3.1.57 \(\int \frac {\text {arccosh}(a x)}{(c-a^2 c x^2)^3} \, dx\) [57]

3.1.57.1 Optimal result

Integrand size = 18, antiderivative size = 164 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{8 a c^3} \]

output

1/12/a/c^3/(a*x-1)^(3/2)/(a*x+1)^(3/2)+1/4*x*arccosh(a*x)/c^3/(-a^2*x^2+1) 
^2+3/8*x*arccosh(a*x)/c^3/(-a^2*x^2+1)+3/4*arccosh(a*x)*arctanh(a*x+(a*x-1 
)^(1/2)*(a*x+1)^(1/2))/a/c^3+3/8*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2 
))/a/c^3-3/8*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-3/8/a/c^3/(a 
*x-1)^(1/2)/(a*x+1)^(1/2)
 

3.1.57.2 Mathematica [A] (warning: unable to verify)

Time = 1.79 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.36 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {-\frac {2 (-2+a x) \sqrt {1+a x}}{(-1+a x)^{3/2}}+\frac {2 \sqrt {-1+a x} (2+a x)}{(1+a x)^{3/2}}+\frac {6 \text {arccosh}(a x)}{(-1+a x)^2}-\frac {6 \text {arccosh}(a x)}{(1+a x)^2}+18 \left (-\frac {1}{\sqrt {\frac {-1+a x}{1+a x}}}+\frac {\text {arccosh}(a x)}{1-a x}\right )+18 \left (\sqrt {\frac {-1+a x}{1+a x}}-\frac {\text {arccosh}(a x)}{1+a x}\right )+9 \text {arccosh}(a x) \left (\text {arccosh}(a x)-4 \log \left (1-e^{\text {arccosh}(a x)}\right )\right )-9 \text {arccosh}(a x) \left (\text {arccosh}(a x)-4 \log \left (1+e^{\text {arccosh}(a x)}\right )\right )+36 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-36 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{96 a c^3} \]

input

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

output

((-2*(-2 + a*x)*Sqrt[1 + a*x])/(-1 + a*x)^(3/2) + (2*Sqrt[-1 + a*x]*(2 + a 
*x))/(1 + a*x)^(3/2) + (6*ArcCosh[a*x])/(-1 + a*x)^2 - (6*ArcCosh[a*x])/(1 
 + a*x)^2 + 18*(-(1/Sqrt[(-1 + a*x)/(1 + a*x)]) + ArcCosh[a*x]/(1 - a*x)) 
+ 18*(Sqrt[(-1 + a*x)/(1 + a*x)] - ArcCosh[a*x]/(1 + a*x)) + 9*ArcCosh[a*x 
]*(ArcCosh[a*x] - 4*Log[1 - E^ArcCosh[a*x]]) - 9*ArcCosh[a*x]*(ArcCosh[a*x 
] - 4*Log[1 + E^ArcCosh[a*x]]) + 36*PolyLog[2, -E^ArcCosh[a*x]] - 36*PolyL 
og[2, E^ArcCosh[a*x]])/(96*a*c^3)
 

3.1.57.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.70 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.96, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {6316, 27, 83, 6316, 83, 6318, 3042, 26, 4670, 2715, 2838}

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.

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

\(\Big \downarrow \) 6316

\(\displaystyle \frac {3 \int \frac {\text {arccosh}(a x)}{c^2 \left (1-a^2 x^2\right )^2}dx}{4 c}-\frac {a \int \frac {x}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{4 c^3}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \int \frac {\text {arccosh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{4 c^3}-\frac {a \int \frac {x}{(a x-1)^{5/2} (a x+1)^{5/2}}dx}{4 c^3}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}\)

\(\Big \downarrow \) 83

\(\displaystyle \frac {3 \int \frac {\text {arccosh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{4 c^3}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}\)

\(\Big \downarrow \) 6316

\(\displaystyle \frac {3 \left (\frac {1}{2} \int \frac {\text {arccosh}(a x)}{1-a^2 x^2}dx+\frac {1}{2} a \int \frac {x}{(a x-1)^{3/2} (a x+1)^{3/2}}dx+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}\)

\(\Big \downarrow \) 83

\(\displaystyle \frac {3 \left (\frac {1}{2} \int \frac {\text {arccosh}(a x)}{1-a^2 x^2}dx+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}\)

\(\Big \downarrow \) 6318

\(\displaystyle \frac {3 \left (-\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {\frac {a x-1}{a x+1}} (a x+1)}d\text {arccosh}(a x)}{2 a}+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (-\frac {\int i \text {arccosh}(a x) \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{2 a}+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {3 \left (-\frac {i \int \text {arccosh}(a x) \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{2 a}+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}\)

\(\Big \downarrow \) 4670

\(\displaystyle \frac {3 \left (-\frac {i \left (i \int \log \left (1-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-i \int \log \left (1+e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {3 \left (-\frac {i \left (i \int e^{-\text {arccosh}(a x)} \log \left (1-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}-i \int e^{-\text {arccosh}(a x)} \log \left (1+e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}+2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{2 a}+\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {3 \left (\frac {x \text {arccosh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {i \left (2 i \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )}{2 a}-\frac {1}{2 a \sqrt {a x-1} \sqrt {a x+1}}\right )}{4 c^3}+\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}\)

input

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

output

1/(12*a*c^3*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)) + (x*ArcCosh[a*x])/(4*c^3*(1 
 - a^2*x^2)^2) + (3*(-1/2*1/(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[ 
a*x])/(2*(1 - a^2*x^2)) - ((I/2)*((2*I)*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*x 
]] + I*PolyLog[2, -E^ArcCosh[a*x]] - I*PolyLog[2, E^ArcCosh[a*x]]))/a))/(4 
*c^3)
 

3.1.57.3.1 Defintions of rubi rules used

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 83

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

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

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]
 

3.1.57.4 Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(\frac {-\frac {9 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )+9 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-15 a x \,\operatorname {arccosh}\left (a x \right )-11 \sqrt {a x -1}\, \sqrt {a x +1}}{24 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}}{a}\)	\(205\)
default	\(\frac {-\frac {9 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )+9 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-15 a x \,\operatorname {arccosh}\left (a x \right )-11 \sqrt {a x -1}\, \sqrt {a x +1}}{24 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}}{a}\)	\(205\)

input

int(arccosh(a*x)/(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)
 

output

1/a*(-1/24*(9*a^3*x^3*arccosh(a*x)+9*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1 
5*a*x*arccosh(a*x)-11*(a*x-1)^(1/2)*(a*x+1)^(1/2))/(a^4*x^4-2*a^2*x^2+1)/c 
^3-3/8/c^3*arccosh(a*x)*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-3/8/c^3*poly 
log(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+3/8/c^3*arccosh(a*x)*ln(1+a*x+(a*x- 
1)^(1/2)*(a*x+1)^(1/2))+3/8/c^3*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2) 
))
 

3.1.57.5 Fricas [F]

\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]

input

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

output

integral(-arccosh(a*x)/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3) 
, x)
 

3.1.57.6 Sympy [F]

\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=- \frac {\int \frac {\operatorname {acosh}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \]

input

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

output

-Integral(acosh(a*x)/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x)/c**3
 

3.1.57.7 Maxima [F]

\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]

input

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

output

-1/64*(10*a^3*x^3 + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 + 6*(a^4*x^ 
4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(a*x - 1) - 3*(a^4*x^4 - 2*a^2*x^2 + 1) 
*log(a*x - 1)^2 - 14*a*x + 4*(6*a^3*x^3 - 10*a*x - 3*(a^4*x^4 - 2*a^2*x^2 
+ 1)*log(a*x + 1) + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*log(a*x + sq 
rt(a*x + 1)*sqrt(a*x - 1)) - 7*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))/(a^ 
5*c^3*x^4 - 2*a^3*c^3*x^2 + a*c^3) + 3/16*(log(a*x - 1)*log(1/2*a*x + 1/2) 
 + dilog(-1/2*a*x + 1/2))/(a*c^3) - 7/64*log(a*x + 1)/(a*c^3) + integrate( 
-1/16*(6*a^3*x^3 - 10*a*x - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1) + 3*( 
a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))/(a^7*c^3*x^7 - 3*a^5*c^3*x^5 + 3*a^ 
3*c^3*x^3 - a*c^3*x + (a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)* 
sqrt(a*x + 1)*sqrt(a*x - 1)), x)
 

3.1.57.8 Giac [F]

\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {\operatorname {arcosh}\left (a x\right )}{{\left (a^{2} c x^{2} - c\right )}^{3}} \,d x } \]

input

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

output

integrate(-arccosh(a*x)/(a^2*c*x^2 - c)^3, x)
 

3.1.57.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^3} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^3} \,d x \]

input

int(acosh(a*x)/(c - a^2*c*x^2)^3,x)
 

output

int(acosh(a*x)/(c - a^2*c*x^2)^3, x)