### 3.772 $$\int e^{\coth ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^4 \, dx$$

Optimal. Leaf size=342 $\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{9/2}}{7 a}+\frac{47 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{9/2}}{42 a}+c^4 x \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{61 c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}}{70 a}-\frac{131 c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}}{280 a}-\frac{91 c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}}{120 a}-\frac{67 c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}{48 a}-\frac{51 c^4 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{16 a}+\frac{35 c^4 \csc ^{-1}(a x)}{16 a}+\frac{c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a}$

[Out]

(-51*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(16*a) - (67*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(48*a) -
(91*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(120*a) - (131*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))/(280
*a) + (61*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2))/(70*a) + (47*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2))
/(42*a) + (8*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(9/2))/(7*a) + c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(9/2)*
x + (35*c^4*ArcCsc[a*x])/(16*a) + (c^4*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a

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Rubi [A]  time = 0.24551, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {6194, 97, 154, 157, 41, 216, 92, 208} $\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{9/2}}{7 a}+\frac{47 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{9/2}}{42 a}+c^4 x \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{61 c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}}{70 a}-\frac{131 c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}}{280 a}-\frac{91 c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}}{120 a}-\frac{67 c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}{48 a}-\frac{51 c^4 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{16 a}+\frac{35 c^4 \csc ^{-1}(a x)}{16 a}+\frac{c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-51*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(16*a) - (67*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(48*a) -
(91*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(120*a) - (131*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))/(280
*a) + (61*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2))/(70*a) + (47*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2))
/(42*a) + (8*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(9/2))/(7*a) + c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(9/2)*
x + (35*c^4*ArcCsc[a*x])/(16*a) + (c^4*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a

Rule 6194

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

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 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^{\coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^4 \, dx &=-\left (c^4 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{7/2} \left (1+\frac{x}{a}\right )^{9/2}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x-c^4 \operatorname{Subst}\left (\int \frac{\left (\frac{1}{a}-\frac{8 x}{a^2}\right ) \left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{7/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x-\frac{1}{7} \left (a c^4\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{7}{a^2}-\frac{47 x}{a^3}\right ) \left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{7/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{47 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}{42 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x-\frac{1}{42} \left (a^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{42}{a^3}-\frac{183 x}{a^4}\right ) \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{61 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{70 a}+\frac{47 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}{42 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x-\frac{1}{210} \left (a^3 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{210}{a^4}-\frac{393 x}{a^5}\right ) \left (1+\frac{x}{a}\right )^{7/2}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{131 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{280 a}+\frac{61 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{70 a}+\frac{47 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}{42 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x+\frac{1}{840} \left (a^4 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{840}{a^5}+\frac{1911 x}{a^6}\right ) \left (1+\frac{x}{a}\right )^{5/2}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{91 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{120 a}-\frac{131 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{280 a}+\frac{61 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{70 a}+\frac{47 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}{42 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x-\frac{\left (a^5 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{2520}{a^6}-\frac{7035 x}{a^7}\right ) \left (1+\frac{x}{a}\right )^{3/2}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2520}\\ &=-\frac{67 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{48 a}-\frac{91 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{120 a}-\frac{131 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{280 a}+\frac{61 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{70 a}+\frac{47 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}{42 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x+\frac{\left (a^6 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{5040}{a^7}+\frac{16065 x}{a^8}\right ) \sqrt{1+\frac{x}{a}}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{5040}\\ &=-\frac{51 c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{16 a}-\frac{67 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{48 a}-\frac{91 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{120 a}-\frac{131 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{280 a}+\frac{61 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{70 a}+\frac{47 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}{42 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x-\frac{\left (a^7 c^4\right ) \operatorname{Subst}\left (\int \frac{\frac{5040}{a^8}-\frac{11025 x}{a^9}}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{5040}\\ &=-\frac{51 c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{16 a}-\frac{67 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{48 a}-\frac{91 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{120 a}-\frac{131 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{280 a}+\frac{61 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{70 a}+\frac{47 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}{42 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x+\frac{\left (35 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 a^2}-\frac{c^4 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{51 c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{16 a}-\frac{67 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{48 a}-\frac{91 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{120 a}-\frac{131 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{280 a}+\frac{61 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{70 a}+\frac{47 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}{42 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x+\frac{c^4 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a}-\frac{x^2}{a}} \, dx,x,\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a^2}+\frac{\left (35 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{16 a^2}\\ &=-\frac{51 c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{16 a}-\frac{67 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{48 a}-\frac{91 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{120 a}-\frac{131 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}{280 a}+\frac{61 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2}}{70 a}+\frac{47 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2}}{42 a}+\frac{8 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x+\frac{35 c^4 \csc ^{-1}(a x)}{16 a}+\frac{c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.274908, size = 120, normalized size = 0.35 $\frac{c^4 \left (\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left (1680 a^7 x^7-2816 a^6 x^6+3045 a^5 x^5+1952 a^4 x^4-1330 a^3 x^3-1056 a^2 x^2+280 a x+240\right )}{x^6}+1680 a^6 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )+3675 a^6 \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{1680 a^7}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^4*((Sqrt[1 - 1/(a^2*x^2)]*(240 + 280*a*x - 1056*a^2*x^2 - 1330*a^3*x^3 + 1952*a^4*x^4 + 3045*a^5*x^5 - 2816
*a^6*x^6 + 1680*a^7*x^7))/x^6 + 3675*a^6*ArcSin[1/(a*x)] + 1680*a^6*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(1680
*a^7)

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Maple [A]  time = 0.165, size = 320, normalized size = 0.9 \begin{align*}{\frac{ \left ( ax-1 \right ){c}^{4}}{1680\,{a}^{8}{x}^{7}} \left ( -1680\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{8}{a}^{8}+1680\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{6}{a}^{6}+3675\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{7}{a}^{7}+3675\,{a}^{7}{x}^{7}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +1680\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{7}{a}^{8}-1995\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{5}{a}^{5}-1136\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}{x}^{4}{a}^{4}+1050\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+816\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-280\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-240\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a^2/x^2)^4,x)

[Out]

1/1680*(a*x-1)*c^4*(-1680*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^8*a^8+1680*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^6*a^6+367
5*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^7*a^7+3675*a^7*x^7*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))+1680*ln((a^2*x+(a
^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^7*a^8-1995*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^5*a^5-1136*(a^2)^(1/2)*
(a^2*x^2-1)^(3/2)*x^4*a^4+1050*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^3*a^3+816*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^2*a^2
-280*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*x*a-240*(a^2*x^2-1)^(3/2)*(a^2)^(1/2))/((a*x-1)/(a*x+1))^(1/2)/((a*x-1)*(a*
x+1))^(1/2)/a^8/x^7/(a^2)^(1/2)

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Maxima [A]  time = 1.60309, size = 513, normalized size = 1.5 \begin{align*} -\frac{1}{840} \,{\left (\frac{3675 \, c^{4} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} - \frac{840 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac{840 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{5355 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{15}{2}} + 31465 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{13}{2}} + 72051 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{11}{2}} + 71801 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} + 4569 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 17619 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 10185 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 1995 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{6 \,{\left (a x - 1\right )} a^{2}}{a x + 1} + \frac{14 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{14 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac{14 \,{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac{14 \,{\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} - \frac{6 \,{\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac{{\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + a^{2}}\right )} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a^2/x^2)^4,x, algorithm="maxima")

[Out]

-1/840*(3675*c^4*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 - 840*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 + 840*
c^4*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - (5355*c^4*((a*x - 1)/(a*x + 1))^(15/2) + 31465*c^4*((a*x - 1)/(a*
x + 1))^(13/2) + 72051*c^4*((a*x - 1)/(a*x + 1))^(11/2) + 71801*c^4*((a*x - 1)/(a*x + 1))^(9/2) + 4569*c^4*((a
*x - 1)/(a*x + 1))^(7/2) + 17619*c^4*((a*x - 1)/(a*x + 1))^(5/2) + 10185*c^4*((a*x - 1)/(a*x + 1))^(3/2) + 199
5*c^4*sqrt((a*x - 1)/(a*x + 1)))/(6*(a*x - 1)*a^2/(a*x + 1) + 14*(a*x - 1)^2*a^2/(a*x + 1)^2 + 14*(a*x - 1)^3*
a^2/(a*x + 1)^3 - 14*(a*x - 1)^5*a^2/(a*x + 1)^5 - 14*(a*x - 1)^6*a^2/(a*x + 1)^6 - 6*(a*x - 1)^7*a^2/(a*x + 1
)^7 - (a*x - 1)^8*a^2/(a*x + 1)^8 + a^2))*a

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Fricas [A]  time = 1.40842, size = 489, normalized size = 1.43 \begin{align*} -\frac{7350 \, a^{7} c^{4} x^{7} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) + 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (1680 \, a^{8} c^{4} x^{8} - 1136 \, a^{7} c^{4} x^{7} + 229 \, a^{6} c^{4} x^{6} + 4997 \, a^{5} c^{4} x^{5} + 622 \, a^{4} c^{4} x^{4} - 2386 \, a^{3} c^{4} x^{3} - 776 \, a^{2} c^{4} x^{2} + 520 \, a c^{4} x + 240 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{1680 \, a^{8} x^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a^2/x^2)^4,x, algorithm="fricas")

[Out]

-1/1680*(7350*a^7*c^4*x^7*arctan(sqrt((a*x - 1)/(a*x + 1))) - 1680*a^7*c^4*x^7*log(sqrt((a*x - 1)/(a*x + 1)) +
1) + 1680*a^7*c^4*x^7*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (1680*a^8*c^4*x^8 - 1136*a^7*c^4*x^7 + 229*a^6*c^4
*x^6 + 4997*a^5*c^4*x^5 + 622*a^4*c^4*x^4 - 2386*a^3*c^4*x^3 - 776*a^2*c^4*x^2 + 520*a*c^4*x + 240*c^4)*sqrt((
a*x - 1)/(a*x + 1)))/(a^8*x^7)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.15955, size = 493, normalized size = 1.44 \begin{align*} -\frac{1}{840} \,{\left (\frac{3675 \, c^{4} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} - \frac{840 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac{840 \, c^{4} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac{1680 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}} + \frac{\frac{1260 \,{\left (a x - 1\right )} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + \frac{18921 \,{\left (a x - 1\right )}^{2} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + \frac{73152 \,{\left (a x - 1\right )}^{3} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + \frac{60151 \,{\left (a x - 1\right )}^{4} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{4}} + \frac{23380 \,{\left (a x - 1\right )}^{5} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{5}} + \frac{3675 \,{\left (a x - 1\right )}^{6} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{6}} - 315 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{7}}\right )} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a^2/x^2)^4,x, algorithm="giac")

[Out]

-1/840*(3675*c^4*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 - 840*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 + 840*
c^4*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/a^2 + 1680*c^4*sqrt((a*x - 1)/(a*x + 1))/(a^2*((a*x - 1)/(a*x + 1)
- 1)) + (1260*(a*x - 1)*c^4*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1) + 18921*(a*x - 1)^2*c^4*sqrt((a*x - 1)/(a*x +
1))/(a*x + 1)^2 + 73152*(a*x - 1)^3*c^4*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^3 + 60151*(a*x - 1)^4*c^4*sqrt((a
*x - 1)/(a*x + 1))/(a*x + 1)^4 + 23380*(a*x - 1)^5*c^4*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^5 + 3675*(a*x - 1)^
6*c^4*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^6 - 315*c^4*sqrt((a*x - 1)/(a*x + 1)))/(a^2*((a*x - 1)/(a*x + 1) + 1
)^7))*a