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

3.2.68.1 Optimal result

Integrand size = 20, antiderivative size = 163 \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {\text {arccosh}(a x)}{a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {\text {arctanh}(a x)}{a c^2}+\frac {\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {\operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {\operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{a c^2} \]

output

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+po 
lylog(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)
 

3.2.68.2 Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.17 \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {-4 \text {arccosh}(a x) \coth \left (\frac {1}{2} \text {arccosh}(a x)\right )-\text {arccosh}(a x)^2 \text {csch}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )-4 \text {arccosh}(a x)^2 \log \left (1-e^{-\text {arccosh}(a x)}\right )+4 \text {arccosh}(a x)^2 \log \left (1+e^{-\text {arccosh}(a x)}\right )+8 \log \left (\tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )\right )-8 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(a x)}\right )+8 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(a x)}\right )-8 \operatorname {PolyLog}\left (3,-e^{-\text {arccosh}(a x)}\right )+8 \operatorname {PolyLog}\left (3,e^{-\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \text {sech}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )+4 \text {arccosh}(a x) \tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )}{8 a c^2} \]

input

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

output

(-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^(-ArcCosh[a*x])] + 4*ArcCosh[a*x]^2*Log[1 
 + E^(-ArcCosh[a*x])] + 8*Log[Tanh[ArcCosh[a*x]/2]] - 8*ArcCosh[a*x]*PolyL 
og[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])] - ArcC 
osh[a*x]^2*Sech[ArcCosh[a*x]/2]^2 + 4*ArcCosh[a*x]*Tanh[ArcCosh[a*x]/2])/( 
8*a*c^2)
 

3.2.68.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.63 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.96, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {6316, 27, 6318, 3042, 26, 4670, 3011, 2720, 6330, 25, 39, 219, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^2} \, dx\)

\(\Big \downarrow \) 6316

\(\displaystyle \frac {\int \frac {\text {arccosh}(a x)^2}{c \left (1-a^2 x^2\right )}dx}{2 c}+\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx}{c^2}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\text {arccosh}(a x)^2}{1-a^2 x^2}dx}{2 c^2}+\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx}{c^2}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 6318

\(\displaystyle \frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx}{c^2}-\frac {\int \frac {\text {arccosh}(a x)^2}{\sqrt {\frac {a x-1}{a x+1}} (a x+1)}d\text {arccosh}(a x)}{2 a c^2}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx}{c^2}-\frac {\int i \text {arccosh}(a x)^2 \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{2 a c^2}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx}{c^2}-\frac {i \int \text {arccosh}(a x)^2 \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{2 a c^2}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 4670

\(\displaystyle -\frac {i \left (2 i \int \text {arccosh}(a x) \log \left (1-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-2 i \int \text {arccosh}(a x) \log \left (1+e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a c^2}+\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx}{c^2}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {i \left (-2 i \left (\int \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\int \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a c^2}+\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx}{c^2}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {i \left (-2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a c^2}+\frac {a \int \frac {x \text {arccosh}(a x)}{(a x-1)^{3/2} (a x+1)^{3/2}}dx}{c^2}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 6330

\(\displaystyle \frac {a \left (\frac {\int -\frac {1}{(1-a x) (a x+1)}dx}{a}-\frac {\text {arccosh}(a x)}{a^2 \sqrt {a x-1} \sqrt {a x+1}}\right )}{c^2}-\frac {i \left (-2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a c^2}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {a \left (-\frac {\int \frac {1}{(1-a x) (a x+1)}dx}{a}-\frac {\text {arccosh}(a x)}{a^2 \sqrt {a x-1} \sqrt {a x+1}}\right )}{c^2}-\frac {i \left (-2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a c^2}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 39

\(\displaystyle \frac {a \left (-\frac {\int \frac {1}{1-a^2 x^2}dx}{a}-\frac {\text {arccosh}(a x)}{a^2 \sqrt {a x-1} \sqrt {a x+1}}\right )}{c^2}-\frac {i \left (-2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a c^2}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {i \left (-2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a c^2}+\frac {a \left (-\frac {\text {arccosh}(a x)}{a^2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arctanh}(a x)}{a^2}\right )}{c^2}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {a \left (-\frac {\text {arccosh}(a x)}{a^2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arctanh}(a x)}{a^2}\right )}{c^2}+\frac {x \text {arccosh}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}-\frac {i \left (2 i \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )-2 i \left (\operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+2 i \left (\operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )\right )}{2 a c^2}\)

input

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

output

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

3.2.68.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 39

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 219

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

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[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 6316

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] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b* 
ArcCosh[c*x])^n, x], x] - Simp[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 6318

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[-(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 6330

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p 
_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + 
e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[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] && E 
qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

3.2.68.4 Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.56

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

input

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

output

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*poly 
log(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*arccosh(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*arct 
anh(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))
 

3.2.68.5 Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]

input

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

output

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

3.2.68.6 Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]

input

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

output

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

3.2.68.7 Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]

input

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

output

-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 + sqr 
t(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)
 

3.2.68.8 Giac [F]

\[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]

input

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

output

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

3.2.68.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^2} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^2} \,d x \]

input

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

output

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