3.556 \(\int e^{\coth ^{-1}(a x)} (c-a^2 c x^2)^4 \, dx\)

Optimal. Leaf size=393 \[ \frac{1}{9} a^8 c^4 x^9 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{11/2}-\frac{7}{72} a^7 c^4 x^8 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{11/2}+\frac{5}{72} a^6 c^4 x^7 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{11/2}-\frac{5}{144} a^5 c^4 x^6 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{11/2}+\frac{1}{144} a^4 c^4 x^5 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{1}{64} a^3 c^4 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{7}{192} a^2 c^4 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}+\frac{35}{384} a c^4 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}+\frac{35}{128} c^4 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{35 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{128 a} \]

[Out]

(35*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/128 + (35*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2)/384
+ (7*a^2*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3)/192 + (a^3*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x
^4)/64 + (a^4*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2)*x^5)/144 - (5*a^5*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^
(11/2)*x^6)/144 + (5*a^6*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(11/2)*x^7)/72 - (7*a^7*c^4*(1 - 1/(a*x))^(5/2)
*(1 + 1/(a*x))^(11/2)*x^8)/72 + (a^8*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(11/2)*x^9)/9 + (35*c^4*ArcTanh[Sqr
t[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(128*a)

________________________________________________________________________________________

Rubi [A]  time = 0.33667, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6191, 6195, 94, 92, 208} \[ \frac{1}{9} a^8 c^4 x^9 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{11/2}-\frac{7}{72} a^7 c^4 x^8 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{11/2}+\frac{5}{72} a^6 c^4 x^7 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{11/2}-\frac{5}{144} a^5 c^4 x^6 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{11/2}+\frac{1}{144} a^4 c^4 x^5 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{1}{64} a^3 c^4 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}+\frac{7}{192} a^2 c^4 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}+\frac{35}{384} a c^4 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}+\frac{35}{128} c^4 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{35 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{128 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/128 + (35*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2)/384
+ (7*a^2*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3)/192 + (a^3*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x
^4)/64 + (a^4*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2)*x^5)/144 - (5*a^5*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^
(11/2)*x^6)/144 + (5*a^6*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(11/2)*x^7)/72 - (7*a^7*c^4*(1 - 1/(a*x))^(5/2)
*(1 + 1/(a*x))^(11/2)*x^8)/72 + (a^8*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(11/2)*x^9)/9 + (35*c^4*ArcTanh[Sqr
t[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(128*a)

Rule 6191

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

Rule 6195

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

Rule 94

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

Rule 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-a^2 c x^2\right )^4 \, dx &=\left (a^8 c^4\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^4 x^8 \, dx\\ &=-\left (\left (a^8 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{7/2} \left (1+\frac{x}{a}\right )^{9/2}}{x^{10}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{11/2} x^9+\frac{1}{9} \left (7 a^7 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{9/2}}{x^9} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{7}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{11/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{11/2} x^9-\frac{1}{72} \left (35 a^6 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{9/2}}{x^8} \, dx,x,\frac{1}{x}\right )\\ &=\frac{5}{72} a^6 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{11/2} x^7-\frac{7}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{11/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{11/2} x^9+\frac{1}{24} \left (5 a^5 c^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{9/2}}{x^7} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5}{144} a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6+\frac{5}{72} a^6 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{11/2} x^7-\frac{7}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{11/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{11/2} x^9-\frac{1}{144} \left (5 a^4 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{9/2}}{x^6 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{144} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{144} a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6+\frac{5}{72} a^6 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{11/2} x^7-\frac{7}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{11/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{11/2} x^9-\frac{1}{16} \left (a^3 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{7/2}}{x^5 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{64} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{1}{144} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{144} a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6+\frac{5}{72} a^6 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{11/2} x^7-\frac{7}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{11/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{11/2} x^9-\frac{1}{64} \left (7 a^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{5/2}}{x^4 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{7}{192} a^2 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{64} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{1}{144} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{144} a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6+\frac{5}{72} a^6 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{11/2} x^7-\frac{7}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{11/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{11/2} x^9-\frac{1}{192} \left (35 a c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^3 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{35}{384} a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{7}{192} a^2 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{64} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{1}{144} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{144} a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6+\frac{5}{72} a^6 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{11/2} x^7-\frac{7}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{11/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{11/2} x^9-\frac{1}{128} \left (35 c^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^2 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{35}{128} c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x+\frac{35}{384} a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{7}{192} a^2 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{64} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{1}{144} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{144} a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6+\frac{5}{72} a^6 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{11/2} x^7-\frac{7}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{11/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{11/2} x^9-\frac{\left (35 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{128 a}\\ &=\frac{35}{128} c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x+\frac{35}{384} a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{7}{192} a^2 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{64} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{1}{144} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{144} a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6+\frac{5}{72} a^6 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{11/2} x^7-\frac{7}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{11/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{11/2} x^9+\frac{\left (35 c^4\right ) \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 )}{128 a^2}\\ &=\frac{35}{128} c^4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x+\frac{35}{384} a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{7}{192} a^2 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3+\frac{1}{64} a^3 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{1}{144} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{144} a^5 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{11/2} x^6+\frac{5}{72} a^6 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{11/2} x^7-\frac{7}{72} a^7 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{11/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{11/2} x^9+\frac{35 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{128 a}\\ \end{align*}

Mathematica [A]  time = 0.179308, size = 111, normalized size = 0.28 \[ \frac{c^4 \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (128 a^8 x^8+144 a^7 x^7-512 a^6 x^6-600 a^5 x^5+768 a^4 x^4+978 a^3 x^3-512 a^2 x^2-837 a x+128\right )+315 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{1152 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^4*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(128 - 837*a*x - 512*a^2*x^2 + 978*a^3*x^3 + 768*a^4*x^4 - 600*a^5*x^5 - 512*a
^6*x^6 + 144*a^7*x^7 + 128*a^8*x^8) + 315*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(1152*a)

________________________________________________________________________________________

Maple [A]  time = 0.149, size = 279, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax-1 \right ){c}^{4}}{1152\,a} \left ( 128\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{6}{a}^{6}+144\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{5}{a}^{5}-384\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}{x}^{4}{a}^{4}-456\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+384\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+522\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+256\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}-315\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-384\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}+315\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a \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)*(-a^2*c*x^2+c)^4,x)

[Out]

1/1152*(a*x-1)*c^4/a*(128*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^6*a^6+144*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^5*a^5-384*
(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*x^4*a^4-456*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^3*a^3+384*(a^2*x^2-1)^(3/2)*(a^2)^(1
/2)*x^2*a^2+522*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*x*a+256*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)-315*(a^2)^(1/2)*(a^2*x^2-1
)^(1/2)*x*a-384*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)+315*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*
a)/((a*x-1)/(a*x+1))^(1/2)/((a*x-1)*(a*x+1))^(1/2)/(a^2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.06928, size = 560, normalized size = 1.42 \begin{align*} \frac{1}{1152} \,{\left (\frac{315 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{315 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{2 \,{\left (315 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{17}{2}} - 2730 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{15}{2}} + 10458 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{13}{2}} - 23202 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{11}{2}} + 32768 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} + 23202 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 10458 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 2730 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 315 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{9 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{36 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{84 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac{126 \,{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac{126 \,{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac{84 \,{\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac{36 \,{\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac{9 \,{\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + \frac{{\left (a x - 1\right )}^{9} a^{2}}{{\left (a x + 1\right )}^{9}} - 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)*(-a^2*c*x^2+c)^4,x, algorithm="maxima")

[Out]

1/1152*(315*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 315*c^4*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 2*(3
15*c^4*((a*x - 1)/(a*x + 1))^(17/2) - 2730*c^4*((a*x - 1)/(a*x + 1))^(15/2) + 10458*c^4*((a*x - 1)/(a*x + 1))^
(13/2) - 23202*c^4*((a*x - 1)/(a*x + 1))^(11/2) + 32768*c^4*((a*x - 1)/(a*x + 1))^(9/2) + 23202*c^4*((a*x - 1)
/(a*x + 1))^(7/2) - 10458*c^4*((a*x - 1)/(a*x + 1))^(5/2) + 2730*c^4*((a*x - 1)/(a*x + 1))^(3/2) - 315*c^4*sqr
t((a*x - 1)/(a*x + 1)))/(9*(a*x - 1)*a^2/(a*x + 1) - 36*(a*x - 1)^2*a^2/(a*x + 1)^2 + 84*(a*x - 1)^3*a^2/(a*x
+ 1)^3 - 126*(a*x - 1)^4*a^2/(a*x + 1)^4 + 126*(a*x - 1)^5*a^2/(a*x + 1)^5 - 84*(a*x - 1)^6*a^2/(a*x + 1)^6 +
36*(a*x - 1)^7*a^2/(a*x + 1)^7 - 9*(a*x - 1)^8*a^2/(a*x + 1)^8 + (a*x - 1)^9*a^2/(a*x + 1)^9 - a^2))*a

________________________________________________________________________________________

Fricas [A]  time = 1.39674, size = 404, normalized size = 1.03 \begin{align*} \frac{315 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 315 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (128 \, a^{9} c^{4} x^{9} + 272 \, a^{8} c^{4} x^{8} - 368 \, a^{7} c^{4} x^{7} - 1112 \, a^{6} c^{4} x^{6} + 168 \, a^{5} c^{4} x^{5} + 1746 \, a^{4} c^{4} x^{4} + 466 \, a^{3} c^{4} x^{3} - 1349 \, a^{2} c^{4} x^{2} - 709 \, a c^{4} x + 128 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{1152 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1152*(315*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 315*c^4*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (128*a^9*c^4
*x^9 + 272*a^8*c^4*x^8 - 368*a^7*c^4*x^7 - 1112*a^6*c^4*x^6 + 168*a^5*c^4*x^5 + 1746*a^4*c^4*x^4 + 466*a^3*c^4
*x^3 - 1349*a^2*c^4*x^2 - 709*a*c^4*x + 128*c^4)*sqrt((a*x - 1)/(a*x + 1)))/a

________________________________________________________________________________________

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)*(-a**2*c*x**2+c)**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.17999, size = 500, normalized size = 1.27 \begin{align*} \frac{1}{1152} \,{\left (\frac{315 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{315 \, c^{4} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} - \frac{2 \,{\left (\frac{2730 \,{\left (a x - 1\right )} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - \frac{10458 \,{\left (a x - 1\right )}^{2} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + \frac{23202 \,{\left (a x - 1\right )}^{3} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + \frac{32768 \,{\left (a x - 1\right )}^{4} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{4}} - \frac{23202 \,{\left (a x - 1\right )}^{5} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{5}} + \frac{10458 \,{\left (a x - 1\right )}^{6} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{6}} - \frac{2730 \,{\left (a x - 1\right )}^{7} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{7}} + \frac{315 \,{\left (a x - 1\right )}^{8} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{8}} - 315 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{9}}\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)*(-a^2*c*x^2+c)^4,x, algorithm="giac")

[Out]

1/1152*(315*c^4*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 315*c^4*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/a^2 -
2*(2730*(a*x - 1)*c^4*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1) - 10458*(a*x - 1)^2*c^4*sqrt((a*x - 1)/(a*x + 1))/(
a*x + 1)^2 + 23202*(a*x - 1)^3*c^4*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^3 + 32768*(a*x - 1)^4*c^4*sqrt((a*x - 1
)/(a*x + 1))/(a*x + 1)^4 - 23202*(a*x - 1)^5*c^4*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^5 + 10458*(a*x - 1)^6*c^4
*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^6 - 2730*(a*x - 1)^7*c^4*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^7 + 315*(a*x
- 1)^8*c^4*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^8 - 315*c^4*sqrt((a*x - 1)/(a*x + 1)))/(a^2*((a*x - 1)/(a*x +
1) - 1)^9))*a