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

Optimal. Leaf size=147 \[ \frac{8 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{15 a^2}+\frac{8 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{15 a^2}-\frac{32 x \sqrt{a^2 x^2+1}}{15 a \sqrt{\sinh ^{-1}(a x)}}-\frac{2 x \sqrt{a^2 x^2+1}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{4}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \sinh ^{-1}(a x)^{3/2}} \]

[Out]

(-2*x*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2)) - 4/(15*a^2*ArcSinh[a*x]^(3/2)) - (8*x^2)/(15*ArcSinh[a*x]^(
3/2)) - (32*x*Sqrt[1 + a^2*x^2])/(15*a*Sqrt[ArcSinh[a*x]]) + (8*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(1
5*a^2) + (8*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(15*a^2)

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Rubi [A]  time = 0.211751, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {5667, 5774, 5665, 3307, 2180, 2204, 2205, 5675} \[ \frac{8 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{15 a^2}+\frac{8 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{15 a^2}-\frac{32 x \sqrt{a^2 x^2+1}}{15 a \sqrt{\sinh ^{-1}(a x)}}-\frac{2 x \sqrt{a^2 x^2+1}}{5 a \sinh ^{-1}(a x)^{5/2}}-\frac{4}{15 a^2 \sinh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \sinh ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2)) - 4/(15*a^2*ArcSinh[a*x]^(3/2)) - (8*x^2)/(15*ArcSinh[a*x]^(
3/2)) - (32*x*Sqrt[1 + a^2*x^2])/(15*a*Sqrt[ArcSinh[a*x]]) + (8*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(1
5*a^2) + (8*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(15*a^2)

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 3307

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

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]

Rule 5675

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

Rubi steps

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

Mathematica [A]  time = 0.295126, size = 118, normalized size = 0.8 \[ -\frac{2 \sinh ^{-1}(a x) \left (4 \sqrt{2} \left (-\sinh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-2 \sinh ^{-1}(a x)\right )+4 \sqrt{2} \sinh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},2 \sinh ^{-1}(a x)\right )+e^{-2 \sinh ^{-1}(a x)} \left (1-4 \sinh ^{-1}(a x)\right )+e^{2 \sinh ^{-1}(a x)} \left (4 \sinh ^{-1}(a x)+1\right )\right )+3 \sinh \left (2 \sinh ^{-1}(a x)\right )}{15 a^2 \sinh ^{-1}(a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(2*ArcSinh[a*x]*((1 - 4*ArcSinh[a*x])/E^(2*ArcSinh[a*x]) + E^(2*ArcSinh[a*x])*(1 + 4*ArcSinh[a*x]) + 4*Sqrt[2
]*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -2*ArcSinh[a*x]] + 4*Sqrt[2]*ArcSinh[a*x]^(3/2)*Gamma[1/2, 2*ArcSinh[a*x]])
+ 3*Sinh[2*ArcSinh[a*x]])/(15*a^2*ArcSinh[a*x]^(5/2))

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Maple [A]  time = 0.092, size = 147, normalized size = 1. \begin{align*}{\frac{\sqrt{2}}{15\,\sqrt{\pi }{a}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}} \left ( -16\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{5/2}\sqrt{2}\sqrt{\pi }\sqrt{{a}^{2}{x}^{2}+1}xa-4\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }{x}^{2}{a}^{2}-3\,\sqrt{2}\sqrt{{\it Arcsinh} \left ( ax \right ) }\sqrt{\pi }\sqrt{{a}^{2}{x}^{2}+1}xa+8\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}\pi \,{\it Erf} \left ( \sqrt{2}\sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) +8\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}\pi \,{\it erfi} \left ( \sqrt{2}\sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) -2\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*2^(1/2)*(-16*arcsinh(a*x)^(5/2)*2^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*x*a-4*arcsinh(a*x)^(3/2)*2^(1/2)*Pi^(1
/2)*x^2*a^2-3*2^(1/2)*arcsinh(a*x)^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*x*a+8*arcsinh(a*x)^3*Pi*erf(2^(1/2)*arcsin
h(a*x)^(1/2))+8*arcsinh(a*x)^3*Pi*erfi(2^(1/2)*arcsinh(a*x)^(1/2))-2*arcsinh(a*x)^(3/2)*2^(1/2)*Pi^(1/2))/Pi^(
1/2)/a^2/arcsinh(a*x)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\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/arcsinh(a*x)^(7/2),x, algorithm="maxima")

[Out]

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

[Out]

integrate(x/arcsinh(a*x)^(7/2), x)