3.99 \(\int \frac{x^4}{\sinh ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=188 \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}+\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac{\sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}-\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac{2 x^4 \sqrt{a^2 x^2+1}}{a \sqrt{\sinh ^{-1}(a x)}} \]

[Out]

(-2*x^4*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]]) - (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(8*a^5) + (3*Sqrt[3*Pi]
*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(16*a^5) - (Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(16*a^5) + (Sqrt[Pi]
*Erfi[Sqrt[ArcSinh[a*x]]])/(8*a^5) - (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(16*a^5) + (Sqrt[5*Pi]*Er
fi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(16*a^5)

________________________________________________________________________________________

Rubi [A]  time = 0.185115, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5665, 3308, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}+\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac{\sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}-\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac{2 x^4 \sqrt{a^2 x^2+1}}{a \sqrt{\sinh ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x^4*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]]) - (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(8*a^5) + (3*Sqrt[3*Pi]
*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(16*a^5) - (Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(16*a^5) + (Sqrt[Pi]
*Erfi[Sqrt[ArcSinh[a*x]]])/(8*a^5) - (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(16*a^5) + (Sqrt[5*Pi]*Er
fi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(16*a^5)

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)^{3/2}} \, dx &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}+\frac{2 \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 )}{a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}-\frac{9 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}-\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}+\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}-\frac{5 \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac{9 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}-\frac{9 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}-\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^5}+\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4 a^5}-\frac{5 \operatorname{Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}+\frac{5 \operatorname{Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}+\frac{9 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}-\frac{9 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}\\ &=-\frac{2 x^4 \sqrt{1+a^2 x^2}}{a \sqrt{\sinh ^{-1}(a x)}}-\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}+\frac{3 \sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}-\frac{\sqrt{5 \pi } \text{erf}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{8 a^5}-\frac{3 \sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{5 \pi } \text{erfi}\left (\sqrt{5} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a^5}\\ \end{align*}

Mathematica [A]  time = 0.286323, size = 216, normalized size = 1.15 \[ \frac{\sqrt{5} \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-5 \sinh ^{-1}(a x)\right )-3 \sqrt{3} \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 \sinh ^{-1}(a x)\right )+2 \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-\sinh ^{-1}(a x)\right )+2 \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},\sinh ^{-1}(a x)\right )-3 \sqrt{3} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 \sinh ^{-1}(a x)\right )+\sqrt{5} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},5 \sinh ^{-1}(a x)\right )-e^{-5 \sinh ^{-1}(a x)}+3 e^{-3 \sinh ^{-1}(a x)}-2 e^{-\sinh ^{-1}(a x)}-2 e^{\sinh ^{-1}(a x)}+3 e^{3 \sinh ^{-1}(a x)}-e^{5 \sinh ^{-1}(a x)}}{16 a^5 \sqrt{\sinh ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-E^(-5*ArcSinh[a*x]) + 3/E^(3*ArcSinh[a*x]) - 2/E^ArcSinh[a*x] - 2*E^ArcSinh[a*x] + 3*E^(3*ArcSinh[a*x]) - E^
(5*ArcSinh[a*x]) + Sqrt[5]*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -5*ArcSinh[a*x]] - 3*Sqrt[3]*Sqrt[-ArcSinh[a*x]]*Gam
ma[1/2, -3*ArcSinh[a*x]] + 2*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -ArcSinh[a*x]] + 2*Sqrt[ArcSinh[a*x]]*Gamma[1/2, A
rcSinh[a*x]] - 3*Sqrt[3]*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 3*ArcSinh[a*x]] + Sqrt[5]*Sqrt[ArcSinh[a*x]]*Gamma[1/2,
5*ArcSinh[a*x]])/(16*a^5*Sqrt[ArcSinh[a*x]])

________________________________________________________________________________________

Maple [F]  time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arcsinh(a*x)^(3/2),x)

[Out]

int(x^4/arcsinh(a*x)^(3/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/arcsinh(a*x)^(3/2), x)

________________________________________________________________________________________

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)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{asinh}^{\frac{3}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**4/asinh(a*x)**(3/2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/arcsinh(a*x)^(3/2), x)