\(\int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) [222]

Optimal result

Integrand size = 24, antiderivative size = 315 \[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {45 x^2 \sqrt {1-a x}}{128 a^3 \sqrt {-1+a x}}+\frac {3 x^4 \sqrt {1-a x}}{128 a \sqrt {-1+a x}}-\frac {45 x \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{64 a^4}-\frac {3 x^3 \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{32 a^2}-\frac {45 \sqrt {1-a x} \text {arccosh}(a x)^2}{128 a^5 \sqrt {-1+a x}}+\frac {9 x^2 \sqrt {1-a x} \text {arccosh}(a x)^2}{16 a^3 \sqrt {-1+a x}}+\frac {3 x^4 \sqrt {1-a x} \text {arccosh}(a x)^2}{16 a \sqrt {-1+a x}}-\frac {3 x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}-\frac {3 \sqrt {1-a x} \text {arccosh}(a x)^4}{32 a^5 \sqrt {-1+a x}} \] Output:

45/128*x^2*(-a*x+1)^(1/2)/a^3/(a*x-1)^(1/2)+3/128*x^4*(-a*x+1)^(1/2)/a/(a* 
x-1)^(1/2)-45/64*x*(-a*x+1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)/a^4-3/32*x^3* 
(-a*x+1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)/a^2-45/128*(-a*x+1)^(1/2)*arccos 
h(a*x)^2/a^5/(a*x-1)^(1/2)+9/16*x^2*(-a*x+1)^(1/2)*arccosh(a*x)^2/a^3/(a*x 
-1)^(1/2)+3/16*x^4*(-a*x+1)^(1/2)*arccosh(a*x)^2/a/(a*x-1)^(1/2)-3/8*x*(-a 
^2*x^2+1)^(1/2)*arccosh(a*x)^3/a^4-1/4*x^3*(-a^2*x^2+1)^(1/2)*arccosh(a*x) 
^3/a^2-3/32*(-a*x+1)^(1/2)*arccosh(a*x)^4/a^5/(a*x-1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.43 \[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (-192 \left (1+2 \text {arccosh}(a x)^2\right ) \cosh (2 \text {arccosh}(a x))-3 \left (1+8 \text {arccosh}(a x)^2\right ) \cosh (4 \text {arccosh}(a x))+4 \text {arccosh}(a x) \left (24 \text {arccosh}(a x)^3+32 \left (3+2 \text {arccosh}(a x)^2\right ) \sinh (2 \text {arccosh}(a x))+\left (3+8 \text {arccosh}(a x)^2\right ) \sinh (4 \text {arccosh}(a x))\right )\right )}{1024 a^5 \sqrt {-((-1+a x) (1+a x))}} \] Input:

Integrate[(x^4*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]
 

Output:

(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(-192*(1 + 2*ArcCosh[a*x]^2)*Cosh[2* 
ArcCosh[a*x]] - 3*(1 + 8*ArcCosh[a*x]^2)*Cosh[4*ArcCosh[a*x]] + 4*ArcCosh[ 
a*x]*(24*ArcCosh[a*x]^3 + 32*(3 + 2*ArcCosh[a*x]^2)*Sinh[2*ArcCosh[a*x]] + 
 (3 + 8*ArcCosh[a*x]^2)*Sinh[4*ArcCosh[a*x]])))/(1024*a^5*Sqrt[-((-1 + a*x 
)*(1 + a*x))])
 

Rubi [A] (verified)

Time = 4.45 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6353, 6298, 6353, 6298, 6307, 6354, 15, 6308, 6354, 15, 6308}

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.

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6307

\(\displaystyle \frac {3 \left (-\frac {3 \sqrt {a x-1} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{2 a \sqrt {1-a x}}+\frac {\sqrt {a x-1} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}\right )}{4 a^2}-\frac {3 \sqrt {a x-1} \left (\frac {1}{4} x^4 \text {arccosh}(a x)^2-\frac {1}{2} a \int \frac {x^4 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\right )}{4 a \sqrt {1-a x}}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}\)

\(\Big \downarrow \) 6354

\(\displaystyle -\frac {3 \sqrt {a x-1} \left (\frac {1}{4} x^4 \text {arccosh}(a x)^2-\frac {1}{2} a \left (\frac {3 \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}-\frac {\int x^3dx}{4 a}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{4 a^2}\right )\right )}{4 a \sqrt {1-a x}}+\frac {3 \left (-\frac {3 \sqrt {a x-1} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}-\frac {\int xdx}{2 a}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}\right )\right )}{2 a \sqrt {1-a x}}+\frac {\sqrt {a x-1} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {3 \sqrt {a x-1} \left (\frac {1}{4} x^4 \text {arccosh}(a x)^2-\frac {1}{2} a \left (\frac {3 \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )\right )}{4 a \sqrt {1-a x}}+\frac {3 \left (-\frac {3 \sqrt {a x-1} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 a \sqrt {1-a x}}+\frac {\sqrt {a x-1} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}\)

\(\Big \downarrow \) 6308

\(\displaystyle -\frac {3 \sqrt {a x-1} \left (\frac {1}{4} x^4 \text {arccosh}(a x)^2-\frac {1}{2} a \left (\frac {3 \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )\right )}{4 a \sqrt {1-a x}}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}+\frac {3 \left (\frac {\sqrt {a x-1} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}-\frac {3 \sqrt {a x-1} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\text {arccosh}(a x)^2}{4 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 a \sqrt {1-a x}}\right )}{4 a^2}\)

\(\Big \downarrow \) 6354

\(\displaystyle -\frac {3 \sqrt {a x-1} \left (\frac {1}{4} x^4 \text {arccosh}(a x)^2-\frac {1}{2} a \left (\frac {3 \left (\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}-\frac {\int xdx}{2 a}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}\right )}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )\right )}{4 a \sqrt {1-a x}}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}+\frac {3 \left (\frac {\sqrt {a x-1} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}-\frac {3 \sqrt {a x-1} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\text {arccosh}(a x)^2}{4 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 a \sqrt {1-a x}}\right )}{4 a^2}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {3 \sqrt {a x-1} \left (\frac {1}{4} x^4 \text {arccosh}(a x)^2-\frac {1}{2} a \left (\frac {3 \left (\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 a^2}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )}{4 a^2}+\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{4 a^2}-\frac {x^4}{16 a}\right )\right )}{4 a \sqrt {1-a x}}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}+\frac {3 \left (\frac {\sqrt {a x-1} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}-\frac {3 \sqrt {a x-1} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\text {arccosh}(a x)^2}{4 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 a \sqrt {1-a x}}\right )}{4 a^2}\)

\(\Big \downarrow \) 6308

\(\displaystyle -\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{4 a^2}+\frac {3 \left (\frac {\sqrt {a x-1} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}-\frac {3 \sqrt {a x-1} \left (\frac {1}{2} x^2 \text {arccosh}(a x)^2-a \left (\frac {\text {arccosh}(a x)^2}{4 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 a \sqrt {1-a x}}\right )}{4 a^2}-\frac {3 \sqrt {a x-1} \left (\frac {1}{4} x^4 \text {arccosh}(a x)^2-\frac {1}{2} a \left (\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{4 a^2}+\frac {3 \left (\frac {\text {arccosh}(a x)^2}{4 a^3}+\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )}{4 a^2}-\frac {x^4}{16 a}\right )\right )}{4 a \sqrt {1-a x}}\)

Input:

Int[(x^4*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]
 

Output:

-1/4*(x^3*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/a^2 - (3*Sqrt[-1 + a*x]*((x^4* 
ArcCosh[a*x]^2)/4 - (a*(-1/16*x^4/a + (x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Ar 
cCosh[a*x])/(4*a^2) + (3*(-1/4*x^2/a + (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Arc 
Cosh[a*x])/(2*a^2) + ArcCosh[a*x]^2/(4*a^3)))/(4*a^2)))/2))/(4*a*Sqrt[1 - 
a*x]) + (3*(-1/2*(x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/a^2 + (Sqrt[-1 + a*x 
]*ArcCosh[a*x]^4)/(8*a^3*Sqrt[1 - a*x]) - (3*Sqrt[-1 + a*x]*((x^2*ArcCosh[ 
a*x]^2)/2 - a*(-1/4*x^2/a + (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/ 
(2*a^2) + ArcCosh[a*x]^2/(4*a^3))))/(2*a*Sqrt[1 - a*x])))/(4*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 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 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && 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]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(519\) vs. \(2(259)=518\).

Time = 0.44 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.65

method	result	size
default	\(-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{4}}{32 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}+8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+4 a x -8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (32 \operatorname {arccosh}\left (a x \right )^{3}-24 \operatorname {arccosh}\left (a x \right )^{2}+12 \,\operatorname {arccosh}\left (a x \right )-3\right )}{2048 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}-6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-3\right )}{32 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (-2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 a^{3} x^{3}+\sqrt {a x -1}\, \sqrt {a x +1}-2 a x \right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}+6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+3\right )}{32 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (-8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+8 a^{5} x^{5}+8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-12 a^{3} x^{3}-\sqrt {a x -1}\, \sqrt {a x +1}+4 a x \right ) \left (32 \operatorname {arccosh}\left (a x \right )^{3}+24 \operatorname {arccosh}\left (a x \right )^{2}+12 \,\operatorname {arccosh}\left (a x \right )+3\right )}{2048 a^{5} \left (a^{2} x^{2}-1\right )}\)	\(520\)

Input:

int(x^4*arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-3/32*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5/(a^2*x^2-1)*arcco 
sh(a*x)^4-1/2048*(-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))*(32*arccosh(a*x)^3-24*arccosh(a*x)^2+12*arccosh(a*x)- 
3)/a^5/(a^2*x^2-1)-1/32*(-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))*(4*arccosh(a*x)^3-6*a 
rccosh(a*x)^2+6*arccosh(a*x)-3)/a^5/(a^2*x^2-1)-1/32*(-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)*(4*arccosh(a*x)^3+6*arccosh(a*x)^2+6*arccosh(a*x)+3)/a^5/(a^2*x^2 
-1)-1/2048*(-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)*(32*arccosh(a*x)^3+24*arccosh(a*x)^2+12*arccosh(a*x)+3)/a^ 
5/(a^2*x^2-1)
 

Fricas [F]

\[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^4*arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)*x^4*arccosh(a*x)^3/(a^2*x^2 - 1), x)
 

Sympy [F]

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

integrate(x**4*acosh(a*x)**3/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral(x**4*acosh(a*x)**3/sqrt(-(a*x - 1)*(a*x + 1)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4*arccosh(a*x)^3/(-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]

\[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^4*arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^4*arccosh(a*x)^3/sqrt(-a^2*x^2 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^4\,{\mathrm {acosh}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \] Input:

int((x^4*acosh(a*x)^3)/(1 - a^2*x^2)^(1/2),x)
 

Output:

int((x^4*acosh(a*x)^3)/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^4 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acosh} \left (a x \right )^{3} x^{4}}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:

int(x^4*acosh(a*x)^3/(-a^2*x^2+1)^(1/2),x)
 

Output:

int((acosh(a*x)**3*x**4)/sqrt( - a**2*x**2 + 1),x)