3.724 \(\int e^{-2 \tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^{7/2} \, dx\)

Optimal. Leaf size=375 \[ \frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{15 (a x+1)}-\frac {x (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{6 (a x+1)}+\frac {23 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{120 (1-a x) (a x+1)}-\frac {7 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)^3}+\frac {2 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \sin ^{-1}(a x)}{(1-a x)^{7/2} (a x+1)^{7/2}}-\frac {25 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{16 (1-a x)^{7/2} (a x+1)^{7/2}}-\frac {3 a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{8 (1-a x)^3 (a x+1)^2}+\frac {19 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)}-\frac {2 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{3 (1-a x)^2 (a x+1)} \]

[Out]

-7/16*a^6*(c-c/a^2/x^2)^(7/2)*x^7/(-a*x+1)^3/(a*x+1)^3-3/8*a^5*(c-c/a^2/x^2)^(7/2)*x^6/(-a*x+1)^3/(a*x+1)^2+1/
15*a*(c-c/a^2/x^2)^(7/2)*x^2/(a*x+1)+19/16*a^4*(c-c/a^2/x^2)^(7/2)*x^5/(-a*x+1)^3/(a*x+1)-2/3*a^3*(c-c/a^2/x^2
)^(7/2)*x^4/(-a*x+1)^2/(a*x+1)+23/120*a^2*(c-c/a^2/x^2)^(7/2)*x^3/(-a*x+1)/(a*x+1)-1/6*(c-c/a^2/x^2)^(7/2)*x*(
-a*x+1)/(a*x+1)+2*a^6*(c-c/a^2/x^2)^(7/2)*x^7*arcsin(a*x)/(-a*x+1)^(7/2)/(a*x+1)^(7/2)-25/16*a^6*(c-c/a^2/x^2)
^(7/2)*x^7*arctanh((-a*x+1)^(1/2)*(a*x+1)^(1/2))/(-a*x+1)^(7/2)/(a*x+1)^(7/2)

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Rubi [A]  time = 0.47, antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ -\frac {7 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)^3}-\frac {3 a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{8 (1-a x)^3 (a x+1)^2}+\frac {19 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)}-\frac {2 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{3 (1-a x)^2 (a x+1)}+\frac {23 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{120 (1-a x) (a x+1)}+\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{15 (a x+1)}-\frac {x (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{6 (a x+1)}+\frac {2 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \sin ^{-1}(a x)}{(1-a x)^{7/2} (a x+1)^{7/2}}-\frac {25 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{16 (1-a x)^{7/2} (a x+1)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c - c/(a^2*x^2))^(7/2)/E^(2*ArcTanh[a*x]),x]

[Out]

(-7*a^6*(c - c/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)^3*(1 + a*x)^3) - (3*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(8*(1
- a*x)^3*(1 + a*x)^2) + (a*(c - c/(a^2*x^2))^(7/2)*x^2)/(15*(1 + a*x)) + (19*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/
(16*(1 - a*x)^3*(1 + a*x)) - (2*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(3*(1 - a*x)^2*(1 + a*x)) + (23*a^2*(c - c/(a
^2*x^2))^(7/2)*x^3)/(120*(1 - a*x)*(1 + a*x)) - ((c - c/(a^2*x^2))^(7/2)*x*(1 - a*x))/(6*(1 + a*x)) + (2*a^6*(
c - c/(a^2*x^2))^(7/2)*x^7*ArcSin[a*x])/((1 - a*x)^(7/2)*(1 + a*x)^(7/2)) - (25*a^6*(c - c/(a^2*x^2))^(7/2)*x^
7*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(16*(1 - a*x)^(7/2)*(1 + a*x)^(7/2))

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 )^{7/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {e^{-2 \tanh ^{-1}(a x)} (1-a x)^{7/2} (1+a x)^{7/2}}{x^7} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^{9/2} (1+a x)^{5/2}}{x^7} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^{7/2} (1+a x)^{3/2} \left (-2 a-7 a^2 x\right )}{x^6} \, dx}{6 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^{5/2} (1+a x)^{3/2} \left (-23 a^2+37 a^3 x\right )}{x^5} \, dx}{30 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}+\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^{3/2} (1+a x)^{3/2} \left (240 a^3-125 a^4 x\right )}{x^4} \, dx}{120 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}+\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {\sqrt {1-a x} (1+a x)^{3/2} \left (-855 a^4+135 a^5 x\right )}{x^3} \, dx}{360 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}+\frac {19 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}-\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}+\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1+a x)^{3/2} \left (270 a^5+585 a^6 x\right )}{x^2 \sqrt {1-a x}} \, dx}{720 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {3 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}+\frac {19 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}-\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}+\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {\sqrt {1+a x} \left (1125 a^6+315 a^7 x\right )}{x \sqrt {1-a x}} \, dx}{720 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {7 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}-\frac {3 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}+\frac {19 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}-\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}+\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {-1125 a^7-1440 a^8 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{720 a (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {7 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}-\frac {3 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}+\frac {19 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}-\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}+\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}+\frac {\left (25 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{16 (1-a x)^{7/2} (1+a x)^{7/2}}+\frac {\left (2 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {7 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}-\frac {3 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}+\frac {19 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}-\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}+\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac {\left (25 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{16 (1-a x)^{7/2} (1+a x)^{7/2}}+\frac {\left (2 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {7 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}-\frac {3 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}+\frac {19 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}-\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}+\frac {23 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}+\frac {2 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \sin ^{-1}(a x)}{(1-a x)^{7/2} (1+a x)^{7/2}}-\frac {25 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{16 (1-a x)^{7/2} (1+a x)^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 150, normalized size = 0.40 \[ -\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \left (-480 a^6 x^6 \log \left (\sqrt {a^2 x^2-1}+a x\right )+375 a^6 x^6 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )+\sqrt {a^2 x^2-1} \left (240 a^6 x^6+736 a^5 x^5+105 a^4 x^4-352 a^3 x^3+70 a^2 x^2+96 a x-40\right )\right )}{240 a^6 x^5 \sqrt {a^2 x^2-1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - c/(a^2*x^2))^(7/2)/E^(2*ArcTanh[a*x]),x]

[Out]

-1/240*(c^3*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(-40 + 96*a*x + 70*a^2*x^2 - 352*a^3*x^3 + 105*a^4*x^4 +
736*a^5*x^5 + 240*a^6*x^6) + 375*a^6*x^6*ArcTan[1/Sqrt[-1 + a^2*x^2]] - 480*a^6*x^6*Log[a*x + Sqrt[-1 + a^2*x
^2]]))/(a^6*x^5*Sqrt[-1 + a^2*x^2])

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fricas [A]  time = 1.22, size = 438, normalized size = 1.17 \[ \left [-\frac {960 \, a^{5} \sqrt {-c} c^{3} x^{5} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 375 \, a^{5} \sqrt {-c} c^{3} x^{5} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (240 \, a^{6} c^{3} x^{6} + 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} - 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} + 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{480 \, a^{6} x^{5}}, -\frac {375 \, a^{5} c^{\frac {7}{2}} x^{5} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 240 \, a^{5} c^{\frac {7}{2}} x^{5} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (240 \, a^{6} c^{3} x^{6} + 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} - 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} + 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{240 \, a^{6} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas")

[Out]

[-1/480*(960*a^5*sqrt(-c)*c^3*x^5*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - 3
75*a^5*sqrt(-c)*c^3*x^5*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*(240*
a^6*c^3*x^6 + 736*a^5*c^3*x^5 + 105*a^4*c^3*x^4 - 352*a^3*c^3*x^3 + 70*a^2*c^3*x^2 + 96*a*c^3*x - 40*c^3)*sqrt
((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5), -1/240*(375*a^5*c^(7/2)*x^5*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a
^2*x^2))/(a^2*c*x^2 - c)) - 240*a^5*c^(7/2)*x^5*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) + (240*a^6*c^3*x^6 + 736*a^5*c^3*x^5 + 105*a^4*c^3*x^4 - 352*a^3*c^3*x^3 + 70*a^2*c^3*x^2 + 96*a*c^
3*x - 40*c^3)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5)]

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giac [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((c-c/a^2/x^2)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.07, size = 795, normalized size = 2.12 \[ -\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}} x \left (-2016 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{7} a^{9} c +2016 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{5} a^{9}-375 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{6} a^{8} c +480 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} x^{6} a^{8} c -105 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{4} a^{8}+2352 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{7} a^{7} c^{2}-560 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{7} a^{7} c^{2}+224 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{3} a^{7}+525 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{6} a^{6} c^{2}-2940 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{7} a^{5} c^{3}+700 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} x^{7} a^{5} c^{3}-630 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{2} a^{6}-875 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{6} a^{4} c^{3}+672 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, x \,a^{5}+4410 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, x^{7} a^{3} c^{4}-1050 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{7} a^{3} c^{4}-280 a^{4} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}+2625 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, x^{6} a^{2} c^{4}-4410 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {9}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x^{6} a +1050 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {9}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) x^{6} a +2625 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x^{6} c^{5}\right )}{1680 a^{2} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}\, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-c/a^2/x^2)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x)

[Out]

-1/1680*(c*(a^2*x^2-1)/a^2/x^2)^(7/2)*x/a^2*(-2016*(c*(a^2*x^2-1)/a^2)^(7/2)*(-c/a^2)^(1/2)*x^7*a^9*c+2016*(c*
(a^2*x^2-1)/a^2)^(9/2)*(-c/a^2)^(1/2)*x^5*a^9-375*(c*(a^2*x^2-1)/a^2)^(7/2)*(-c/a^2)^(1/2)*x^6*a^8*c+480*(-c/a
^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(7/2)*x^6*a^8*c-105*(c*(a^2*x^2-1)/a^2)^(9/2)*(-c/a^2)^(1/2)*x^4*a^8+2352*(c
*(a^2*x^2-1)/a^2)^(5/2)*(-c/a^2)^(1/2)*x^7*a^7*c^2-560*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(5/2)*x^7*a^7*c^
2+224*(c*(a^2*x^2-1)/a^2)^(9/2)*(-c/a^2)^(1/2)*x^3*a^7+525*(c*(a^2*x^2-1)/a^2)^(5/2)*(-c/a^2)^(1/2)*x^6*a^6*c^
2-2940*(c*(a^2*x^2-1)/a^2)^(3/2)*(-c/a^2)^(1/2)*x^7*a^5*c^3+700*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(3/2)*x
^7*a^5*c^3-630*(c*(a^2*x^2-1)/a^2)^(9/2)*(-c/a^2)^(1/2)*x^2*a^6-875*(c*(a^2*x^2-1)/a^2)^(3/2)*(-c/a^2)^(1/2)*x
^6*a^4*c^3+672*(c*(a^2*x^2-1)/a^2)^(9/2)*(-c/a^2)^(1/2)*x*a^5+4410*(c*(a^2*x^2-1)/a^2)^(1/2)*(-c/a^2)^(1/2)*x^
7*a^3*c^4-1050*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)*x^7*a^3*c^4-280*a^4*(c*(a^2*x^2-1)/a^2)^(9/2)*(-c/
a^2)^(1/2)+2625*(c*(a^2*x^2-1)/a^2)^(1/2)*(-c/a^2)^(1/2)*x^6*a^2*c^4-4410*(-c/a^2)^(1/2)*c^(9/2)*ln(x*c^(1/2)+
(c*(a^2*x^2-1)/a^2)^(1/2))*x^6*a+1050*(-c/a^2)^(1/2)*c^(9/2)*ln((c^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)+c*x)/c^
(1/2))*x^6*a+2625*ln(2*((-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*a^2-c)/a^2/x)*x^6*c^5)/(c*(a^2*x^2-1)/a^2)^(7
/2)/(-c/a^2)^(1/2)/c

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}{{\left (a x + 1\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(7/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima")

[Out]

-integrate((a^2*x^2 - 1)*(c - c/(a^2*x^2))^(7/2)/(a*x + 1)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((c - c/(a^2*x^2))^(7/2)*(a^2*x^2 - 1))/(a*x + 1)^2,x)

[Out]

-int(((c - c/(a^2*x^2))^(7/2)*(a^2*x^2 - 1))/(a*x + 1)^2, x)

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sympy [C]  time = 24.01, size = 1059, normalized size = 2.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a**2/x**2)**(7/2)/(a*x+1)**2*(-a**2*x**2+1),x)

[Out]

-c**3*Piecewise((sqrt(c)*sqrt(a**2*x**2 - 1)/a - I*sqrt(c)*log(a*x)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) + sqrt(
c)*asin(1/(a*x))/a, Abs(a**2*x**2) > 1), (I*sqrt(c)*sqrt(-a**2*x**2 + 1)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) -
I*sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1)/a, True)) + 2*c**3*Piecewise((-a*sqrt(c)*x/sqrt(a**2*x**2 - 1) + sqrt(
c)*acosh(a*x) + sqrt(c)/(a*x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (I*a*sqrt(c)*x/sqrt(-a**2*x**2 + 1) -
I*sqrt(c)*asin(a*x) - I*sqrt(c)/(a*x*sqrt(-a**2*x**2 + 1)), True))/a + c**3*Piecewise((I*a*sqrt(c)*acosh(1/(a*
x))/2 + I*sqrt(c)/(2*x*sqrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(2*a**2*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**
2*x**2) > 1), (-a*sqrt(c)*asin(1/(a*x))/2 - sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**2 - 4*c**3*Piecew
ise((0, Eq(c, 0)), (a**2*(c - c/(a**2*x**2))**(3/2)/(3*c), True))/a**3 + c**3*Piecewise((I*a**3*sqrt(c)*acosh(
1/(a*x))/8 - I*a**2*sqrt(c)/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*I*sqrt(c)/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - I
*sqrt(c)/(4*a**2*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-a**3*sqrt(c)*asin(1/(a*x))/8 + a**2*
sqrt(c)/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*sqrt(c)/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + sqrt(c)/(4*a**2*x**5*sqrt
(1 - 1/(a**2*x**2))), True))/a**4 + 2*c**3*Piecewise((2*a**3*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x) + a*sqrt(c)*sq
rt(a**2*x**2 - 1)/(15*x**3) - sqrt(c)*sqrt(a**2*x**2 - 1)/(5*a*x**5), Abs(a**2*x**2) > 1), (2*I*a**3*sqrt(c)*s
qrt(-a**2*x**2 + 1)/(15*x) + I*a*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x**3) - I*sqrt(c)*sqrt(-a**2*x**2 + 1)/(5*a*
x**5), True))/a**5 - c**3*Piecewise((I*a**5*sqrt(c)*acosh(1/(a*x))/16 - I*a**4*sqrt(c)/(16*x*sqrt(-1 + 1/(a**2
*x**2))) + I*a**2*sqrt(c)/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) + 5*I*sqrt(c)/(24*x**5*sqrt(-1 + 1/(a**2*x**2)))
- I*sqrt(c)/(6*a**2*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-a**5*sqrt(c)*asin(1/(a*x))/16 + a
**4*sqrt(c)/(16*x*sqrt(1 - 1/(a**2*x**2))) - a**2*sqrt(c)/(48*x**3*sqrt(1 - 1/(a**2*x**2))) - 5*sqrt(c)/(24*x*
*5*sqrt(1 - 1/(a**2*x**2))) + sqrt(c)/(6*a**2*x**7*sqrt(1 - 1/(a**2*x**2))), True))/a**6

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