\(\int x^4 \text {arcsinh}(a x)^4 \, dx\) [33]

Optimal result

Integrand size = 10, antiderivative size = 244 \[ \int x^4 \text {arcsinh}(a x)^4 \, dx=\frac {16576 x}{5625 a^4}-\frac {1088 x^3}{16875 a^2}+\frac {24 x^5}{3125}-\frac {16576 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{5625 a^5}+\frac {1088 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{5625 a^3}-\frac {24 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{625 a}+\frac {32 x \text {arcsinh}(a x)^2}{25 a^4}-\frac {16 x^3 \text {arcsinh}(a x)^2}{75 a^2}+\frac {12}{125} x^5 \text {arcsinh}(a x)^2-\frac {32 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{75 a^5}+\frac {16 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^4 \]

[Out]

16576/5625*x/a^4-1088/16875*x^3/a^2+24/3125*x^5+32/25*x*arcsinh(a*x)^2/a^4-16/75*x^3*arcsinh(a*x)^2/a^2+12/125
*x^5*arcsinh(a*x)^2+1/5*x^5*arcsinh(a*x)^4-16576/5625*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a^5+1088/5625*x^2*arcsinh
(a*x)*(a^2*x^2+1)^(1/2)/a^3-24/625*x^4*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a-32/75*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)
/a^5+16/75*x^2*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)/a^3-4/25*x^4*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5776, 5812, 5798, 5772, 8, 30} \[ \int x^4 \text {arcsinh}(a x)^4 \, dx=\frac {32 x \text {arcsinh}(a x)^2}{25 a^4}+\frac {16576 x}{5625 a^4}-\frac {16 x^3 \text {arcsinh}(a x)^2}{75 a^2}-\frac {4 x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{25 a}-\frac {24 x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{625 a}-\frac {1088 x^3}{16875 a^2}-\frac {32 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{75 a^5}-\frac {16576 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{5625 a^5}+\frac {16 x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{75 a^3}+\frac {1088 x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{5625 a^3}+\frac {1}{5} x^5 \text {arcsinh}(a x)^4+\frac {12}{125} x^5 \text {arcsinh}(a x)^2+\frac {24 x^5}{3125} \]

[In]

Int[x^4*ArcSinh[a*x]^4,x]

[Out]

(16576*x)/(5625*a^4) - (1088*x^3)/(16875*a^2) + (24*x^5)/3125 - (16576*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(5625*a
^5) + (1088*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(5625*a^3) - (24*x^4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(625*a) +
(32*x*ArcSinh[a*x]^2)/(25*a^4) - (16*x^3*ArcSinh[a*x]^2)/(75*a^2) + (12*x^5*ArcSinh[a*x]^2)/125 - (32*Sqrt[1
+ a^2*x^2]*ArcSinh[a*x]^3)/(75*a^5) + (16*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(75*a^3) - (4*x^4*Sqrt[1 + a^2
*x^2]*ArcSinh[a*x]^3)/(25*a) + (x^5*ArcSinh[a*x]^4)/5

Rule 8

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

Rule 30

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

Rule 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[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 5798

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

Rule 5812

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] + (-Dist[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] - Dist[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
]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \text {arcsinh}(a x)^4-\frac {1}{5} (4 a) \int \frac {x^5 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {4 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^4+\frac {12}{25} \int x^4 \text {arcsinh}(a x)^2 \, dx+\frac {16 \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{25 a} \\ & = \frac {12}{125} x^5 \text {arcsinh}(a x)^2+\frac {16 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^4-\frac {32 \int \frac {x \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{75 a^3}-\frac {16 \int x^2 \text {arcsinh}(a x)^2 \, dx}{25 a^2}-\frac {1}{125} (24 a) \int \frac {x^5 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {24 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{625 a}-\frac {16 x^3 \text {arcsinh}(a x)^2}{75 a^2}+\frac {12}{125} x^5 \text {arcsinh}(a x)^2-\frac {32 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{75 a^5}+\frac {16 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^4+\frac {24 \int x^4 \, dx}{625}+\frac {32 \int \text {arcsinh}(a x)^2 \, dx}{25 a^4}+\frac {96 \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{625 a}+\frac {32 \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{75 a} \\ & = \frac {24 x^5}{3125}+\frac {1088 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{5625 a^3}-\frac {24 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{625 a}+\frac {32 x \text {arcsinh}(a x)^2}{25 a^4}-\frac {16 x^3 \text {arcsinh}(a x)^2}{75 a^2}+\frac {12}{125} x^5 \text {arcsinh}(a x)^2-\frac {32 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{75 a^5}+\frac {16 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^4-\frac {64 \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{625 a^3}-\frac {64 \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{225 a^3}-\frac {64 \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{25 a^3}-\frac {32 \int x^2 \, dx}{625 a^2}-\frac {32 \int x^2 \, dx}{225 a^2} \\ & = -\frac {1088 x^3}{16875 a^2}+\frac {24 x^5}{3125}-\frac {16576 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{5625 a^5}+\frac {1088 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{5625 a^3}-\frac {24 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{625 a}+\frac {32 x \text {arcsinh}(a x)^2}{25 a^4}-\frac {16 x^3 \text {arcsinh}(a x)^2}{75 a^2}+\frac {12}{125} x^5 \text {arcsinh}(a x)^2-\frac {32 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{75 a^5}+\frac {16 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^4+\frac {64 \int 1 \, dx}{625 a^4}+\frac {64 \int 1 \, dx}{225 a^4}+\frac {64 \int 1 \, dx}{25 a^4} \\ & = \frac {16576 x}{5625 a^4}-\frac {1088 x^3}{16875 a^2}+\frac {24 x^5}{3125}-\frac {16576 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{5625 a^5}+\frac {1088 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{5625 a^3}-\frac {24 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{625 a}+\frac {32 x \text {arcsinh}(a x)^2}{25 a^4}-\frac {16 x^3 \text {arcsinh}(a x)^2}{75 a^2}+\frac {12}{125} x^5 \text {arcsinh}(a x)^2-\frac {32 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{75 a^5}+\frac {16 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.61 \[ \int x^4 \text {arcsinh}(a x)^4 \, dx=\frac {8 a x \left (31080-680 a^2 x^2+81 a^4 x^4\right )-120 \sqrt {1+a^2 x^2} \left (2072-136 a^2 x^2+27 a^4 x^4\right ) \text {arcsinh}(a x)+900 a x \left (120-20 a^2 x^2+9 a^4 x^4\right ) \text {arcsinh}(a x)^2-4500 \sqrt {1+a^2 x^2} \left (8-4 a^2 x^2+3 a^4 x^4\right ) \text {arcsinh}(a x)^3+16875 a^5 x^5 \text {arcsinh}(a x)^4}{84375 a^5} \]

[In]

Integrate[x^4*ArcSinh[a*x]^4,x]

[Out]

(8*a*x*(31080 - 680*a^2*x^2 + 81*a^4*x^4) - 120*Sqrt[1 + a^2*x^2]*(2072 - 136*a^2*x^2 + 27*a^4*x^4)*ArcSinh[a*
x] + 900*a*x*(120 - 20*a^2*x^2 + 9*a^4*x^4)*ArcSinh[a*x]^2 - 4500*Sqrt[1 + a^2*x^2]*(8 - 4*a^2*x^2 + 3*a^4*x^4
)*ArcSinh[a*x]^3 + 16875*a^5*x^5*ArcSinh[a*x]^4)/(84375*a^5)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {\frac {a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )^{4}}{5}-\frac {32 \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{75}-\frac {4 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{25}+\frac {16 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{75}+\frac {32 a x \operatorname {arcsinh}\left (a x \right )^{2}}{25}-\frac {16576 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}}{5625}+\frac {16576 a x}{5625}+\frac {12 a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )^{2}}{125}-\frac {24 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}}{625}+\frac {1088 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}}{5625}+\frac {24 a^{5} x^{5}}{3125}-\frac {1088 a^{3} x^{3}}{16875}-\frac {16 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{2}}{75}}{a^{5}}\)	\(210\)
default	\(\frac {\frac {a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )^{4}}{5}-\frac {32 \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{75}-\frac {4 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{25}+\frac {16 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}}{75}+\frac {32 a x \operatorname {arcsinh}\left (a x \right )^{2}}{25}-\frac {16576 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}}{5625}+\frac {16576 a x}{5625}+\frac {12 a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )^{2}}{125}-\frac {24 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}}{625}+\frac {1088 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}}{5625}+\frac {24 a^{5} x^{5}}{3125}-\frac {1088 a^{3} x^{3}}{16875}-\frac {16 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{2}}{75}}{a^{5}}\)	\(210\)

[In]

int(x^4*arcsinh(a*x)^4,x,method=_RETURNVERBOSE)

[Out]

1/a^5*(1/5*a^5*x^5*arcsinh(a*x)^4-32/75*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)-4/25*a^4*x^4*arcsinh(a*x)^3*(a^2*x^2+
1)^(1/2)+16/75*a^2*x^2*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)+32/25*a*x*arcsinh(a*x)^2-16576/5625*arcsinh(a*x)*(a^2*
x^2+1)^(1/2)+16576/5625*a*x+12/125*a^5*x^5*arcsinh(a*x)^2-24/625*a^4*x^4*arcsinh(a*x)*(a^2*x^2+1)^(1/2)+1088/5
625*arcsinh(a*x)*(a^2*x^2+1)^(1/2)*a^2*x^2+24/3125*a^5*x^5-1088/16875*a^3*x^3-16/75*a^3*x^3*arcsinh(a*x)^2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.77 \[ \int x^4 \text {arcsinh}(a x)^4 \, dx=\frac {16875 \, a^{5} x^{5} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} + 648 \, a^{5} x^{5} - 5440 \, a^{3} x^{3} - 4500 \, {\left (3 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 8\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 900 \, {\left (9 \, a^{5} x^{5} - 20 \, a^{3} x^{3} + 120 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 120 \, {\left (27 \, a^{4} x^{4} - 136 \, a^{2} x^{2} + 2072\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + 248640 \, a x}{84375 \, a^{5}} \]

[In]

integrate(x^4*arcsinh(a*x)^4,x, algorithm="fricas")

[Out]

1/84375*(16875*a^5*x^5*log(a*x + sqrt(a^2*x^2 + 1))^4 + 648*a^5*x^5 - 5440*a^3*x^3 - 4500*(3*a^4*x^4 - 4*a^2*x
^2 + 8)*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^3 + 900*(9*a^5*x^5 - 20*a^3*x^3 + 120*a*x)*log(a*x + sq
rt(a^2*x^2 + 1))^2 - 120*(27*a^4*x^4 - 136*a^2*x^2 + 2072)*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1)) + 24
8640*a*x)/a^5

Sympy [A] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.99 \[ \int x^4 \text {arcsinh}(a x)^4 \, dx=\begin {cases} \frac {x^{5} \operatorname {asinh}^{4}{\left (a x \right )}}{5} + \frac {12 x^{5} \operatorname {asinh}^{2}{\left (a x \right )}}{125} + \frac {24 x^{5}}{3125} - \frac {4 x^{4} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{25 a} - \frac {24 x^{4} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{625 a} - \frac {16 x^{3} \operatorname {asinh}^{2}{\left (a x \right )}}{75 a^{2}} - \frac {1088 x^{3}}{16875 a^{2}} + \frac {16 x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{75 a^{3}} + \frac {1088 x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{5625 a^{3}} + \frac {32 x \operatorname {asinh}^{2}{\left (a x \right )}}{25 a^{4}} + \frac {16576 x}{5625 a^{4}} - \frac {32 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{75 a^{5}} - \frac {16576 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{5625 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**4*asinh(a*x)**4,x)

[Out]

Piecewise((x**5*asinh(a*x)**4/5 + 12*x**5*asinh(a*x)**2/125 + 24*x**5/3125 - 4*x**4*sqrt(a**2*x**2 + 1)*asinh(
a*x)**3/(25*a) - 24*x**4*sqrt(a**2*x**2 + 1)*asinh(a*x)/(625*a) - 16*x**3*asinh(a*x)**2/(75*a**2) - 1088*x**3/
(16875*a**2) + 16*x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)**3/(75*a**3) + 1088*x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)/
(5625*a**3) + 32*x*asinh(a*x)**2/(25*a**4) + 16576*x/(5625*a**4) - 32*sqrt(a**2*x**2 + 1)*asinh(a*x)**3/(75*a*
*5) - 16576*sqrt(a**2*x**2 + 1)*asinh(a*x)/(5625*a**5), Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.82 \[ \int x^4 \text {arcsinh}(a x)^4 \, dx=\frac {1}{5} \, x^{5} \operatorname {arsinh}\left (a x\right )^{4} - \frac {4}{75} \, {\left (\frac {3 \, \sqrt {a^{2} x^{2} + 1} x^{4}}{a^{2}} - \frac {4 \, \sqrt {a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {a^{2} x^{2} + 1}}{a^{6}}\right )} a \operatorname {arsinh}\left (a x\right )^{3} - \frac {4}{84375} \, {\left (2 \, a {\left (\frac {15 \, {\left (27 \, \sqrt {a^{2} x^{2} + 1} a^{2} x^{4} - 136 \, \sqrt {a^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {a^{2} x^{2} + 1}}{a^{2}}\right )} \operatorname {arsinh}\left (a x\right )}{a^{5}} - \frac {81 \, a^{4} x^{5} - 680 \, a^{2} x^{3} + 31080 \, x}{a^{6}}\right )} - \frac {225 \, {\left (9 \, a^{4} x^{5} - 20 \, a^{2} x^{3} + 120 \, x\right )} \operatorname {arsinh}\left (a x\right )^{2}}{a^{5}}\right )} a \]

[In]

integrate(x^4*arcsinh(a*x)^4,x, algorithm="maxima")

[Out]

1/5*x^5*arcsinh(a*x)^4 - 4/75*(3*sqrt(a^2*x^2 + 1)*x^4/a^2 - 4*sqrt(a^2*x^2 + 1)*x^2/a^4 + 8*sqrt(a^2*x^2 + 1)
/a^6)*a*arcsinh(a*x)^3 - 4/84375*(2*a*(15*(27*sqrt(a^2*x^2 + 1)*a^2*x^4 - 136*sqrt(a^2*x^2 + 1)*x^2 + 2072*sqr
t(a^2*x^2 + 1)/a^2)*arcsinh(a*x)/a^5 - (81*a^4*x^5 - 680*a^2*x^3 + 31080*x)/a^6) - 225*(9*a^4*x^5 - 20*a^2*x^3
+ 120*x)*arcsinh(a*x)^2/a^5)*a

Giac [F(-2)]

Exception generated. \[ \int x^4 \text {arcsinh}(a x)^4 \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^4*arcsinh(a*x)^4,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^4 \text {arcsinh}(a x)^4 \, dx=\int x^4\,{\mathrm {asinh}\left (a\,x\right )}^4 \,d x \]

[In]

int(x^4*asinh(a*x)^4,x)

[Out]

int(x^4*asinh(a*x)^4, x)