\(\int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx\) [341]

Optimal result

Integrand size = 23, antiderivative size = 187 \[ \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {45 x^2}{128 a^3}-\frac {3 x^4}{128 a}-\frac {45 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{64 a^4}+\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{32 a^2}+\frac {45 \text {arcsinh}(a x)^2}{128 a^5}+\frac {9 x^2 \text {arcsinh}(a x)^2}{16 a^3}-\frac {3 x^4 \text {arcsinh}(a x)^2}{16 a}-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{8 a^4}+\frac {x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{4 a^2}+\frac {3 \text {arcsinh}(a x)^4}{32 a^5} \] Output:

45/128*x^2/a^3-3/128*x^4/a-45/64*x*(a^2*x^2+1)^(1/2)*arcsinh(a*x)/a^4+3/32 
*x^3*(a^2*x^2+1)^(1/2)*arcsinh(a*x)/a^2+45/128*arcsinh(a*x)^2/a^5+9/16*x^2 
*arcsinh(a*x)^2/a^3-3/16*x^4*arcsinh(a*x)^2/a-3/8*x*(a^2*x^2+1)^(1/2)*arcs 
inh(a*x)^3/a^4+1/4*x^3*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^3/a^2+3/32*arcsinh(a 
*x)^4/a^5
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.65 \[ \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {45 a^2 x^2-3 a^4 x^4+6 a x \sqrt {1+a^2 x^2} \left (-15+2 a^2 x^2\right ) \text {arcsinh}(a x)+\left (45+72 a^2 x^2-24 a^4 x^4\right ) \text {arcsinh}(a x)^2+16 a x \sqrt {1+a^2 x^2} \left (-3+2 a^2 x^2\right ) \text {arcsinh}(a x)^3+12 \text {arcsinh}(a x)^4}{128 a^5} \] Input:

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

Output:

(45*a^2*x^2 - 3*a^4*x^4 + 6*a*x*Sqrt[1 + a^2*x^2]*(-15 + 2*a^2*x^2)*ArcSin 
h[a*x] + (45 + 72*a^2*x^2 - 24*a^4*x^4)*ArcSinh[a*x]^2 + 16*a*x*Sqrt[1 + a 
^2*x^2]*(-3 + 2*a^2*x^2)*ArcSinh[a*x]^3 + 12*ArcSinh[a*x]^4)/(128*a^5)
 

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.45, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6227, 6191, 6227, 15, 6191, 6198, 6227, 15, 6198}

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.

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 6198

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6198

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

Input:

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

Output:

(x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(4*a^2) - (3*((x^4*ArcSinh[a*x]^2)/ 
4 - (a*(-1/16*x^4/a + (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(4*a^2) - (3*(- 
1/4*x^2/a + (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a^2) - ArcSinh[a*x]^2/(4 
*a^3)))/(4*a^2)))/2))/(4*a) - (3*((x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(2* 
a^2) - ArcSinh[a*x]^4/(8*a^3) - (3*((x^2*ArcSinh[a*x]^2)/2 - a*(-1/4*x^2/a 
 + (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a^2) - ArcSinh[a*x]^2/(4*a^3))))/ 
(2*a)))/(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 6191

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + 
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 6198

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

rule 6227

Int[((a_.) + ArcSinh[(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*ArcSinh[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*ArcSinh[c*x])^n, x], x] 
 - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int 
[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] 
) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ 
m, 1] && NeQ[m + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 1.72 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {32 \operatorname {arcsinh}\left (x a \right )^{3} \sqrt {a^{2} x^{2}+1}\, x^{3} a^{3}-24 x^{4} a^{4} \operatorname {arcsinh}\left (x a \right )^{2}+12 x^{3} a^{3} \operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}-3 a^{4} x^{4}-48 \operatorname {arcsinh}\left (x a \right )^{3} \sqrt {a^{2} x^{2}+1}\, x a +72 a^{2} x^{2} \operatorname {arcsinh}\left (x a \right )^{2}+12 \operatorname {arcsinh}\left (x a \right )^{4}-90 \,\operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}\, x a +45 a^{2} x^{2}+45 \operatorname {arcsinh}\left (x a \right )^{2}+45}{128 a^{5}}\)	\(156\)

Input:

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

Output:

1/128*(32*arcsinh(x*a)^3*(a^2*x^2+1)^(1/2)*x^3*a^3-24*x^4*a^4*arcsinh(x*a) 
^2+12*x^3*a^3*arcsinh(x*a)*(a^2*x^2+1)^(1/2)-3*a^4*x^4-48*arcsinh(x*a)^3*( 
a^2*x^2+1)^(1/2)*x*a+72*a^2*x^2*arcsinh(x*a)^2+12*arcsinh(x*a)^4-90*arcsin 
h(x*a)*(a^2*x^2+1)^(1/2)*x*a+45*a^2*x^2+45*arcsinh(x*a)^2+45)/a^5
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.89 \[ \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=-\frac {3 \, a^{4} x^{4} - 16 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 45 \, a^{2} x^{2} - 12 \, \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} + 3 \, {\left (8 \, a^{4} x^{4} - 24 \, a^{2} x^{2} - 15\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 6 \, {\left (2 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{128 \, a^{5}} \] Input:

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

Output:

-1/128*(3*a^4*x^4 - 16*(2*a^3*x^3 - 3*a*x)*sqrt(a^2*x^2 + 1)*log(a*x + sqr 
t(a^2*x^2 + 1))^3 - 45*a^2*x^2 - 12*log(a*x + sqrt(a^2*x^2 + 1))^4 + 3*(8* 
a^4*x^4 - 24*a^2*x^2 - 15)*log(a*x + sqrt(a^2*x^2 + 1))^2 - 6*(2*a^3*x^3 - 
 15*a*x)*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1)))/a^5
 

Sympy [A] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.99 \[ \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {3 x^{4} \operatorname {asinh}^{2}{\left (a x \right )}}{16 a} - \frac {3 x^{4}}{128 a} + \frac {x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{4 a^{2}} + \frac {3 x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{32 a^{2}} + \frac {9 x^{2} \operatorname {asinh}^{2}{\left (a x \right )}}{16 a^{3}} + \frac {45 x^{2}}{128 a^{3}} - \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{8 a^{4}} - \frac {45 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{64 a^{4}} + \frac {3 \operatorname {asinh}^{4}{\left (a x \right )}}{32 a^{5}} + \frac {45 \operatorname {asinh}^{2}{\left (a x \right )}}{128 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-3*x**4*asinh(a*x)**2/(16*a) - 3*x**4/(128*a) + x**3*sqrt(a**2* 
x**2 + 1)*asinh(a*x)**3/(4*a**2) + 3*x**3*sqrt(a**2*x**2 + 1)*asinh(a*x)/( 
32*a**2) + 9*x**2*asinh(a*x)**2/(16*a**3) + 45*x**2/(128*a**3) - 3*x*sqrt( 
a**2*x**2 + 1)*asinh(a*x)**3/(8*a**4) - 45*x*sqrt(a**2*x**2 + 1)*asinh(a*x 
)/(64*a**4) + 3*asinh(a*x)**4/(32*a**5) + 45*asinh(a*x)**2/(128*a**5), Ne( 
a, 0)), (0, True))
 

Maxima [F]

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

integrate(x^4*arcsinh(a*x)^3/(a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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