3.43 \(\int \frac{x^5}{\sinh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=43 \[ \frac{5 \text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{32 a^6}-\frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^6}+\frac{\text{Shi}\left (6 \sinh ^{-1}(a x)\right )}{32 a^6} \]

[Out]

(5*SinhIntegral[2*ArcSinh[a*x]])/(32*a^6) - SinhIntegral[4*ArcSinh[a*x]]/(8*a^6) + SinhIntegral[6*ArcSinh[a*x]
]/(32*a^6)

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Rubi [A]  time = 0.0826723, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5669, 5448, 3298} \[ \frac{5 \text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{32 a^6}-\frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^6}+\frac{\text{Shi}\left (6 \sinh ^{-1}(a x)\right )}{32 a^6} \]

Antiderivative was successfully verified.

[In]

Int[x^5/ArcSinh[a*x],x]

[Out]

(5*SinhIntegral[2*ArcSinh[a*x]])/(32*a^6) - SinhIntegral[4*ArcSinh[a*x]]/(8*a^6) + SinhIntegral[6*ArcSinh[a*x]
]/(32*a^6)

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^5}{\sinh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^5(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{5 \sinh (2 x)}{32 x}-\frac{\sinh (4 x)}{8 x}+\frac{\sinh (6 x)}{32 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^6}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sinh (6 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^6}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (4 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^6}+\frac{5 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^6}\\ &=\frac{5 \text{Shi}\left (2 \sinh ^{-1}(a x)\right )}{32 a^6}-\frac{\text{Shi}\left (4 \sinh ^{-1}(a x)\right )}{8 a^6}+\frac{\text{Shi}\left (6 \sinh ^{-1}(a x)\right )}{32 a^6}\\ \end{align*}

Mathematica [A]  time = 0.110397, size = 33, normalized size = 0.77 \[ \frac{5 \text{Shi}\left (2 \sinh ^{-1}(a x)\right )-4 \text{Shi}\left (4 \sinh ^{-1}(a x)\right )+\text{Shi}\left (6 \sinh ^{-1}(a x)\right )}{32 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/ArcSinh[a*x],x]

[Out]

(5*SinhIntegral[2*ArcSinh[a*x]] - 4*SinhIntegral[4*ArcSinh[a*x]] + SinhIntegral[6*ArcSinh[a*x]])/(32*a^6)

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Maple [A]  time = 0.033, size = 33, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{6}} \left ({\frac{5\,{\it Shi} \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{32}}-{\frac{{\it Shi} \left ( 4\,{\it Arcsinh} \left ( ax \right ) \right ) }{8}}+{\frac{{\it Shi} \left ( 6\,{\it Arcsinh} \left ( ax \right ) \right ) }{32}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/arcsinh(a*x),x)

[Out]

1/a^6*(5/32*Shi(2*arcsinh(a*x))-1/8*Shi(4*arcsinh(a*x))+1/32*Shi(6*arcsinh(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/arcsinh(a*x),x, algorithm="maxima")

[Out]

integrate(x^5/arcsinh(a*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{5}}{\operatorname{arsinh}\left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/arcsinh(a*x),x, algorithm="fricas")

[Out]

integral(x^5/arcsinh(a*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\operatorname{asinh}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/asinh(a*x),x)

[Out]

Integral(x**5/asinh(a*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/arcsinh(a*x),x, algorithm="giac")

[Out]

integrate(x^5/arcsinh(a*x), x)