3.2.68 \(\int \frac {\cosh ^{-1}(a x)^2}{(c-a^2 c x^2)^2} \, dx\) [168]

Optimal. Leaf size=163 \[ -\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\tanh ^{-1}(a x)}{a c^2}+\frac {\cosh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\text {PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\text {PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]

[Out]

1/2*x*arccosh(a*x)^2/c^2/(-a^2*x^2+1)+arccosh(a*x)^2*arctanh(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2-arctanh(a*
x)/a/c^2+arccosh(a*x)*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2-arccosh(a*x)*polylog(2,a*x+(a*x-1)^(1/
2)*(a*x+1)^(1/2))/a/c^2-polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2+polylog(3,a*x+(a*x-1)^(1/2)*(a*x+1)^
(1/2))/a/c^2-arccosh(a*x)/a/c^2/(a*x-1)^(1/2)/(a*x+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.20, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5901, 5903, 4267, 2611, 2320, 6724, 5915, 35, 213} \begin {gather*} \frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\tanh ^{-1}(a x)}{a c^2}+\frac {\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcCosh[a*x]/(a*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])) + (x*ArcCosh[a*x]^2)/(2*c^2*(1 - a^2*x^2)) + (ArcCosh[a*x
]^2*ArcTanh[E^ArcCosh[a*x]])/(a*c^2) - ArcTanh[a*x]/(a*c^2) + (ArcCosh[a*x]*PolyLog[2, -E^ArcCosh[a*x]])/(a*c^
2) - (ArcCosh[a*x]*PolyLog[2, E^ArcCosh[a*x]])/(a*c^2) - PolyLog[3, -E^ArcCosh[a*x]]/(a*c^2) + PolyLog[3, E^Ar
cCosh[a*x]]/(a*c^2)

Rule 35

Int[1/(((a_) + (b_.)*(x_))*((c_) + (d_.)*(x_))), x_Symbol] :> Int[1/(a*c + b*d*x^2), x] /; FreeQ[{a, b, c, d},
x] && EqQ[b*c + a*d, 0]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5901

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(
p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a
+ b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[x*(
1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 5903

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[-(c*d)^(-1), Subst[Int[
(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Dist[b*
(n/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(1 + c*x)^(p + 1/2)*(-
1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c
*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {\cosh ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {a \int \frac {x \cosh ^{-1}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{c^2}+\frac {\int \frac {\cosh ^{-1}(a x)^2}{c-a^2 c x^2} \, dx}{2 c}\\ &=-\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{c^2}-\frac {\text {Subst}\left (\int x^2 \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}\\ &=-\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\tanh ^{-1}(a x)}{a c^2}+\frac {\text {Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}-\frac {\text {Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}\\ &=-\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\tanh ^{-1}(a x)}{a c^2}+\frac {\cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\text {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}+\frac {\text {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}\\ &=-\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\tanh ^{-1}(a x)}{a c^2}+\frac {\cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}\\ &=-\frac {\cosh ^{-1}(a x)}{a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\tanh ^{-1}(a x)}{a c^2}+\frac {\cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {\text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.58, size = 191, normalized size = 1.17 \begin {gather*} \frac {-4 \cosh ^{-1}(a x) \coth \left (\frac {1}{2} \cosh ^{-1}(a x)\right )-\cosh ^{-1}(a x)^2 \text {csch}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )-4 \cosh ^{-1}(a x)^2 \log \left (1-e^{-\cosh ^{-1}(a x)}\right )+4 \cosh ^{-1}(a x)^2 \log \left (1+e^{-\cosh ^{-1}(a x)}\right )+8 \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )\right )-8 \cosh ^{-1}(a x) \text {PolyLog}\left (2,-e^{-\cosh ^{-1}(a x)}\right )+8 \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{-\cosh ^{-1}(a x)}\right )-8 \text {PolyLog}\left (3,-e^{-\cosh ^{-1}(a x)}\right )+8 \text {PolyLog}\left (3,e^{-\cosh ^{-1}(a x)}\right )-\cosh ^{-1}(a x)^2 \text {sech}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )+4 \cosh ^{-1}(a x) \tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )}{8 a c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*ArcCosh[a*x]*Coth[ArcCosh[a*x]/2] - ArcCosh[a*x]^2*Csch[ArcCosh[a*x]/2]^2 - 4*ArcCosh[a*x]^2*Log[1 - E^(-A
rcCosh[a*x])] + 4*ArcCosh[a*x]^2*Log[1 + E^(-ArcCosh[a*x])] + 8*Log[Tanh[ArcCosh[a*x]/2]] - 8*ArcCosh[a*x]*Pol
yLog[2, -E^(-ArcCosh[a*x])] + 8*ArcCosh[a*x]*PolyLog[2, E^(-ArcCosh[a*x])] - 8*PolyLog[3, -E^(-ArcCosh[a*x])]
+ 8*PolyLog[3, E^(-ArcCosh[a*x])] - ArcCosh[a*x]^2*Sech[ArcCosh[a*x]/2]^2 + 4*ArcCosh[a*x]*Tanh[ArcCosh[a*x]/2
])/(8*a*c^2)

________________________________________________________________________________________

Maple [A]
time = 3.15, size = 255, normalized size = 1.56

method	result	size
derivativedivides	\(\frac {-\frac {\mathrm {arccosh}\left (a x \right ) \left (a x \,\mathrm {arccosh}\left (a x \right )+2 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 \left (a^{2} x^{2}-1\right ) c^{2}}-\frac {\mathrm {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {\mathrm {arccosh}\left (a x \right ) \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {\polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {\mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {\mathrm {arccosh}\left (a x \right ) \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {\polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {2 \arctanh \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}}{a}\)	\(255\)
default	\(\frac {-\frac {\mathrm {arccosh}\left (a x \right ) \left (a x \,\mathrm {arccosh}\left (a x \right )+2 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 \left (a^{2} x^{2}-1\right ) c^{2}}-\frac {\mathrm {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {\mathrm {arccosh}\left (a x \right ) \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {\polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {\mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {\mathrm {arccosh}\left (a x \right ) \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {\polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {2 \arctanh \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}}{a}\)	\(255\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*(-1/2/(a^2*x^2-1)*arccosh(a*x)*(a*x*arccosh(a*x)+2*(a*x-1)^(1/2)*(a*x+1)^(1/2))/c^2-1/2/c^2*arccosh(a*x)^2
*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-1/c^2*arccosh(a*x)*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+1/c^2*pol
ylog(3,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+1/2/c^2*arccosh(a*x)^2*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+1/c^2*arc
cosh(a*x)*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-1/c^2*polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-2/c^2*
arctanh(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [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(arccosh(a*x)^2/(-a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)^2/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

integrate(arccosh(a*x)^2/(a^2*c*x^2 - c)^2, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^2/(c - a^2*c*x^2)^2,x)

[Out]

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

________________________________________________________________________________________