\(\int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) [223]

Optimal result

Integrand size = 24, antiderivative size = 243 \[ \int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {40 x \sqrt {1-a x}}{9 a^3 \sqrt {-1+a x}}+\frac {2 x^3 \sqrt {1-a x}}{27 a \sqrt {-1+a x}}-\frac {40 \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{9 a^4}-\frac {2 x^2 \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{9 a^2}+\frac {2 x \sqrt {1-a x} \text {arccosh}(a x)^2}{a^3 \sqrt {-1+a x}}+\frac {x^3 \sqrt {1-a x} \text {arccosh}(a x)^2}{3 a \sqrt {-1+a x}}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{3 a^2} \] Output:

40/9*x*(-a*x+1)^(1/2)/a^3/(a*x-1)^(1/2)+2/27*x^3*(-a*x+1)^(1/2)/a/(a*x-1)^ 
(1/2)-40/9*(-a*x+1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)/a^4-2/9*x^2*(-a*x+1)^ 
(1/2)*(a*x+1)^(1/2)*arccosh(a*x)/a^2+2*x*(-a*x+1)^(1/2)*arccosh(a*x)^2/a^3 
/(a*x-1)^(1/2)+1/3*x^3*(-a*x+1)^(1/2)*arccosh(a*x)^2/a/(a*x-1)^(1/2)-2/3*( 
-a^2*x^2+1)^(1/2)*arccosh(a*x)^3/a^4-1/3*x^2*(-a^2*x^2+1)^(1/2)*arccosh(a* 
x)^3/a^2
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.58 \[ \int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (2 a x \left (60+a^2 x^2\right )-6 \sqrt {-1+a x} \sqrt {1+a x} \left (20+a^2 x^2\right ) \text {arccosh}(a x)+9 a x \left (6+a^2 x^2\right ) \text {arccosh}(a x)^2-9 \sqrt {-1+a x} \sqrt {1+a x} \left (2+a^2 x^2\right ) \text {arccosh}(a x)^3\right )}{27 a^4 \sqrt {-1+a x} \sqrt {1+a x}} \] Input:

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

Output:

(Sqrt[1 - a^2*x^2]*(2*a*x*(60 + a^2*x^2) - 6*Sqrt[-1 + a*x]*Sqrt[1 + a*x]* 
(20 + a^2*x^2)*ArcCosh[a*x] + 9*a*x*(6 + a^2*x^2)*ArcCosh[a*x]^2 - 9*Sqrt[ 
-1 + a*x]*Sqrt[1 + a*x]*(2 + a^2*x^2)*ArcCosh[a*x]^3))/(27*a^4*Sqrt[-1 + a 
*x]*Sqrt[1 + a*x])
 

Rubi [A] (verified)

Time = 2.06 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6353, 6298, 6329, 6294, 6330, 24, 6354, 15, 6330, 24}

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.

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6329

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

\(\Big \downarrow \) 6294

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

\(\Big \downarrow \) 6330

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 15

\(\displaystyle -\frac {\sqrt {a x-1} \left (\frac {1}{3} x^3 \text {arccosh}(a x)^2-\frac {2}{3} a \left (\frac {2 \int \frac {x \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{3 a^2}-\frac {x^3}{9 a}\right )\right )}{a \sqrt {1-a x}}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{a^2}-\frac {3 \sqrt {a x-1} \left (x \text {arccosh}(a x)^2-2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{a^2}-\frac {x}{a}\right )\right )}{a \sqrt {1-a x}}\right )}{3 a^2}\)

\(\Big \downarrow \) 6330

\(\displaystyle -\frac {\sqrt {a x-1} \left (\frac {1}{3} x^3 \text {arccosh}(a x)^2-\frac {2}{3} a \left (\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{a^2}-\frac {\int 1dx}{a}\right )}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{3 a^2}-\frac {x^3}{9 a}\right )\right )}{a \sqrt {1-a x}}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{a^2}-\frac {3 \sqrt {a x-1} \left (x \text {arccosh}(a x)^2-2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{a^2}-\frac {x}{a}\right )\right )}{a \sqrt {1-a x}}\right )}{3 a^2}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{a^2}-\frac {3 \sqrt {a x-1} \left (x \text {arccosh}(a x)^2-2 a \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{a^2}-\frac {x}{a}\right )\right )}{a \sqrt {1-a x}}\right )}{3 a^2}-\frac {\sqrt {a x-1} \left (\frac {1}{3} x^3 \text {arccosh}(a x)^2-\frac {2}{3} a \left (\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{3 a^2}+\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{a^2}-\frac {x}{a}\right )}{3 a^2}-\frac {x^3}{9 a}\right )\right )}{a \sqrt {1-a x}}\)

Input:

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

Output:

-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/a^2 - (Sqrt[-1 + a*x]*((x^3*Ar 
cCosh[a*x]^2)/3 - (2*a*(-1/9*x^3/a + (x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Arc 
Cosh[a*x])/(3*a^2) + (2*(-(x/a) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a* 
x])/a^2))/(3*a^2)))/3))/(a*Sqrt[1 - a*x]) + (2*(-((Sqrt[1 - a^2*x^2]*ArcCo 
sh[a*x]^3)/a^2) - (3*Sqrt[-1 + a*x]*(x*ArcCosh[a*x]^2 - 2*a*(-(x/a) + (Sqr 
t[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/a^2)))/(a*Sqrt[1 - a*x])))/(3*a^2)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 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 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]
 

rule 6354

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 
1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( 
m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1)))   Int[(f*x)^(m 
 - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f 
*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*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*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, 
p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N 
eQ[m + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.54

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}+4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +1\right ) \left (9 \operatorname {arccosh}\left (a x \right )^{3}-9 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-2\right )}{216 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x -1}\, \sqrt {a x +1}\, a x +a^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (a x \right )^{3}-3 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-6\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x -1\right ) \left (\operatorname {arccosh}\left (a x \right )^{3}+3 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+6\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}-4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}+3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +1\right ) \left (9 \operatorname {arccosh}\left (a x \right )^{3}+9 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+2\right )}{216 a^{4} \left (a^{2} x^{2}-1\right )}\)	\(375\)
orering	\(\text {Expression too large to display}\)	\(1192\)

Input:

int(x^3*arccosh(a*x)^3/(-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)^3-9*arccos 
h(a*x)^2+6*arccosh(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)^3-3*arccosh(a*x)^2+6*arc 
cosh(a*x)-6)/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)^3+3*arccosh(a*x)^2+6*arccosh(a*x)+6)/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)^3+9*arccosh(a*x)^2+6*arccosh(a*x)+2)/a^4/(a^2*x^2-1)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.84 \[ \int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {9 \, {\left (a^{4} x^{4} + a^{2} x^{2} - 2\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 9 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 6 \, {\left (a^{4} x^{4} + 19 \, a^{2} x^{2} - 20\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (a^{3} x^{3} + 60 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1}}{27 \, {\left (a^{6} x^{2} - a^{4}\right )}} \] Input:

integrate(x^3*arccosh(a*x)^3/(-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))^3 - 9*(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 + 6*(a^4*x^4 + 19*a^2*x^2 - 20)*sqrt(-a^2*x^2 + 1 
)*log(a*x + sqrt(a^2*x^2 - 1)) - 2*(a^3*x^3 + 60*a*x)*sqrt(a^2*x^2 - 1)*sq 
rt(-a^2*x^2 + 1))/(a^6*x^2 - a^4)
 

Sympy [F]

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

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

Output:

Integral(x**3*acosh(a*x)**3/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 = 131, normalized size of antiderivative = 0.54 \[ \int \frac {x^3 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arcosh}\left (a x\right )^{3} + \frac {2}{27} \, a {\left (\frac {3 \, {\left (-i \, \sqrt {a^{2} x^{2} - 1} x^{2} - \frac {20 i \, \sqrt {a^{2} x^{2} - 1}}{a^{2}}\right )} \operatorname {arcosh}\left (a x\right )}{a^{3}} + \frac {i \, a^{2} x^{3} + 60 i \, x}{a^{4}}\right )} + \frac {{\left (i \, a^{2} x^{3} + 6 i \, x\right )} \operatorname {arcosh}\left (a x\right )^{2}}{3 \, a^{3}} \] Input:

integrate(x^3*arccosh(a*x)^3/(-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)^ 
3 + 2/27*a*(3*(-I*sqrt(a^2*x^2 - 1)*x^2 - 20*I*sqrt(a^2*x^2 - 1)/a^2)*arcc 
osh(a*x)/a^3 + (I*a^2*x^3 + 60*I*x)/a^4) + 1/3*(I*a^2*x^3 + 6*I*x)*arccosh 
(a*x)^2/a^3
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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