\(\int e^{-\coth ^{-1}(a x)} (c-a^2 c x^2)^2 \, dx\) [592]

Optimal result

Integrand size = 22, antiderivative size = 233 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {3}{8} c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {1}{8} a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {1}{4} a^2 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{4} a^3 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5-\frac {3 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{8 a} \]

[Out]

-1/4*a^3*c^2*(1-1/a/x)^(3/2)*(1+1/a/x)^(5/2)*x^4+1/5*a^4*c^2*(1-1/a/x)^(5/2)*(1+1/a/x)^(5/2)*x^5-3/8*c^2*arcta
nh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a-1/8*a*c^2*(1+1/a/x)^(3/2)*x^2*(1-1/a/x)^(1/2)+1/4*a^2*c^2*(1+1/a/x)^(5/2
)*x^3*(1-1/a/x)^(1/2)-3/8*c^2*x*(1-1/a/x)^(1/2)*(1+1/a/x)^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6326, 6330, 96, 94, 214} \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {1}{5} a^4 c^2 x^5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {1}{4} a^3 c^2 x^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {1}{4} a^2 c^2 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {3 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{8 a}-\frac {1}{8} a c^2 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {3}{8} c^2 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} \]

[In]

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

[Out]

(-3*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/8 - (a*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2)/8 + (a^2*
c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3)/4 - (a^3*c^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)*x^4)/4 + (
a^4*c^2*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)*x^5)/5 - (3*c^2*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/
(8*a)

Rule 94

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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 6326

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 6330

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]

Rubi steps \begin{align*} \text {integral}& = \left (a^4 c^2\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^2 x^4 \, dx \\ & = -\left (\left (a^4 c^2\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{3/2}}{x^6} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\left (a^3 c^2\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{3/2}}{x^5} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {1}{4} a^3 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5-\frac {1}{4} \left (3 a^2 c^2\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}}{x^4} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{4} a^2 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{4} a^3 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {1}{4} \left (a c^2\right ) \text {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 {1}{8} a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {1}{4} a^2 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{4} a^3 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {1}{8} \left (3 c^2\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^2 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {3}{8} c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {1}{8} a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {1}{4} a^2 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{4} a^3 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {\left (3 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 a} \\ & = -\frac {3}{8} c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {1}{8} a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {1}{4} a^2 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{4} a^3 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5-\frac {\left (3 c^2\right ) \text {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 )}{8 a^2} \\ & = -\frac {3}{8} c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {1}{8} a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {1}{4} a^2 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{4} a^3 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5-\frac {3 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{8 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.34 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {c^2 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (8+25 a x-16 a^2 x^2-10 a^3 x^3+8 a^4 x^4\right )-15 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{40 a} \]

[In]

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

[Out]

(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(8 + 25*a*x - 16*a^2*x^2 - 10*a^3*x^3 + 8*a^4*x^4) - 15*Log[(1 + Sqrt[1 - 1/(a
^2*x^2)])*x]))/(40*a)

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.55

method	result	size
risch	\(\frac {\left (8 a^{4} x^{4}-10 a^{3} x^{3}-16 a^{2} x^{2}+25 a x +8\right ) \left (a x +1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}}{40 a}-\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{8 \sqrt {a^{2}}\, \left (a x -1\right )}\)	\(128\)
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{2} \left (24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-30 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +16 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+45 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -40 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-45 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{120 a \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\)	\(183\)

[In]

int((-a^2*c*x^2+c)^2*((a*x-1)/(a*x+1))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/40*(8*a^4*x^4-10*a^3*x^3-16*a^2*x^2+25*a*x+8)*(a*x+1)/a*c^2*((a*x-1)/(a*x+1))^(1/2)-3/8*ln(a^2*x/(a^2)^(1/2)
+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)*c^2*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x-1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.54 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (8 \, a^{5} c^{2} x^{5} - 2 \, a^{4} c^{2} x^{4} - 26 \, a^{3} c^{2} x^{3} + 9 \, a^{2} c^{2} x^{2} + 33 \, a c^{2} x + 8 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{40 \, a} \]

[In]

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

[Out]

-1/40*(15*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*c^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (8*a^5*c^2*x^5
- 2*a^4*c^2*x^4 - 26*a^3*c^2*x^3 + 9*a^2*c^2*x^2 + 33*a*c^2*x + 8*c^2)*sqrt((a*x - 1)/(a*x + 1)))/a

Sympy [F]

\[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=c^{2} \left (\int \left (- 2 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx + \int a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.11 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {1}{40} \, a {\left (\frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (15 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - 70 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 128 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 70 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 15 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {5 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {10 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {10 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - a^{2}}\right )} \]

[In]

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

[Out]

-1/40*a*(15*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 15*c^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - 2*(15
*c^2*((a*x - 1)/(a*x + 1))^(9/2) - 70*c^2*((a*x - 1)/(a*x + 1))^(7/2) - 128*c^2*((a*x - 1)/(a*x + 1))^(5/2) +
70*c^2*((a*x - 1)/(a*x + 1))^(3/2) - 15*c^2*sqrt((a*x - 1)/(a*x + 1)))/(5*(a*x - 1)*a^2/(a*x + 1) - 10*(a*x -
1)^2*a^2/(a*x + 1)^2 + 10*(a*x - 1)^3*a^2/(a*x + 1)^3 - 5*(a*x - 1)^4*a^2/(a*x + 1)^4 + (a*x - 1)^5*a^2/(a*x +
1)^5 - a^2))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.54 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {3 \, c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{8 \, {\left | a \right |}} + \frac {1}{40} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (25 \, c^{2} \mathrm {sgn}\left (a x + 1\right ) - 2 \, {\left (8 \, a c^{2} \mathrm {sgn}\left (a x + 1\right ) - {\left (4 \, a^{3} c^{2} x \mathrm {sgn}\left (a x + 1\right ) - 5 \, a^{2} c^{2} \mathrm {sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x + \frac {8 \, c^{2} \mathrm {sgn}\left (a x + 1\right )}{a}\right )} \]

[In]

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

[Out]

3/8*c^2*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + 1/40*sqrt(a^2*x^2 - 1)*((25*c^2*sgn(a*x
+ 1) - 2*(8*a*c^2*sgn(a*x + 1) - (4*a^3*c^2*x*sgn(a*x + 1) - 5*a^2*c^2*sgn(a*x + 1))*x)*x)*x + 8*c^2*sgn(a*x +
1)/a)

Mupad [B] (verification not implemented)

Time = 3.97 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.92 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {\frac {3\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {7\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{2}+\frac {32\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{5}+\frac {7\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{2}-\frac {3\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{4}}{a-\frac {5\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {10\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {10\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {5\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}}-\frac {3\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a} \]

[In]

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

[Out]

((3*c^2*((a*x - 1)/(a*x + 1))^(1/2))/4 - (7*c^2*((a*x - 1)/(a*x + 1))^(3/2))/2 + (32*c^2*((a*x - 1)/(a*x + 1))
^(5/2))/5 + (7*c^2*((a*x - 1)/(a*x + 1))^(7/2))/2 - (3*c^2*((a*x - 1)/(a*x + 1))^(9/2))/4)/(a - (5*a*(a*x - 1)
)/(a*x + 1) + (10*a*(a*x - 1)^2)/(a*x + 1)^2 - (10*a*(a*x - 1)^3)/(a*x + 1)^3 + (5*a*(a*x - 1)^4)/(a*x + 1)^4
- (a*(a*x - 1)^5)/(a*x + 1)^5) - (3*c^2*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(4*a)