3.44 \(\int \frac{x^4}{\sinh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=41 \[ \frac{\text{Chi}\left (\sinh ^{-1}(a x)\right )}{8 a^5}-\frac{3 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{16 a^5}+\frac{\text{Chi}\left (5 \sinh ^{-1}(a x)\right )}{16 a^5} \]

[Out]

CoshIntegral[ArcSinh[a*x]]/(8*a^5) - (3*CoshIntegral[3*ArcSinh[a*x]])/(16*a^5) + CoshIntegral[5*ArcSinh[a*x]]/
(16*a^5)

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Rubi [A]  time = 0.0837777, antiderivative size = 41, 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, 3301} \[ \frac{\text{Chi}\left (\sinh ^{-1}(a x)\right )}{8 a^5}-\frac{3 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{16 a^5}+\frac{\text{Chi}\left (5 \sinh ^{-1}(a x)\right )}{16 a^5} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcSinh[a*x],x]

[Out]

CoshIntegral[ArcSinh[a*x]]/(8*a^5) - (3*CoshIntegral[3*ArcSinh[a*x]])/(16*a^5) + CoshIntegral[5*ArcSinh[a*x]]/
(16*a^5)

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 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \frac{x^4}{\sinh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^4(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\cosh (x)}{8 x}-\frac{3 \cosh (3 x)}{16 x}+\frac{\cosh (5 x)}{16 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (5 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}\\ &=\frac{\text{Chi}\left (\sinh ^{-1}(a x)\right )}{8 a^5}-\frac{3 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )}{16 a^5}+\frac{\text{Chi}\left (5 \sinh ^{-1}(a x)\right )}{16 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0098321, size = 31, normalized size = 0.76 \[ \frac{2 \text{Chi}\left (\sinh ^{-1}(a x)\right )-3 \text{Chi}\left (3 \sinh ^{-1}(a x)\right )+\text{Chi}\left (5 \sinh ^{-1}(a x)\right )}{16 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/ArcSinh[a*x],x]

[Out]

(2*CoshIntegral[ArcSinh[a*x]] - 3*CoshIntegral[3*ArcSinh[a*x]] + CoshIntegral[5*ArcSinh[a*x]])/(16*a^5)

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Maple [A]  time = 0.025, size = 31, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{{\it Chi} \left ({\it Arcsinh} \left ( ax \right ) \right ) }{8}}-{\frac{3\,{\it Chi} \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{16}}+{\frac{{\it Chi} \left ( 5\,{\it Arcsinh} \left ( ax \right ) \right ) }{16}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^5*(1/8*Chi(arcsinh(a*x))-3/16*Chi(3*arcsinh(a*x))+1/16*Chi(5*arcsinh(a*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 )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/arcsinh(a*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{4}}{\operatorname{arsinh}\left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^4/arcsinh(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**4/asinh(a*x), x)

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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 )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/arcsinh(a*x), x)