### 3.838 $$\int e^{2 \coth ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^{7/2} \, dx$$

Optimal. Leaf size=372 $-\frac{57 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)^3}+\frac{41 a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{24 (1-a x)^3 (a x+1)^2}+\frac{57 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{80 (1-a x)^3 (a x+1)}+\frac{11 a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{30 (1-a x)^3}-\frac{13 a^2 x^3 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{40 (1-a x)^3}+\frac{a x^2 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{15 (1-a x)^2}+\frac{x (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{6 (1-a x)}+\frac{2 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \sin ^{-1}(a x)}{(1-a x)^{7/2} (a x+1)^{7/2}}+\frac{25 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{16 (1-a x)^{7/2} (a x+1)^{7/2}}$

[Out]

(11*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(30*(1 - a*x)^3) - (57*a^6*(c - c/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)^3*(
1 + a*x)^3) + (41*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(24*(1 - a*x)^3*(1 + a*x)^2) + (57*a^4*(c - c/(a^2*x^2))^(7
/2)*x^5)/(80*(1 - a*x)^3*(1 + a*x)) - (13*a^2*(c - c/(a^2*x^2))^(7/2)*x^3*(1 + a*x))/(40*(1 - a*x)^3) + (a*(c
- c/(a^2*x^2))^(7/2)*x^2*(1 + a*x))/(15*(1 - a*x)^2) + ((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))

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Rubi [A]  time = 0.542188, antiderivative size = 372, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.458, Rules used = {6167, 6159, 6129, 97, 149, 154, 157, 41, 216, 92, 208} $-\frac{57 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)^3}+\frac{41 a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{24 (1-a x)^3 (a x+1)^2}+\frac{57 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{80 (1-a x)^3 (a x+1)}+\frac{11 a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{30 (1-a x)^3}-\frac{13 a^2 x^3 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{40 (1-a x)^3}+\frac{a x^2 (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{15 (1-a x)^2}+\frac{x (a x+1) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{6 (1-a x)}+\frac{2 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \sin ^{-1}(a x)}{(1-a x)^{7/2} (a x+1)^{7/2}}+\frac{25 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{16 (1-a x)^{7/2} (a x+1)^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(11*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(30*(1 - a*x)^3) - (57*a^6*(c - c/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)^3*(
1 + a*x)^3) + (41*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(24*(1 - a*x)^3*(1 + a*x)^2) + (57*a^4*(c - c/(a^2*x^2))^(7
/2)*x^5)/(80*(1 - a*x)^3*(1 + a*x)) - (13*a^2*(c - c/(a^2*x^2))^(7/2)*x^3*(1 + a*x))/(40*(1 - a*x)^3) + (a*(c
- c/(a^2*x^2))^(7/2)*x^2*(1 + a*x))/(15*(1 - a*x)^2) + ((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 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

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]

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 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 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 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 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 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]

Rubi steps

\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \, dx &=-\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)^{5/2} (1+a x)^{9/2}}{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)^{3/2} (1+a x)^{7/2} \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 (1+a x)}{15 (1-a x)^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{\sqrt{1-a x} (1+a x)^{7/2} \left (-39 a^2+33 a^3 x\right )}{x^5} \, dx}{30 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac{13 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^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} \left (132 a^3-93 a^4 x\right )}{x^4 \sqrt{1-a x}} \, dx}{120 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac{11 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}-\frac{13 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^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)^{5/2} \left (513 a^4-411 a^5 x\right )}{x^3 \sqrt{1-a x}} \, dx}{360 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac{11 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}+\frac{57 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}-\frac{13 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^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)^{3/2} \left (1230 a^5-1335 a^6 x\right )}{x^2 \sqrt{1-a x}} \, dx}{720 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac{11 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}+\frac{41 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}+\frac{57 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}-\frac{13 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^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{\sqrt{1+a x} \left (1125 a^6-2565 a^7 x\right )}{x \sqrt{1-a x}} \, dx}{720 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac{11 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}-\frac{57 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{41 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}+\frac{57 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}-\frac{13 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^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{-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{11 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}-\frac{57 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{41 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}+\frac{57 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}-\frac{13 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}+\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{11 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}-\frac{57 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{41 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}+\frac{57 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}-\frac{13 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}+\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{11 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}-\frac{57 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{41 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}+\frac{57 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}-\frac{13 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}+\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*}

Mathematica [A]  time = 0.15939, size = 150, normalized size = 0.4 $\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1} \left (240 a^6 x^6-736 a^5 x^5+105 a^4 x^4+352 a^3 x^3+70 a^2 x^2-96 a x-40\right )+480 a^6 x^6 \log \left (\sqrt{a^2 x^2-1}+a x\right )+375 a^6 x^6 \tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{240 a^6 x^5 \sqrt{a^2 x^2-1}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(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]]))/
(240*a^6*x^5*Sqrt[-1 + a^2*x^2])

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Maple [B]  time = 0.277, size = 795, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{7}{2}}}{a x - 1}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.84289, size = 972, normalized size = 2.61 \begin{align*} \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 ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a^2/x^2)^(7/2),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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Sympy [C]  time = 36.1422, size = 1059, normalized size = 2.85 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a**2/x**2)**(7/2),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*Piecewi
se((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*s
qrt(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)*sqr
t(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)*sq
rt(-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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Giac [A]  time = 68.889, size = 757, normalized size = 2.03 \begin{align*} -\frac{1}{120} \,{\left (\frac{375 \, c^{\frac{7}{2}} \arctan \left (-\frac{\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}}{\sqrt{c}}\right ) \mathrm{sgn}\left (x\right )}{a^{2}} + \frac{240 \, c^{\frac{7}{2}} \log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} - c} \right |}\right ) \mathrm{sgn}\left (x\right )}{a{\left | a \right |}} - \frac{120 \, \sqrt{a^{2} c x^{2} - c} c^{3} \mathrm{sgn}\left (x\right )}{a^{2}} + \frac{105 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{11} c^{4}{\left | a \right |} \mathrm{sgn}\left (x\right ) + 1440 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{10} a c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 595 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{9} c^{5}{\left | a \right |} \mathrm{sgn}\left (x\right ) + 4320 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{8} a c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 150 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{7} c^{6}{\left | a \right |} \mathrm{sgn}\left (x\right ) + 7360 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{6} a c^{\frac{13}{2}} \mathrm{sgn}\left (x\right ) + 150 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{5} c^{7}{\left | a \right |} \mathrm{sgn}\left (x\right ) + 6720 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{4} a c^{\frac{15}{2}} \mathrm{sgn}\left (x\right ) - 595 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{3} c^{8}{\left | a \right |} \mathrm{sgn}\left (x\right ) + 2976 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} a c^{\frac{17}{2}} \mathrm{sgn}\left (x\right ) - 105 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )} c^{9}{\left | a \right |} \mathrm{sgn}\left (x\right ) + 736 \, a c^{\frac{19}{2}} \mathrm{sgn}\left (x\right )}{{\left ({\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{6} a^{2}{\left | a \right |}}\right )}{\left | a \right |} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*(375*c^(7/2)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x)/a^2 + 240*c^(7/2)*log(abs(-s
qrt(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn(x)/(a*abs(a)) - 120*sqrt(a^2*c*x^2 - c)*c^3*sgn(x)/a^2 + (105*(sqrt(a
^2*c)*x - sqrt(a^2*c*x^2 - c))^11*c^4*abs(a)*sgn(x) + 1440*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^10*a*c^(9/2)*
sgn(x) + 595*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^9*c^5*abs(a)*sgn(x) + 4320*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2
- c))^8*a*c^(11/2)*sgn(x) - 150*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^7*c^6*abs(a)*sgn(x) + 7360*(sqrt(a^2*c)*
x - sqrt(a^2*c*x^2 - c))^6*a*c^(13/2)*sgn(x) + 150*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^5*c^7*abs(a)*sgn(x) +
6720*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^4*a*c^(15/2)*sgn(x) - 595*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^3*
c^8*abs(a)*sgn(x) + 2976*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2*a*c^(17/2)*sgn(x) - 105*(sqrt(a^2*c)*x - sqrt
(a^2*c*x^2 - c))*c^9*abs(a)*sgn(x) + 736*a*c^(19/2)*sgn(x))/(((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^6*a
^2*abs(a)))*abs(a)