3.791 \(\int e^{3 \coth ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^2 \, dx\)

Optimal. Leaf size=195 \[ c^2 x \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}}{3 a}-\frac{11 c^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}{6 a}-\frac{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{2 a}-\frac{c^2 \csc ^{-1}(a x)}{2 a}+\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]

[Out]

(-5*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(2*a) - (11*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(6*a) - (4
*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(3*a) + c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x - (c^2*ArcCsc[
a*x])/(2*a) + (3*c^2*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a

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Rubi [A]  time = 0.134631, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6194, 97, 154, 157, 41, 216, 92, 208} \[ c^2 x \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}}{3 a}-\frac{11 c^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}{6 a}-\frac{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{2 a}-\frac{c^2 \csc ^{-1}(a x)}{2 a}+\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^2,x]

[Out]

(-5*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(2*a) - (11*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(6*a) - (4
*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(3*a) + c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x - (c^2*ArcCsc[
a*x])/(2*a) + (3*c^2*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^{3 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^2 \, dx &=-\left (c^2 \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x-c^2 \operatorname{Subst}\left (\int \frac{\left (\frac{3}{a}-\frac{4 x}{a^2}\right ) \left (1+\frac{x}{a}\right )^{5/2}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{3 a}+c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x+\frac{1}{3} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{9}{a^2}+\frac{11 x}{a^3}\right ) \left (1+\frac{x}{a}\right )^{3/2}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{11 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{6 a}-\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{3 a}+c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x-\frac{1}{6} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{18}{a^3}-\frac{15 x}{a^4}\right ) \sqrt{1+\frac{x}{a}}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{2 a}-\frac{11 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{6 a}-\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{3 a}+c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x+\frac{1}{6} \left (a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{-\frac{18}{a^4}-\frac{3 x}{a^5}}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{2 a}-\frac{11 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{6 a}-\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{3 a}+c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}-\frac{\left (3 c^2\right ) \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{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{2 a}-\frac{11 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{6 a}-\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{3 a}+c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}+\frac{\left (3 c^2\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 )}{a^2}\\ &=-\frac{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{2 a}-\frac{11 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}{6 a}-\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}{3 a}+c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x-\frac{c^2 \csc ^{-1}(a x)}{2 a}+\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.136157, size = 94, normalized size = 0.48 \[ \frac{c^2 \left (\sqrt{1-\frac{1}{a^2 x^2}} \left (6 a^3 x^3-16 a^2 x^2-9 a x-2\right )+18 a^2 x^2 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )-3 a^2 x^2 \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{6 a^3 x^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^2,x]

[Out]

(c^2*(Sqrt[1 - 1/(a^2*x^2)]*(-2 - 9*a*x - 16*a^2*x^2 + 6*a^3*x^3) - 3*a^2*x^2*ArcSin[1/(a*x)] + 18*a^2*x^2*Log
[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a^3*x^2)

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Maple [A]  time = 0.175, size = 233, normalized size = 1.2 \begin{align*}{\frac{ \left ( ax-1 \right ) ^{2}{c}^{2}}{ \left ( 6\,ax+6 \right ){a}^{4}{x}^{3}} \left ( -18\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{4}{a}^{4}+18\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-3\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+18\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}-3\,{a}^{3}{x}^{3}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +9\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+2\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\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))^(3/2)*(c-c/a^2/x^2)^2,x)

[Out]

1/6*(a*x-1)^2*c^2*(-18*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^4*a^4+18*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^2*a^2-3*(a^2*x
^2-1)^(1/2)*(a^2)^(1/2)*x^3*a^3+18*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^3*a^4-3*a^3*x^3*(a^
2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))+9*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*x*a+2*(a^2*x^2-1)^(3/2)*(a^2)^(1/2))/((a*
x-1)/(a*x+1))^(3/2)/(a*x+1)/((a*x-1)*(a*x+1))^(1/2)/a^4/x^3/(a^2)^(1/2)

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Maxima [A]  time = 1.56766, size = 301, normalized size = 1.54 \begin{align*} \frac{1}{3} \, a{\left (\frac{3 \, c^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{9 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{9 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac{15 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 37 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 17 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 21 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{2 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{2 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac{{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^2,x, algorithm="maxima")

[Out]

1/3*a*(3*c^2*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 9*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 9*c^2*log(
sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 + (15*c^2*((a*x - 1)/(a*x + 1))^(7/2) + 37*c^2*((a*x - 1)/(a*x + 1))^(5/2)
+ 17*c^2*((a*x - 1)/(a*x + 1))^(3/2) - 21*c^2*sqrt((a*x - 1)/(a*x + 1)))/(2*(a*x - 1)*a^2/(a*x + 1) - 2*(a*x -
1)^3*a^2/(a*x + 1)^3 - (a*x - 1)^4*a^2/(a*x + 1)^4 + a^2))

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Fricas [A]  time = 1.67075, size = 362, normalized size = 1.86 \begin{align*} \frac{6 \, a^{3} c^{2} x^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + 18 \, a^{3} c^{2} x^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 18 \, a^{3} c^{2} x^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (6 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} - 25 \, a^{2} c^{2} x^{2} - 11 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^2,x, algorithm="fricas")

[Out]

1/6*(6*a^3*c^2*x^3*arctan(sqrt((a*x - 1)/(a*x + 1))) + 18*a^3*c^2*x^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 18*
a^3*c^2*x^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (6*a^4*c^2*x^4 - 10*a^3*c^2*x^3 - 25*a^2*c^2*x^2 - 11*a*c^2*x
- 2*c^2)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{c^{2} \left (\int - \frac{2 a^{2}}{\frac{a x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int \frac{a^{4}}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int \frac{1}{\frac{a x^{5} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a**2/x**2)**2,x)

[Out]

c**2*(Integral(-2*a**2/(a*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x**2*sqrt(a*x/(a*x + 1) - 1/(a*x
+ 1))/(a*x + 1)), x) + Integral(a**4/(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - sqrt(a*x/(a*x + 1) - 1
/(a*x + 1))/(a*x + 1)), x) + Integral(1/(a*x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x**4*sqrt(a*x/(a
*x + 1) - 1/(a*x + 1))/(a*x + 1)), x))/a**4

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Giac [A]  time = 1.21752, size = 311, normalized size = 1.59 \begin{align*} \frac{1}{3} \, a{\left (\frac{3 \, c^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{9 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{9 \, c^{2} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} - \frac{6 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}} - \frac{\frac{28 \,{\left (a x - 1\right )} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + \frac{9 \,{\left (a x - 1\right )}^{2} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 27 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^2,x, algorithm="giac")

[Out]

1/3*a*(3*c^2*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 9*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 9*c^2*log(
abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/a^2 - 6*c^2*sqrt((a*x - 1)/(a*x + 1))/(a^2*((a*x - 1)/(a*x + 1) - 1)) - (2
8*(a*x - 1)*c^2*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1) + 9*(a*x - 1)^2*c^2*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1)^2
+ 27*c^2*sqrt((a*x - 1)/(a*x + 1)))/(a^2*((a*x - 1)/(a*x + 1) + 1)^3))