\(\int \frac {x^4 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) [208]

Optimal result

Integrand size = 24, antiderivative size = 243 \[ \int \frac {x^4 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {15 x \sqrt {1-a x} \sqrt {1+a x}}{64 a^4}-\frac {x^3 \sqrt {1-a x} \sqrt {1+a x}}{32 a^2}-\frac {15 \sqrt {1-a x} \text {arccosh}(a x)}{64 a^5 \sqrt {-1+a x}}+\frac {3 x^2 \sqrt {1-a x} \text {arccosh}(a x)}{8 a^3 \sqrt {-1+a x}}+\frac {x^4 \sqrt {1-a x} \text {arccosh}(a x)}{8 a \sqrt {-1+a x}}-\frac {3 x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{4 a^2}-\frac {\sqrt {1-a x} \text {arccosh}(a x)^3}{8 a^5 \sqrt {-1+a x}} \] Output:

-15/64*x*(-a*x+1)^(1/2)*(a*x+1)^(1/2)/a^4-1/32*x^3*(-a*x+1)^(1/2)*(a*x+1)^ 
(1/2)/a^2-15/64*(-a*x+1)^(1/2)*arccosh(a*x)/a^5/(a*x-1)^(1/2)+3/8*x^2*(-a* 
x+1)^(1/2)*arccosh(a*x)/a^3/(a*x-1)^(1/2)+1/8*x^4*(-a*x+1)^(1/2)*arccosh(a 
*x)/a/(a*x-1)^(1/2)-3/8*x*(-a^2*x^2+1)^(1/2)*arccosh(a*x)^2/a^4-1/4*x^3*(- 
a^2*x^2+1)^(1/2)*arccosh(a*x)^2/a^2-1/8*(-a*x+1)^(1/2)*arccosh(a*x)^3/a^5/ 
(a*x-1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.48 \[ \int \frac {x^4 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (32 \text {arccosh}(a x)^3-4 \text {arccosh}(a x) (16 \cosh (2 \text {arccosh}(a x))+\cosh (4 \text {arccosh}(a x)))+32 \sinh (2 \text {arccosh}(a x))+\sinh (4 \text {arccosh}(a x))+8 \text {arccosh}(a x)^2 (8 \sinh (2 \text {arccosh}(a x))+\sinh (4 \text {arccosh}(a x)))\right )}{256 a^5 \sqrt {1-a^2 x^2}} \] Input:

Integrate[(x^4*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]
 

Output:

(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(32*ArcCosh[a*x]^3 - 4*ArcCosh[a*x]* 
(16*Cosh[2*ArcCosh[a*x]] + Cosh[4*ArcCosh[a*x]]) + 32*Sinh[2*ArcCosh[a*x]] 
 + Sinh[4*ArcCosh[a*x]] + 8*ArcCosh[a*x]^2*(8*Sinh[2*ArcCosh[a*x]] + Sinh[ 
4*ArcCosh[a*x]])))/(256*a^5*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.21, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6353, 6298, 111, 27, 101, 43, 6353, 6298, 101, 43, 6307}

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.

\(\displaystyle \int \frac {x^4 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\)

\(\Big \downarrow \) 6353

\(\displaystyle \frac {3 \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {\sqrt {a x-1} \int x^3 \text {arccosh}(a x)dx}{2 a \sqrt {1-a x}}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{4 a^2}\)

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 43

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 43

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

\(\Big \downarrow \) 6307

\(\displaystyle -\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{4 a^2}+\frac {3 \left (\frac {\sqrt {a x-1} \text {arccosh}(a x)^3}{6 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{2 a^2}-\frac {\sqrt {a x-1} \left (\frac {1}{2} x^2 \text {arccosh}(a x)-\frac {1}{2} a \left (\frac {\text {arccosh}(a x)}{2 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )\right )}{a \sqrt {1-a x}}\right )}{4 a^2}-\frac {\sqrt {a x-1} \left (\frac {1}{4} x^4 \text {arccosh}(a x)-\frac {1}{4} a \left (\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{4 a^2}+\frac {3 \left (\frac {\text {arccosh}(a x)}{2 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}\right )\right )}{2 a \sqrt {1-a x}}\)

Input:

Int[(x^4*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]
 

Output:

-1/4*(x^3*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/a^2 - (Sqrt[-1 + a*x]*((x^4*Ar 
cCosh[a*x])/4 - (a*((x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(4*a^2) + (3*((x*Sq 
rt[-1 + a*x]*Sqrt[1 + a*x])/(2*a^2) + ArcCosh[a*x]/(2*a^3)))/(4*a^2)))/4)) 
/(2*a*Sqrt[1 - a*x]) + (3*(-1/2*(x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/a^2 + 
 (Sqrt[-1 + a*x]*ArcCosh[a*x]^3)/(6*a^3*Sqrt[1 - a*x]) - (Sqrt[-1 + a*x]*( 
(x^2*ArcCosh[a*x])/2 - (a*((x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a^2) + ArcC 
osh[a*x]/(2*a^3)))/2))/(a*Sqrt[1 - a*x])))/(4*a^2)
 

Defintions of rubi rules used

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 43

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

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 111

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

rule 6298

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + 
 c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& NeQ[m, -1]
 

rule 6307

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

rule 6353

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(199)=398\).

Time = 0.39 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.01

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{3}}{8 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}+8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+4 a x -8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (8 \operatorname {arccosh}\left (a x \right )^{2}-4 \,\operatorname {arccosh}\left (a x \right )+1\right )}{512 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (2 \operatorname {arccosh}\left (a x \right )^{2}-2 \,\operatorname {arccosh}\left (a x \right )+1\right )}{16 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (-2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 a^{3} x^{3}+\sqrt {a x -1}\, \sqrt {a x +1}-2 a x \right ) \left (2 \operatorname {arccosh}\left (a x \right )^{2}+2 \,\operatorname {arccosh}\left (a x \right )+1\right )}{16 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (-8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+8 a^{5} x^{5}+8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-12 a^{3} x^{3}-\sqrt {a x -1}\, \sqrt {a x +1}+4 a x \right ) \left (8 \operatorname {arccosh}\left (a x \right )^{2}+4 \,\operatorname {arccosh}\left (a x \right )+1\right )}{512 a^{5} \left (a^{2} x^{2}-1\right )}\)	\(488\)

Input:

int(x^4*arccosh(a*x)^2/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/8*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5/(a^2*x^2-1)*arccos 
h(a*x)^3-1/512*(-a^2*x^2+1)^(1/2)*(8*a^5*x^5-12*a^3*x^3+8*(a*x-1)^(1/2)*(a 
*x+1)^(1/2)*a^4*x^4+4*a*x-8*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+(a*x-1)^(1 
/2)*(a*x+1)^(1/2))*(8*arccosh(a*x)^2-4*arccosh(a*x)+1)/a^5/(a^2*x^2-1)-1/1 
6*(-a^2*x^2+1)^(1/2)*(2*a^3*x^3-2*a*x+2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2 
)-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(2*arccosh(a*x)^2-2*arccosh(a*x)+1)/a^5/(a^ 
2*x^2-1)-1/16*(-a^2*x^2+1)^(1/2)*(-2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2 
*a^3*x^3+(a*x-1)^(1/2)*(a*x+1)^(1/2)-2*a*x)*(2*arccosh(a*x)^2+2*arccosh(a* 
x)+1)/a^5/(a^2*x^2-1)-1/512*(-a^2*x^2+1)^(1/2)*(-8*(a*x-1)^(1/2)*(a*x+1)^( 
1/2)*a^4*x^4+8*a^5*x^5+8*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-12*a^3*x^3-(a 
*x-1)^(1/2)*(a*x+1)^(1/2)+4*a*x)*(8*arccosh(a*x)^2+4*arccosh(a*x)+1)/a^5/( 
a^2*x^2-1)
 

Fricas [F]

\[ \int \frac {x^4 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^4*arccosh(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)*x^4*arccosh(a*x)^2/(a^2*x^2 - 1), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4*arccosh(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

\[ \int \frac {x^4 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^4*arccosh(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^4*arccosh(a*x)^2/sqrt(-a^2*x^2 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^4\,{\mathrm {acosh}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \] Input:

int((x^4*acosh(a*x)^2)/(1 - a^2*x^2)^(1/2),x)
 

Output:

int((x^4*acosh(a*x)^2)/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^4 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acosh} \left (a x \right )^{2} x^{4}}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:

int(x^4*acosh(a*x)^2/(-a^2*x^2+1)^(1/2),x)
 

Output:

int((acosh(a*x)**2*x**4)/sqrt( - a**2*x**2 + 1),x)