3.111 \(\int \frac{x^4}{\sinh ^{-1}(a x)^{7/2}} \, dx\)

Optimal. Leaf size=285 \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{30 a^5}+\frac{9 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{20 a^5}-\frac{5 \sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{30 a^5}-\frac{9 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{20 a^5}+\frac{5 \sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}-\frac{40 x^4 \sqrt{a^2 x^2+1}}{3 a \sqrt{\sinh ^{-1}(a x)}}-\frac{2 x^4 \sqrt{a^2 x^2+1}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{a^2 x^2+1}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}} \]

[Out]

(-2*x^4*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2)) - (16*x^3)/(15*a^2*ArcSinh[a*x]^(3/2)) - (4*x^5)/(3*ArcSin
h[a*x]^(3/2)) - (32*x^2*Sqrt[1 + a^2*x^2])/(5*a^3*Sqrt[ArcSinh[a*x]]) - (40*x^4*Sqrt[1 + a^2*x^2])/(3*a*Sqrt[A
rcSinh[a*x]]) - (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(30*a^5) + (9*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(
20*a^5) - (5*Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(12*a^5) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(30*a^
5) - (9*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(20*a^5) + (5*Sqrt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]]
)/(12*a^5)

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Rubi [A]  time = 0.543711, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5667, 5774, 5665, 3308, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{30 a^5}+\frac{9 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{20 a^5}-\frac{5 \sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{30 a^5}-\frac{9 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{20 a^5}+\frac{5 \sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}-\frac{40 x^4 \sqrt{a^2 x^2+1}}{3 a \sqrt{\sinh ^{-1}(a x)}}-\frac{2 x^4 \sqrt{a^2 x^2+1}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{a^2 x^2+1}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcSinh[a*x]^(7/2),x]

[Out]

(-2*x^4*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2)) - (16*x^3)/(15*a^2*ArcSinh[a*x]^(3/2)) - (4*x^5)/(3*ArcSin
h[a*x]^(3/2)) - (32*x^2*Sqrt[1 + a^2*x^2])/(5*a^3*Sqrt[ArcSinh[a*x]]) - (40*x^4*Sqrt[1 + a^2*x^2])/(3*a*Sqrt[A
rcSinh[a*x]]) - (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(30*a^5) + (9*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(
20*a^5) - (5*Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(12*a^5) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(30*a^
5) - (9*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(20*a^5) + (5*Sqrt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]]
)/(12*a^5)

Rule 5667

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

Rule 5774

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

Rule 5665

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 + c^2*x^2]*(a + b*ArcSi
nh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n +
1), Sinh[x]^(m - 1)*(m + (m + 1)*Sinh[x]^2), x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0]
&& GeQ[n, -2] && LtQ[n, -1]

Rule 3308

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{x^4}{\sinh ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}+\frac{8 \int \frac{x^3}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{5/2}} \, dx}{5 a}+(2 a) \int \frac{x^5}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}}+\frac{20}{3} \int \frac{x^4}{\sinh ^{-1}(a x)^{3/2}} \, dx+\frac{16 \int \frac{x^2}{\sinh ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1+a^2 x^2}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1+a^2 x^2}}{3 a \sqrt{\sinh ^{-1}(a x)}}+\frac{32 \operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{4 \sqrt{x}}+\frac{3 \sinh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}+\frac{40 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{8 \sqrt{x}}-\frac{9 \sinh (3 x)}{16 \sqrt{x}}+\frac{5 \sinh (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1+a^2 x^2}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1+a^2 x^2}}{3 a \sqrt{\sinh ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^5}+\frac{24 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}-\frac{15 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1+a^2 x^2}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1+a^2 x^2}}{3 a \sqrt{\sinh ^{-1}(a x)}}+\frac{4 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}-\frac{4 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}-\frac{5 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^5}-\frac{25 \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^5}-\frac{12 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}+\frac{12 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a^5}+\frac{15 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^5}-\frac{15 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1+a^2 x^2}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1+a^2 x^2}}{3 a \sqrt{\sinh ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^5}-\frac{8 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^5}-\frac{5 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3 a^5}+\frac{5 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3 a^5}-\frac{25 \operatorname{Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{6 a^5}+\frac{25 \operatorname{Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{6 a^5}-\frac{24 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^5}+\frac{24 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{5 a^5}+\frac{15 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{2 a^5}-\frac{15 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{2 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{16 x^3}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{4 x^5}{3 \sinh ^{-1}(a x)^{3/2}}-\frac{32 x^2 \sqrt{1+a^2 x^2}}{5 a^3 \sqrt{\sinh ^{-1}(a x)}}-\frac{40 x^4 \sqrt{1+a^2 x^2}}{3 a \sqrt{\sinh ^{-1}(a x)}}-\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{30 a^5}+\frac{9 \sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{20 a^5}-\frac{5 \sqrt{5 \pi } \text{erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}+\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{30 a^5}-\frac{9 \sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{20 a^5}+\frac{5 \sqrt{5 \pi } \text{erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{12 a^5}\\ \end{align*}

Mathematica [A]  time = 0.673252, size = 334, normalized size = 1.17 \[ \frac{100 \sqrt{5} \left (-\sinh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-5 \sinh ^{-1}(a x)\right )-108 \sqrt{3} \left (-\sinh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-3 \sinh ^{-1}(a x)\right )+8 \left (-\sinh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-\sinh ^{-1}(a x)\right )+e^{-\sinh ^{-1}(a x)} \left (8 e^{\sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},\sinh ^{-1}(a x)\right )-8 \sinh ^{-1}(a x)^2+4 \sinh ^{-1}(a x)-6\right )+9 e^{-3 \sinh ^{-1}(a x)} \left (-12 \sqrt{3} e^{3 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},3 \sinh ^{-1}(a x)\right )+12 \sinh ^{-1}(a x)^2-2 \sinh ^{-1}(a x)+1\right )+e^{-5 \sinh ^{-1}(a x)} \left (100 \sqrt{5} e^{5 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},5 \sinh ^{-1}(a x)\right )-100 \sinh ^{-1}(a x)^2+10 \sinh ^{-1}(a x)-3\right )-2 e^{\sinh ^{-1}(a x)} \left (4 \sinh ^{-1}(a x)^2+2 \sinh ^{-1}(a x)+3\right )+9 e^{3 \sinh ^{-1}(a x)} \left (12 \sinh ^{-1}(a x)^2+2 \sinh ^{-1}(a x)+1\right )-e^{5 \sinh ^{-1}(a x)} \left (100 \sinh ^{-1}(a x)^2+10 \sinh ^{-1}(a x)+3\right )}{240 a^5 \sinh ^{-1}(a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/ArcSinh[a*x]^(7/2),x]

[Out]

(-2*E^ArcSinh[a*x]*(3 + 2*ArcSinh[a*x] + 4*ArcSinh[a*x]^2) + 9*E^(3*ArcSinh[a*x])*(1 + 2*ArcSinh[a*x] + 12*Arc
Sinh[a*x]^2) - E^(5*ArcSinh[a*x])*(3 + 10*ArcSinh[a*x] + 100*ArcSinh[a*x]^2) + 100*Sqrt[5]*(-ArcSinh[a*x])^(5/
2)*Gamma[1/2, -5*ArcSinh[a*x]] - 108*Sqrt[3]*(-ArcSinh[a*x])^(5/2)*Gamma[1/2, -3*ArcSinh[a*x]] + 8*(-ArcSinh[a
*x])^(5/2)*Gamma[1/2, -ArcSinh[a*x]] + (-6 + 4*ArcSinh[a*x] - 8*ArcSinh[a*x]^2 + 8*E^ArcSinh[a*x]*ArcSinh[a*x]
^(5/2)*Gamma[1/2, ArcSinh[a*x]])/E^ArcSinh[a*x] + (9*(1 - 2*ArcSinh[a*x] + 12*ArcSinh[a*x]^2 - 12*Sqrt[3]*E^(3
*ArcSinh[a*x])*ArcSinh[a*x]^(5/2)*Gamma[1/2, 3*ArcSinh[a*x]]))/E^(3*ArcSinh[a*x]) + (-3 + 10*ArcSinh[a*x] - 10
0*ArcSinh[a*x]^2 + 100*Sqrt[5]*E^(5*ArcSinh[a*x])*ArcSinh[a*x]^(5/2)*Gamma[1/2, 5*ArcSinh[a*x]])/E^(5*ArcSinh[
a*x]))/(240*a^5*ArcSinh[a*x]^(5/2))

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Maple [F]  time = 0.18, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arcsinh(a*x)^(7/2),x)

[Out]

int(x^4/arcsinh(a*x)^(7/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{arsinh}\left (a x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsinh(a*x)^(7/2),x, algorithm="maxima")

[Out]

integrate(x^4/arcsinh(a*x)^(7/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsinh(a*x)^(7/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/asinh(a*x)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{arsinh}\left (a x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsinh(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(x^4/arcsinh(a*x)^(7/2), x)