3.8.77 \(\int \frac {e^{\coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^2} \, dx\) [777]

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

[Out]

arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c^2-4/3/a/c^2/(1-1/a/x)^(3/2)/(1+1/a/x)^(1/2)+x/c^2/(1-1/a/x)^(3/2)
/(1+1/a/x)^(1/2)-11/3/a/c^2/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)+8/3*(1-1/a/x)^(1/2)/a/c^2/(1+1/a/x)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6329, 105, 157, 12, 94, 214} \begin {gather*} \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} \end {gather*}

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 12

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

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 105

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] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 157

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 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 6329

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]

Rubi steps

\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx &=-\frac {\text {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 {\text {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 \text {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 \text {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 \text {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 {\text {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 {\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 )}{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*}

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Mathematica [A]
time = 0.08, size = 83, normalized size = 0.46 \begin {gather*} \frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (8-5 a x-7 a^2 x^2+3 a^3 x^3\right )}{3 (-1+a x)^2 (1+a x)}+\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^2} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(529\) vs. \(2(154)=308\).
time = 0.15, size = 530, normalized size = 2.94

method	result	size
risch	\(\frac {a x -1}{a \,c^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}-\frac {\sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{6 a^{7} \left (x -\frac {1}{a}\right )^{2}}-\frac {19 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{12 a^{6} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{4 a^{6} \left (x +\frac {1}{a}\right )}\right ) a^{4} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(218\)
default	\(-\frac {-45 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+21 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+45 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} x^{4}+24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+11 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+90 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}+48 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-5 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a x -90 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-48 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-19 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-45 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x -24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +45 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}+24 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )}{24 a \left (a x -1\right )^{2} \sqrt {a^{2}}\, \left (a x +1\right )^{2} c^{2} \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}\)	\(530\)

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,method=_RETURNVERBOSE)

[Out]

-1/24*(-45*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a^5*x^5-24*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)
^(1/2))*a^6*x^5+21*((a*x+1)*(a*x-1))^(3/2)*(a^2)^(1/2)*a^3*x^3+45*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)*a^4*x^4+
24*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^5*x^4+11*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(3/2)*
a^2*x^2+90*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)*a^3*x^3+48*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)
^(1/2))*a^4*x^3-5*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(3/2)*a*x-90*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a^2*x^2-48*ln
((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^3*x^2-19*((a*x+1)*(a*x-1))^(3/2)*(a^2)^(1/2)-45*((
a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a*x-24*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^2*x+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*x-1
)^2/(a^2)^(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 = 0.27, size = 160, normalized size = 0.89 \begin {gather*} \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 {gather*}

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 = 0.37, size = 134, normalized size = 0.74 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

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

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Mupad [B]
time = 0.07, size = 128, normalized size = 0.71 \begin {gather*} \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{4\,a\,c^2}-\frac {\frac {17\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {14\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}+\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*x - 1)/(a*x + 1))^(1/2)/(4*a*c^2) - ((17*(a*x - 1))/(3*(a*x + 1)) - (14*(a*x - 1)^2)/(a*x + 1)^2 + 1/3)/(4
*a*c^2*((a*x - 1)/(a*x + 1))^(3/2) - 4*a*c^2*((a*x - 1)/(a*x + 1))^(5/2)) + (2*atanh(((a*x - 1)/(a*x + 1))^(1/
2)))/(a*c^2)

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