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

Optimal. Leaf size=393 \[ \frac{1}{9} a^8 c^4 x^9 \left (1-\frac{1}{a x}\right )^{9/2} \left (\frac{1}{a x}+1\right )^{9/2}-\frac{1}{8} a^7 c^4 x^8 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{1}{8} a^6 c^4 x^7 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{9/2}-\frac{5}{48} a^5 c^4 x^6 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{1}{16} a^4 c^4 x^5 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}-\frac{1}{64} a^3 c^4 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{7}{192} a^2 c^4 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{35}{384} a c^4 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{35}{128} c^4 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{35 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{128 a} \]

[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)/16 - (5*a^5*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x)
)^(9/2)*x^6)/48 + (a^6*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(9/2)*x^7)/8 - (a^7*c^4*(1 - 1/(a*x))^(7/2)*(1 +
1/(a*x))^(9/2)*x^8)/8 + (a^8*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(9/2)*x^9)/9 - (35*c^4*ArcTanh[Sqrt[1 - 1/(
a*x)]*Sqrt[1 + 1/(a*x)]])/(128*a)

________________________________________________________________________________________

Rubi [A]  time = 0.321884, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6191, 6195, 94, 92, 208} \[ \frac{1}{9} a^8 c^4 x^9 \left (1-\frac{1}{a x}\right )^{9/2} \left (\frac{1}{a x}+1\right )^{9/2}-\frac{1}{8} a^7 c^4 x^8 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{1}{8} a^6 c^4 x^7 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{9/2}-\frac{5}{48} a^5 c^4 x^6 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{9/2}+\frac{1}{16} a^4 c^4 x^5 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}-\frac{1}{64} a^3 c^4 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{7}{192} a^2 c^4 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{35}{384} a c^4 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{35}{128} c^4 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{35 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{128 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^4/E^ArcCoth[a*x],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)/16 - (5*a^5*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x)
)^(9/2)*x^6)/48 + (a^6*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(9/2)*x^7)/8 - (a^7*c^4*(1 - 1/(a*x))^(7/2)*(1 +
1/(a*x))^(9/2)*x^8)/8 + (a^8*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(9/2)*x^9)/9 - (35*c^4*ArcTanh[Sqrt[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 )^{9/2} \left (1+\frac{x}{a}\right )^{7/2}}{x^{10}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{9/2} x^9+\left (a^7 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{7/2} \left (1+\frac{x}{a}\right )^{7/2}}{x^9} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{8} a^7 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{9/2} x^9-\frac{1}{8} \left (7 a^6 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{7/2}}{x^8} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{8} a^6 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{8} a^7 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{9/2} x^9+\frac{1}{8} \left (5 a^5 c^4\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{7/2}}{x^7} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6+\frac{1}{8} a^6 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{8} a^7 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{9/2} x^9-\frac{1}{16} \left (5 a^4 c^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}}{x^6} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{16} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6+\frac{1}{8} a^6 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{8} a^7 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{9/2} 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}{16} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6+\frac{1}{8} a^6 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{8} a^7 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{9/2} 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}{16} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6+\frac{1}{8} a^6 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{8} a^7 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{9/2} 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}{16} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6+\frac{1}{8} a^6 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{8} a^7 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{9/2} 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}{16} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6+\frac{1}{8} a^6 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{8} a^7 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{9/2} 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}{16} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6+\frac{1}{8} a^6 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{8} a^7 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{9/2} 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}{16} a^4 c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{5}{48} a^5 c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{9/2} x^6+\frac{1}{8} a^6 c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{9/2} x^7-\frac{1}{8} a^7 c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{9/2} x^8+\frac{1}{9} a^8 c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{9/2} 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 = 0.199277, 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[(c - a^2*c*x^2)^4/E^ArcCoth[a*x],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 = 0.154, size = 279, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax+1 \right ){c}^{4}}{1152\,a}\sqrt{{\frac{ax-1}{ax+1}}} \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{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1152*((a*x-1)/(a*x+1))^(1/2)*(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)+31
5*(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^2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.12695, 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((-a^2*c*x^2+c)^4*((a*x-1)/(a*x+1))^(1/2),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*(
315*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*sq
rt((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 = 1.65731, size = 401, normalized size = 1.02 \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} - 16 \, a^{8} c^{4} x^{8} - 656 \, a^{7} c^{4} x^{7} + 88 \, a^{6} c^{4} x^{6} + 1368 \, a^{5} c^{4} x^{5} - 210 \, a^{4} c^{4} x^{4} - 1490 \, a^{3} c^{4} x^{3} + 325 \, a^{2} c^{4} x^{2} + 965 \, 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((-a^2*c*x^2+c)^4*((a*x-1)/(a*x+1))^(1/2),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 - 16*a^8*c^4*x^8 - 656*a^7*c^4*x^7 + 88*a^6*c^4*x^6 + 1368*a^5*c^4*x^5 - 210*a^4*c^4*x^4 - 1490*a^3*c^4*
x^3 + 325*a^2*c^4*x^2 + 965*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((-a**2*c*x**2+c)**4*((a*x-1)/(a*x+1))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.18317, size = 265, normalized size = 0.67 \begin{align*} \frac{35 \, c^{4} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{128 \,{\left | a \right |}} + \frac{1}{1152} \, \sqrt{a^{2} x^{2} - 1}{\left (\frac{128 \, c^{4} \mathrm{sgn}\left (a x + 1\right )}{a} +{\left (837 \, c^{4} \mathrm{sgn}\left (a x + 1\right ) - 2 \,{\left (256 \, a c^{4} \mathrm{sgn}\left (a x + 1\right ) +{\left (489 \, a^{2} c^{4} \mathrm{sgn}\left (a x + 1\right ) - 4 \,{\left (96 \, a^{3} c^{4} \mathrm{sgn}\left (a x + 1\right ) +{\left (75 \, a^{4} c^{4} \mathrm{sgn}\left (a x + 1\right ) - 2 \,{\left (32 \, a^{5} c^{4} \mathrm{sgn}\left (a x + 1\right ) -{\left (8 \, a^{7} c^{4} x \mathrm{sgn}\left (a x + 1\right ) - 9 \, a^{6} c^{4} \mathrm{sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/128*c^4*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + 1/1152*sqrt(a^2*x^2 - 1)*(128*c^4*sgn
(a*x + 1)/a + (837*c^4*sgn(a*x + 1) - 2*(256*a*c^4*sgn(a*x + 1) + (489*a^2*c^4*sgn(a*x + 1) - 4*(96*a^3*c^4*sg
n(a*x + 1) + (75*a^4*c^4*sgn(a*x + 1) - 2*(32*a^5*c^4*sgn(a*x + 1) - (8*a^7*c^4*x*sgn(a*x + 1) - 9*a^6*c^4*sgn
(a*x + 1))*x)*x)*x)*x)*x)*x)*x)