3.695 \(\int e^{2 \tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^{9/2} \, dx\)

Optimal. Leaf size=450 \[ -\frac {a x^2 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{28 (1-a x)^2}-\frac {x (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{8 (1-a x)}+\frac {71 a^2 x^3 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{336 (1-a x)^3}-\frac {501 a^8 x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{128 (1-a x)^4 (a x+1)^4}+\frac {2 a^8 x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \sin ^{-1}(a x)}{(1-a x)^{9/2} (a x+1)^{9/2}}+\frac {245 a^8 x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{128 (1-a x)^{9/2} (a x+1)^{9/2}}+\frac {373 a^7 x^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{192 (1-a x)^4 (a x+1)^3}+\frac {501 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{640 (1-a x)^4 (a x+1)^2}+\frac {661 a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{1680 (1-a x)^4 (a x+1)}+\frac {295 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{1344 (1-a x)^4}-\frac {127 a^3 x^4 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{420 (1-a x)^4} \]

[Out]

295/1344*a^4*(c-c/a^2/x^2)^(9/2)*x^5/(-a*x+1)^4-501/128*a^8*(c-c/a^2/x^2)^(9/2)*x^9/(-a*x+1)^4/(a*x+1)^4+373/1
92*a^7*(c-c/a^2/x^2)^(9/2)*x^8/(-a*x+1)^4/(a*x+1)^3+501/640*a^6*(c-c/a^2/x^2)^(9/2)*x^7/(-a*x+1)^4/(a*x+1)^2+6
61/1680*a^5*(c-c/a^2/x^2)^(9/2)*x^6/(-a*x+1)^4/(a*x+1)-127/420*a^3*(c-c/a^2/x^2)^(9/2)*x^4*(a*x+1)/(-a*x+1)^4+
71/336*a^2*(c-c/a^2/x^2)^(9/2)*x^3*(a*x+1)/(-a*x+1)^3-1/28*a*(c-c/a^2/x^2)^(9/2)*x^2*(a*x+1)/(-a*x+1)^2-1/8*(c
-c/a^2/x^2)^(9/2)*x*(a*x+1)/(-a*x+1)+2*a^8*(c-c/a^2/x^2)^(9/2)*x^9*arcsin(a*x)/(-a*x+1)^(9/2)/(a*x+1)^(9/2)+24
5/128*a^8*(c-c/a^2/x^2)^(9/2)*x^9*arctanh((-a*x+1)^(1/2)*(a*x+1)^(1/2))/(-a*x+1)^(9/2)/(a*x+1)^(9/2)

________________________________________________________________________________________

Rubi [A]  time = 0.56, antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6159, 6129, 97, 149, 154, 157, 41, 216, 92, 208} \[ -\frac {501 a^8 x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{128 (1-a x)^4 (a x+1)^4}+\frac {373 a^7 x^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{192 (1-a x)^4 (a x+1)^3}+\frac {501 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{640 (1-a x)^4 (a x+1)^2}+\frac {661 a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{1680 (1-a x)^4 (a x+1)}+\frac {295 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{1344 (1-a x)^4}-\frac {127 a^3 x^4 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{420 (1-a x)^4}+\frac {71 a^2 x^3 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{336 (1-a x)^3}-\frac {a x^2 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{28 (1-a x)^2}-\frac {x (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}{8 (1-a x)}+\frac {2 a^8 x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \sin ^{-1}(a x)}{(1-a x)^{9/2} (a x+1)^{9/2}}+\frac {245 a^8 x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{128 (1-a x)^{9/2} (a x+1)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(9/2),x]

[Out]

(295*a^4*(c - c/(a^2*x^2))^(9/2)*x^5)/(1344*(1 - a*x)^4) - (501*a^8*(c - c/(a^2*x^2))^(9/2)*x^9)/(128*(1 - a*x
)^4*(1 + a*x)^4) + (373*a^7*(c - c/(a^2*x^2))^(9/2)*x^8)/(192*(1 - a*x)^4*(1 + a*x)^3) + (501*a^6*(c - c/(a^2*
x^2))^(9/2)*x^7)/(640*(1 - a*x)^4*(1 + a*x)^2) + (661*a^5*(c - c/(a^2*x^2))^(9/2)*x^6)/(1680*(1 - a*x)^4*(1 +
a*x)) - (127*a^3*(c - c/(a^2*x^2))^(9/2)*x^4*(1 + a*x))/(420*(1 - a*x)^4) + (71*a^2*(c - c/(a^2*x^2))^(9/2)*x^
3*(1 + a*x))/(336*(1 - a*x)^3) - (a*(c - c/(a^2*x^2))^(9/2)*x^2*(1 + a*x))/(28*(1 - a*x)^2) - ((c - c/(a^2*x^2
))^(9/2)*x*(1 + a*x))/(8*(1 - a*x)) + (2*a^8*(c - c/(a^2*x^2))^(9/2)*x^9*ArcSin[a*x])/((1 - a*x)^(9/2)*(1 + a*
x)^(9/2)) + (245*a^8*(c - c/(a^2*x^2))^(9/2)*x^9*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(128*(1 - a*x)^(9/2)*(1
+ a*x)^(9/2))

Rule 41

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

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 97

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)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 149

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*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 6129

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

Rule 6159

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbol] :> Dist[(x^(2*p)*(c + d/x^2)^p)/(
(1 - a*x)^p*(1 + a*x)^p), Int[(u*(1 - a*x)^p*(1 + a*x)^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] &&  !GtQ[c, 0]

Rubi steps

\begin {align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^{9/2} (1+a x)^{9/2}}{x^9} \, dx}{(1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1-a x)^{7/2} (1+a x)^{11/2}}{x^9} \, dx}{(1-a x)^{9/2} (1+a x)^{9/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1-a x)^{5/2} (1+a x)^{9/2} \left (2 a-9 a^2 x\right )}{x^8} \, dx}{8 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1-a x)^{3/2} (1+a x)^{9/2} \left (-71 a^2+61 a^3 x\right )}{x^7} \, dx}{56 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {\sqrt {1-a x} (1+a x)^{9/2} \left (508 a^3-295 a^4 x\right )}{x^6} \, dx}{336 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1+a x)^{9/2} \left (-1475 a^4+967 a^5 x\right )}{x^5 \sqrt {1-a x}} \, dx}{1680 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1+a x)^{7/2} \left (-7932 a^5+5343 a^6 x\right )}{x^4 \sqrt {1-a x}} \, dx}{6720 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1+a x)^{5/2} \left (-31563 a^6+23961 a^7 x\right )}{x^3 \sqrt {1-a x}} \, dx}{20160 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}+\frac {501 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {(1+a x)^{3/2} \left (-78330 a^7+79485 a^8 x\right )}{x^2 \sqrt {1-a x}} \, dx}{40320 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}+\frac {373 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac {501 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {\sqrt {1+a x} \left (-77175 a^8+157815 a^9 x\right )}{x \sqrt {1-a x}} \, dx}{40320 (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac {501 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac {373 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac {501 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {77175 a^9-80640 a^{10} x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{40320 a (1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac {501 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac {373 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac {501 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}-\frac {\left (245 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{128 (1-a x)^{9/2} (1+a x)^{9/2}}+\frac {\left (2 a^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{(1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac {501 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac {373 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac {501 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {\left (245 a^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{128 (1-a x)^{9/2} (1+a x)^{9/2}}+\frac {\left (2 a^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{(1-a x)^{9/2} (1+a x)^{9/2}}\\ &=\frac {295 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}{1344 (1-a x)^4}-\frac {501 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}{128 (1-a x)^4 (1+a x)^4}+\frac {373 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^8}{192 (1-a x)^4 (1+a x)^3}+\frac {501 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}{640 (1-a x)^4 (1+a x)^2}+\frac {661 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}{1680 (1-a x)^4 (1+a x)}-\frac {127 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4 (1+a x)}{420 (1-a x)^4}+\frac {71 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3 (1+a x)}{336 (1-a x)^3}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2 (1+a x)}{28 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x (1+a x)}{8 (1-a x)}+\frac {2 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9 \sin ^{-1}(a x)}{(1-a x)^{9/2} (1+a x)^{9/2}}+\frac {245 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9 \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{128 (1-a x)^{9/2} (1+a x)^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 166, normalized size = 0.37 \[ -\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}} \left (26880 a^8 x^8 \log \left (\sqrt {a^2 x^2-1}+a x\right )+25725 a^8 x^8 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )+\sqrt {a^2 x^2-1} \left (13440 a^8 x^8-45056 a^7 x^7+14595 a^6 x^6+31232 a^5 x^5+770 a^4 x^4-16896 a^3 x^3-4760 a^2 x^2+3840 a x+1680\right )\right )}{13440 a^8 x^7 \sqrt {a^2 x^2-1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(9/2),x]

[Out]

-1/13440*(c^4*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(1680 + 3840*a*x - 4760*a^2*x^2 - 16896*a^3*x^3 + 770*
a^4*x^4 + 31232*a^5*x^5 + 14595*a^6*x^6 - 45056*a^7*x^7 + 13440*a^8*x^8) + 25725*a^8*x^8*ArcTan[1/Sqrt[-1 + a^
2*x^2]] + 26880*a^8*x^8*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(a^8*x^7*Sqrt[-1 + a^2*x^2])

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fricas [A]  time = 0.77, size = 482, normalized size = 1.07 \[ \left [\frac {53760 \, a^{7} \sqrt {-c} c^{4} x^{7} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 25725 \, a^{7} \sqrt {-c} c^{4} x^{7} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (13440 \, a^{8} c^{4} x^{8} - 45056 \, a^{7} c^{4} x^{7} + 14595 \, a^{6} c^{4} x^{6} + 31232 \, a^{5} c^{4} x^{5} + 770 \, a^{4} c^{4} x^{4} - 16896 \, a^{3} c^{4} x^{3} - 4760 \, a^{2} c^{4} x^{2} + 3840 \, a c^{4} x + 1680 \, c^{4}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{26880 \, a^{8} x^{7}}, -\frac {25725 \, a^{7} c^{\frac {9}{2}} x^{7} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 13440 \, a^{7} c^{\frac {9}{2}} x^{7} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (13440 \, a^{8} c^{4} x^{8} - 45056 \, a^{7} c^{4} x^{7} + 14595 \, a^{6} c^{4} x^{6} + 31232 \, a^{5} c^{4} x^{5} + 770 \, a^{4} c^{4} x^{4} - 16896 \, a^{3} c^{4} x^{3} - 4760 \, a^{2} c^{4} x^{2} + 3840 \, a c^{4} x + 1680 \, c^{4}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{13440 \, a^{8} x^{7}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(9/2),x, algorithm="fricas")

[Out]

[1/26880*(53760*a^7*sqrt(-c)*c^4*x^7*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c))
+ 25725*a^7*sqrt(-c)*c^4*x^7*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*
(13440*a^8*c^4*x^8 - 45056*a^7*c^4*x^7 + 14595*a^6*c^4*x^6 + 31232*a^5*c^4*x^5 + 770*a^4*c^4*x^4 - 16896*a^3*c
^4*x^3 - 4760*a^2*c^4*x^2 + 3840*a*c^4*x + 1680*c^4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7), -1/13440*(257
25*a^7*c^(9/2)*x^7*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 13440*a^7*c^(9/2)*x^7
*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) + (13440*a^8*c^4*x^8 - 45056*a^7*c^4
*x^7 + 14595*a^6*c^4*x^6 + 31232*a^5*c^4*x^5 + 770*a^4*c^4*x^4 - 16896*a^3*c^4*x^3 - 4760*a^2*c^4*x^2 + 3840*a
*c^4*x + 1680*c^4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^8*x^7)]

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giac [A]  time = 137.06, size = 707, normalized size = 1.57 \[ \frac {1}{6720} \, {\left (\frac {25725 \, c^{\frac {9}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x)}{a^{2}} + \frac {13440 \, c^{\frac {9}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\relax (x)}{a {\left | a \right |}} - \frac {6720 \, \sqrt {a^{2} c x^{2} - c} c^{4} \mathrm {sgn}\relax (x)}{a^{2}} + \frac {14595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{15} c^{5} {\left | a \right |} \mathrm {sgn}\relax (x) + 107520 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{14} a c^{\frac {11}{2}} \mathrm {sgn}\relax (x) + 76055 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{13} c^{6} {\left | a \right |} \mathrm {sgn}\relax (x) + 430080 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{12} a c^{\frac {13}{2}} \mathrm {sgn}\relax (x) + 64435 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{11} c^{7} {\left | a \right |} \mathrm {sgn}\relax (x) + 1111040 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{10} a c^{\frac {15}{2}} \mathrm {sgn}\relax (x) + 110495 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{9} c^{8} {\left | a \right |} \mathrm {sgn}\relax (x) + 1576960 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{8} a c^{\frac {17}{2}} \mathrm {sgn}\relax (x) - 110495 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} c^{9} {\left | a \right |} \mathrm {sgn}\relax (x) + 1412096 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a c^{\frac {19}{2}} \mathrm {sgn}\relax (x) - 64435 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} c^{10} {\left | a \right |} \mathrm {sgn}\relax (x) + 831488 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {21}{2}} \mathrm {sgn}\relax (x) - 76055 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{11} {\left | a \right |} \mathrm {sgn}\relax (x) + 252928 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {23}{2}} \mathrm {sgn}\relax (x) - 14595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{12} {\left | a \right |} \mathrm {sgn}\relax (x) + 45056 \, a c^{\frac {25}{2}} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{8} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(9/2),x, algorithm="giac")

[Out]

1/6720*(25725*c^(9/2)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x)/a^2 + 13440*c^(9/2)*log(ab
s(-sqrt(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn(x)/(a*abs(a)) - 6720*sqrt(a^2*c*x^2 - c)*c^4*sgn(x)/a^2 + (14595*
(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^15*c^5*abs(a)*sgn(x) + 107520*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^14*a
*c^(11/2)*sgn(x) + 76055*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^13*c^6*abs(a)*sgn(x) + 430080*(sqrt(a^2*c)*x -
sqrt(a^2*c*x^2 - c))^12*a*c^(13/2)*sgn(x) + 64435*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^11*c^7*abs(a)*sgn(x) +
1111040*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^10*a*c^(15/2)*sgn(x) + 110495*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 -
c))^9*c^8*abs(a)*sgn(x) + 1576960*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^8*a*c^(17/2)*sgn(x) - 110495*(sqrt(a^
2*c)*x - sqrt(a^2*c*x^2 - c))^7*c^9*abs(a)*sgn(x) + 1412096*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^6*a*c^(19/2)
*sgn(x) - 64435*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^5*c^10*abs(a)*sgn(x) + 831488*(sqrt(a^2*c)*x - sqrt(a^2*
c*x^2 - c))^4*a*c^(21/2)*sgn(x) - 76055*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^3*c^11*abs(a)*sgn(x) + 252928*(s
qrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2*a*c^(23/2)*sgn(x) - 14595*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))*c^12*abs
(a)*sgn(x) + 45056*a*c^(25/2)*sgn(x))/(((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^8*a^2*abs(a)))*abs(a)

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maple [B]  time = 0.12, size = 965, normalized size = 2.14 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(9/2),x)

[Out]

-1/40320*(c*(a^2*x^2-1)/a^2/x^2)^(9/2)*x/a^2*(-5040*a^4*(c*(a^2*x^2-1)/a^2)^(11/2)*(-c/a^2)^(1/2)+77175*ln(2*(
(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*a^2-c)/a^2/x)*x^8*c^6+23808*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(11/2)
*x^7*a^11-17535*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(11/2)*x^6*a^10-13056*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1
1/2)*x^5*a^9-6510*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(11/2)*x^4*a^8-6912*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1
1/2)*x^3*a^7-10920*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(11/2)*x^2*a^6-11520*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^
(11/2)*x*a^5+58590*(-c/a^2)^(1/2)*c^(11/2)*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*x^8*a+22050*(-c/a^2)^(1/2)*
c^(11/2)*ln((c^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)+c*x)/c^(1/2))*x^8*a-23808*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2
)^(9/2)*x^9*a^11*c+8575*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(9/2)*x^8*a^10*c+8960*(-c/a^2)^(1/2)*((a*x-1)*(a*x+
1)*c/a^2)^(9/2)*x^8*a^10*c+26784*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(7/2)*x^9*a^9*c^2+10080*(-c/a^2)^(1/2)*((a
*x-1)*(a*x+1)*c/a^2)^(7/2)*x^9*a^9*c^2-11025*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(7/2)*x^8*a^8*c^2-31248*(-c/a^
2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(5/2)*x^9*a^7*c^3-11760*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(5/2)*x^9*a^7*c^3+
15435*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(5/2)*x^8*a^6*c^3+39060*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*x^9*
a^5*c^4+14700*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(3/2)*x^9*a^5*c^4-25725*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2
)^(3/2)*x^8*a^4*c^4-58590*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^9*a^3*c^5-22050*(-c/a^2)^(1/2)*((a*x-1)*(
a*x+1)*c/a^2)^(1/2)*x^9*a^3*c^5+77175*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^8*a^2*c^5)/(-c/a^2)^(1/2)/(c*
(a^2*x^2-1)/a^2)^(9/2)/c

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a x + 1\right )}^{2} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {9}{2}}}{a^{2} x^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(9/2),x, algorithm="maxima")

[Out]

-integrate((a*x + 1)^2*(c - c/(a^2*x^2))^(9/2)/(a^2*x^2 - 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int -\frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{9/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((c - c/(a^2*x^2))^(9/2)*(a*x + 1)^2)/(a^2*x^2 - 1),x)

[Out]

int(-((c - c/(a^2*x^2))^(9/2)*(a*x + 1)^2)/(a^2*x^2 - 1), x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(9/2),x)

[Out]

Exception raised: TypeError

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