### 3.777 $$\int \frac{e^{\coth ^{-1}(a x)}}{(c-\frac{c}{a^2 x^2})^2} \, dx$$

Optimal. Leaf size=180 $\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{\frac{1}{a x}+1}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^2}$

[Out]

-4/(3*a*c^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]) - 11/(3*a*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (8*Sqr
t[1 - 1/(a*x)])/(3*a*c^2*Sqrt[1 + 1/(a*x)]) + x/(c^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]) + ArcTanh[Sqrt[1 -
1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^2)

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Rubi [A]  time = 0.115933, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.3, Rules used = {6194, 103, 152, 12, 92, 208} $\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{\frac{1}{a x}+1}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-4/(3*a*c^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]) - 11/(3*a*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (8*Sqr
t[1 - 1/(a*x)])/(3*a*c^2*Sqrt[1 + 1/(a*x)]) + x/(c^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]) + ArcTanh[Sqrt[1 -
1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^2)

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 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 \frac{e^{\coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{a}-\frac{3 x}{a^2}}{x \left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{a \operatorname{Subst}\left (\int \frac{\frac{3}{a^2}+\frac{8 x}{a^3}}{x \left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 c^2}\\ &=-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{a^2 \operatorname{Subst}\left (\int \frac{-\frac{3}{a^3}-\frac{11 x}{a^4}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 c^2}\\ &=-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{a^3 \operatorname{Subst}\left (\int -\frac{3}{a^4 x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{3 c^2}\\ &=-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a c^2}\\ &=-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\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 c^2}\\ &=-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a c^2}\\ \end{align*}

Mathematica [A]  time = 0.153513, size = 83, normalized size = 0.46 $\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (3 a^3 x^3-7 a^2 x^2-5 a x+8\right )}{3 (a x-1)^2 (a x+1)}+\log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a c^2}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(8 - 5*a*x - 7*a^2*x^2 + 3*a^3*x^3))/(3*(-1 + a*x)^2*(1 + a*x)) + Log[(1 + Sqrt[1
- 1/(a^2*x^2)])*x])/(a*c^2)

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Maple [B]  time = 0.149, size = 530, normalized size = 2.9 \begin{align*} -{\frac{1}{24\,a \left ( ax+1 \right ) ^{2} \left ( ax-1 \right ) ^{2}{c}^{2}} \left ( -45\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{5}{a}^{5}-24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{5}{a}^{6}+21\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{3}{a}^{3}+45\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{4}{a}^{4}+24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{4}{a}^{5}+11\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}+90\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}+48\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}-5\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-90\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}-48\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-19\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-45\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}+45\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+24\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \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)^2,x)

[Out]

-1/24*(-45*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^5*a^5-24*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)
^(1/2))*x^5*a^6+21*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3+45*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4+
24*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^4*a^5+11*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*
x^2*a^2+90*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3+48*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)
^(1/2))*x^3*a^4-5*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a-90*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^2*a^2-48*ln
((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^2*a^3-19*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)-45*(a
^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a-24*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x*a^2+45*
(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)+24*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2)))/a/(a^2)^
(1/2)/(a*x+1)^2/(a*x-1)^2/c^2/((a*x-1)*(a*x+1))^(1/2)/((a*x-1)/(a*x+1))^(1/2)

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Maxima [A]  time = 1.0066, size = 216, normalized size = 1.2 \begin{align*} \frac{1}{12} \, a{\left (\frac{\frac{17 \,{\left (a x - 1\right )}}{a x + 1} - \frac{42 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - a^{2} c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}} + \frac{12 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac{12 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}} + \frac{3 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}}\right )} \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)^2,x, algorithm="maxima")

[Out]

1/12*a*((17*(a*x - 1)/(a*x + 1) - 42*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c^2*((a*x - 1)/(a*x + 1))^(5/2) - a^2*c
^2*((a*x - 1)/(a*x + 1))^(3/2)) + 12*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^2) - 12*log(sqrt((a*x - 1)/(a*x
+ 1)) - 1)/(a^2*c^2) + 3*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^2))

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Fricas [A]  time = 1.37424, size = 306, normalized size = 1.7 \begin{align*} \frac{3 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 3 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (3 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 5 \, a x + 8\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \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)^2,x, algorithm="fricas")

[Out]

1/3*(3*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 3*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(
a*x + 1)) - 1) + (3*a^3*x^3 - 7*a^2*x^2 - 5*a*x + 8)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a
*c^2)

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

[Out]

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

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Giac [A]  time = 1.14055, size = 231, normalized size = 1.28 \begin{align*} \frac{1}{12} \, a{\left (\frac{12 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac{12 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{2}} - \frac{{\left (a x + 1\right )}{\left (\frac{18 \,{\left (a x - 1\right )}}{a x + 1} + 1\right )}}{{\left (a x - 1\right )} a^{2} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}} + \frac{3 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}} - \frac{24 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}}\right )} \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)^2,x, algorithm="giac")

[Out]

1/12*a*(12*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^2) - 12*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a^2*c^2)
- (a*x + 1)*(18*(a*x - 1)/(a*x + 1) + 1)/((a*x - 1)*a^2*c^2*sqrt((a*x - 1)/(a*x + 1))) + 3*sqrt((a*x - 1)/(a*
x + 1))/(a^2*c^2) - 24*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^2*((a*x - 1)/(a*x + 1) - 1)))