\(\int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) [209]

Optimal result

Integrand size = 24, antiderivative size = 198 \[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {4 \sqrt {1-a x} \sqrt {1+a x}}{27 a^4}-\frac {2 x^2 \sqrt {1-a x} \sqrt {1+a x}}{27 a^2}-\frac {4 \sqrt {1-a^2 x^2}}{3 a^4}+\frac {4 x \sqrt {1-a x} \text {arccosh}(a x)}{3 a^3 \sqrt {-1+a x}}+\frac {2 x^3 \sqrt {1-a x} \text {arccosh}(a x)}{9 a \sqrt {-1+a x}}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{3 a^2} \] Output:

-4/27*(-a*x+1)^(1/2)*(a*x+1)^(1/2)/a^4-2/27*x^2*(-a*x+1)^(1/2)*(a*x+1)^(1/ 
2)/a^2-4/3*(-a^2*x^2+1)^(1/2)/a^4+4/3*x*(-a*x+1)^(1/2)*arccosh(a*x)/a^3/(a 
*x-1)^(1/2)+2/9*x^3*(-a*x+1)^(1/2)*arccosh(a*x)/a/(a*x-1)^(1/2)-2/3*(-a^2* 
x^2+1)^(1/2)*arccosh(a*x)^2/a^4-1/3*x^2*(-a^2*x^2+1)^(1/2)*arccosh(a*x)^2/ 
a^2
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.62 \[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\left (-\frac {40}{27 a^4}-\frac {2 x^2}{27 a^2}\right ) \sqrt {1-a^2 x^2}+\frac {2 x \sqrt {1-a^2 x^2} \left (6+a^2 x^2\right ) \text {arccosh}(a x)}{9 a^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \text {arccosh}(a x)^2}{3 a^4} \] Input:

Integrate[(x^3*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]
 

Output:

(-40/(27*a^4) - (2*x^2)/(27*a^2))*Sqrt[1 - a^2*x^2] + (2*x*Sqrt[1 - a^2*x^ 
2]*(6 + a^2*x^2)*ArcCosh[a*x])/(9*a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (Sqr 
t[1 - a^2*x^2]*(2 + a^2*x^2)*ArcCosh[a*x]^2)/(3*a^4)
 

Rubi [A] (warning: unable to verify)

Time = 1.26 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6353, 6298, 111, 27, 83, 6329, 6294, 83}

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.

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 83

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

\(\Big \downarrow \) 6329

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

\(\Big \downarrow \) 6294

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

\(\Big \downarrow \) 83

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

Input:

Int[(x^3*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]
 

Output:

-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/a^2 - (2*Sqrt[-1 + a*x]*(-1/3* 
(a*((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a^4) + (x^2*Sqrt[-1 + a*x]*Sqrt[1 
+ a*x])/(3*a^2))) + (x^3*ArcCosh[a*x])/3))/(3*a*Sqrt[1 - a*x]) + (2*(-((Sq 
rt[1 - a^2*x^2]*ArcCosh[a*x]^2)/a^2) - (2*Sqrt[-1 + a*x]*(-((Sqrt[-1 + a*x 
]*Sqrt[1 + a*x])/a) + x*ArcCosh[a*x]))/(a*Sqrt[1 - a*x])))/(3*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 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 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 6294

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A 
rcCosh[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt 
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

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 6329

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*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, d, e, p}, x] && EqQ[c^2*d + e, 0] && 
GtQ[n, 0] && NeQ[p, -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. \(342\) vs. \(2(162)=324\).

Time = 0.46 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.73

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}+4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +1\right ) \left (9 \operatorname {arccosh}\left (a x \right )^{2}-6 \,\operatorname {arccosh}\left (a x \right )+2\right )}{216 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x -1}\, \sqrt {a x +1}\, a x +a^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (a x \right )^{2}-2 \,\operatorname {arccosh}\left (a x \right )+2\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x -1\right ) \left (\operatorname {arccosh}\left (a x \right )^{2}+2 \,\operatorname {arccosh}\left (a x \right )+2\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}-4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}+3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +1\right ) \left (9 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+2\right )}{216 a^{4} \left (a^{2} x^{2}-1\right )}\)	\(343\)
orering	\(\frac {\left (19 a^{6} x^{6}+100 a^{4} x^{4}-380 a^{2} x^{2}+240\right ) \operatorname {arccosh}\left (a x \right )^{2}}{27 a^{6} x^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (a^{4} x^{4}+12 a^{2} x^{2}-20\right ) \left (\frac {3 x^{2} \operatorname {arccosh}\left (a x \right )^{2}}{\sqrt {-a^{2} x^{2}+1}}+\frac {2 x^{3} \operatorname {arccosh}\left (a x \right ) a}{\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}}+\frac {x^{4} \operatorname {arccosh}\left (a x \right )^{2} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{9 a^{6} x^{4}}+\frac {\left (a^{2} x^{2}+20\right ) \left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (\frac {6 x \operatorname {arccosh}\left (a x \right )^{2}}{\sqrt {-a^{2} x^{2}+1}}+\frac {12 x^{2} \operatorname {arccosh}\left (a x \right ) a}{\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}}+\frac {7 x^{3} \operatorname {arccosh}\left (a x \right )^{2} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {2 x^{3} a^{2}}{\left (a x -1\right ) \left (a x +1\right ) \sqrt {-a^{2} x^{2}+1}}+\frac {4 x^{4} \operatorname {arccosh}\left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \sqrt {a x -1}\, \sqrt {a x +1}}-\frac {x^{3} \operatorname {arccosh}\left (a x \right ) a^{2}}{\sqrt {-a^{2} x^{2}+1}\, \left (a x -1\right )^{\frac {3}{2}} \sqrt {a x +1}}-\frac {x^{3} \operatorname {arccosh}\left (a x \right ) a^{2}}{\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \left (a x +1\right )^{\frac {3}{2}}}+\frac {3 x^{5} \operatorname {arccosh}\left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{27 a^{6} x^{3}}\)	\(463\)

Input:

int(x^3*arccosh(a*x)^2/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/216*(-a^2*x^2+1)^(1/2)*(4*a^4*x^4-5*a^2*x^2+4*a^3*x^3*(a*x-1)^(1/2)*(a* 
x+1)^(1/2)-3*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x+1)*(9*arccosh(a*x)^2-6*arccos 
h(a*x)+2)/a^4/(a^2*x^2-1)-3/8*(-a^2*x^2+1)^(1/2)*((a*x-1)^(1/2)*(a*x+1)^(1 
/2)*a*x+a^2*x^2-1)*(arccosh(a*x)^2-2*arccosh(a*x)+2)/a^4/(a^2*x^2-1)-3/8*( 
-a^2*x^2+1)^(1/2)*(a^2*x^2-(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x-1)*(arccosh(a*x 
)^2+2*arccosh(a*x)+2)/a^4/(a^2*x^2-1)-1/216*(-a^2*x^2+1)^(1/2)*(4*a^4*x^4- 
5*a^2*x^2-4*a^3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)+3*(a*x-1)^(1/2)*(a*x+1)^(1 
/2)*a*x+1)*(9*arccosh(a*x)^2+6*arccosh(a*x)+2)/a^4/(a^2*x^2-1)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.76 \[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {9 \, {\left (a^{4} x^{4} + a^{2} x^{2} - 2\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + 2 \, {\left (a^{4} x^{4} + 19 \, a^{2} x^{2} - 20\right )} \sqrt {-a^{2} x^{2} + 1}}{27 \, {\left (a^{6} x^{2} - a^{4}\right )}} \] Input:

integrate(x^3*arccosh(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

-1/27*(9*(a^4*x^4 + a^2*x^2 - 2)*sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 
 - 1))^2 - 6*(a^3*x^3 + 6*a*x)*sqrt(a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1)*log(a* 
x + sqrt(a^2*x^2 - 1)) + 2*(a^4*x^4 + 19*a^2*x^2 - 20)*sqrt(-a^2*x^2 + 1)) 
/(a^6*x^2 - a^4)
 

Sympy [F]

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

integrate(x**3*acosh(a*x)**2/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral(x**3*acosh(a*x)**2/sqrt(-(a*x - 1)*(a*x + 1)), x)
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.53 \[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arcosh}\left (a x\right )^{2} + \frac {2 \, {\left (-i \, \sqrt {a^{2} x^{2} - 1} x^{2} - \frac {20 i \, \sqrt {a^{2} x^{2} - 1}}{a^{2}}\right )}}{27 \, a^{2}} + \frac {2 \, {\left (i \, a^{2} x^{3} + 6 i \, x\right )} \operatorname {arcosh}\left (a x\right )}{9 \, a^{3}} \] Input:

integrate(x^3*arccosh(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

-1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arccosh(a*x)^ 
2 + 2/27*(-I*sqrt(a^2*x^2 - 1)*x^2 - 20*I*sqrt(a^2*x^2 - 1)/a^2)/a^2 + 2/9 
*(I*a^2*x^3 + 6*I*x)*arccosh(a*x)/a^3
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3*arccosh(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^3\,{\mathrm {acosh}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \] Input:

int((x^3*acosh(a*x)^2)/(1 - a^2*x^2)^(1/2),x)
 

Output:

int((x^3*acosh(a*x)^2)/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acosh} \left (a x \right )^{2} x^{3}}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:

int(x^3*acosh(a*x)^2/(-a^2*x^2+1)^(1/2),x)
 

Output:

int((acosh(a*x)**2*x**3)/sqrt( - a**2*x**2 + 1),x)