3.45 \(\int \frac{x^3}{\sinh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=29 \[ \frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^4}-\frac{\text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{4 a^4} \]

[Out]

-SinhIntegral[2*ArcSinh[a*x]]/(4*a^4) + SinhIntegral[4*ArcSinh[a*x]]/(8*a^4)

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Rubi [A]  time = 0.0679608, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5669, 5448, 3298} \[ \frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^4}-\frac{\text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{4 a^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/ArcSinh[a*x],x]

[Out]

-SinhIntegral[2*ArcSinh[a*x]]/(4*a^4) + SinhIntegral[4*ArcSinh[a*x]]/(8*a^4)

Rule 5669

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Sinh[x]^m*Cosh[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \frac{x^3}{\sinh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^3(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{\sinh (2 x)}{4 x}+\frac{\sinh (4 x)}{8 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sinh (4 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac{\text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{4 a^4}+\frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^4}\\ \end{align*}

Mathematica [A]  time = 0.0717786, size = 24, normalized size = 0.83 \[ \frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )-2 \text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{8 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/ArcSinh[a*x],x]

[Out]

(-2*SinhIntegral[2*ArcSinh[a*x]] + SinhIntegral[4*ArcSinh[a*x]])/(8*a^4)

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Maple [A]  time = 0.024, size = 24, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{4}} \left ( -{\frac{{\it Shi} \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{4}}+{\frac{{\it Shi} \left ( 4\,{\it Arcsinh} \left ( ax \right ) \right ) }{8}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^4*(-1/4*Shi(2*arcsinh(a*x))+1/8*Shi(4*arcsinh(a*x)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3/arcsinh(a*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}}{\operatorname{arsinh}\left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/arcsinh(a*x),x, algorithm="fricas")

[Out]

integral(x^3/arcsinh(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/asinh(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3/arcsinh(a*x), x)