3 n (a x+1)^{\frac {n-1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-1)}}{a^6 (n+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}+\frac {3 n (a x+1)^{\frac {n+1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1}{2} (-n-1)}}{a^6 (n+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {2^{\frac {n+3}{2}} n \left (1-a^2 x^2\right )^{5/2} \, _2F_1\left (\frac {1}{2} (-n-1),\frac {1}{2} (-n-1);\frac {1-n}{2};\frac {1}{2} (1-a x)\right ) (1-a x)^{\frac {1}{2} (-n-1)}}{a^6 (n+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {2 n (a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1-n}{2}}}{a^6 \left (1-n^2\right ) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}+\frac {3 n (a x+1)^{\frac {n-1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1-n}{2}}}{a^6 \left (1-n^2\right ) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}-\frac {3 (n+4) \left (-n^2+2 n+1\right ) (a x+1)^{\frac {n-1}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {1-n}{2}}}{a^6 \left (n^4-10 n^2+9\right ) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}+\frac {2 n (a x+1)^{\frac {n-3}{2}} \left (1-a^2 x^2\right )^{5/2} (1-a x)^{\frac {3-n}{2}}}{a^6 (n+1) \left (n^2-4 n+3\right ) \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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]))

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(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] && SumSimplerQ[p, 1]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 94

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

Rule 100

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))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && 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 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Dist[h/b, Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(a + b*x)^m*(
c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (SumSimplerQ[m, 1] || ( !SumS
implerQ[n, 1] &&  !SumSimplerQ[p, 1]))

Rule 6150

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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 6160

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

Rubi steps

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

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Mathematica [A]  time = 6.32, size = 227, normalized size = 0.22 \[ \frac {\left (a^2 x^2-1\right )^2 \left (-\frac {4 \left (a^2 x^2-1\right ) \left (\frac {2 n e^{(n+1) \tanh ^{-1}(a x)} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-e^{2 \tanh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-(n+1) e^{n \tanh ^{-1}(a x)}\right )}{n+1}-\frac {e^{n \tanh ^{-1}(a x)} \left (3 \left (n^2-1\right ) \sqrt {1-a^2 x^2} \cosh \left (3 \tanh ^{-1}(a x)\right )-2 a n^3 x-2 a \left (n^2-1\right ) n x \cosh \left (2 \tanh ^{-1}(a x)\right )+10 a n x+n^2-9\right )}{n^4-10 n^2+9}-\frac {8 (a n x-1) e^{n \tanh ^{-1}(a x)}}{n^2-1}\right )}{4 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-1 + a^2*x^2)^2*((-8*E^(n*ArcTanh[a*x])*(-1 + a*n*x))/(-1 + n^2) - (E^(n*ArcTanh[a*x])*(-9 + n^2 + 10*a*n*x
- 2*a*n^3*x - 2*a*n*(-1 + n^2)*x*Cosh[2*ArcTanh[a*x]] + 3*(-1 + n^2)*Sqrt[1 - a^2*x^2]*Cosh[3*ArcTanh[a*x]]))/
(9 - 10*n^2 + n^4) - (4*(-1 + a^2*x^2)*(-(E^(n*ArcTanh[a*x])*(1 + n)) + (2*E^((1 + n)*ArcTanh[a*x])*n*Hypergeo
metric2F1[1, (1 + n)/2, (3 + n)/2, -E^(2*ArcTanh[a*x])])/Sqrt[1 - a^2*x^2]))/(1 + n)))/(4*a^6*(c - c/(a^2*x^2)
)^(5/2)*x^5)

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fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{6} x^{6} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(a^6*x^6*((a*x + 1)/(a*x - 1))^(1/2*n)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{\left (c -\frac {c}{a^{2} x^{2}}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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