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

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

[Out]

SinhIntegral[ArcCosh[a*x]]/(8*a^5) + (3*SinhIntegral[3*ArcCosh[a*x]])/(16*a^5) + SinhIntegral[5*ArcCosh[a*x]]/
(16*a^5)

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Rubi [A]  time = 0.0801527, 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 = {5670, 5448, 3298} \[ \frac{\text{Shi}\left (\cosh ^{-1}(a x)\right )}{8 a^5}+\frac{3 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{16 a^5}+\frac{\text{Shi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

SinhIntegral[ArcCosh[a*x]]/(8*a^5) + (3*SinhIntegral[3*ArcCosh[a*x]])/(16*a^5) + SinhIntegral[5*ArcCosh[a*x]]/
(16*a^5)

Rule 5670

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Cosh[x]^m*Sinh[x], x], x, ArcCosh[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^4}{\cosh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh ^4(x) \sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{8 x}+\frac{3 \sinh (3 x)}{16 x}+\frac{\sinh (5 x)}{16 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sinh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}\\ &=\frac{\text{Shi}\left (\cosh ^{-1}(a x)\right )}{8 a^5}+\frac{3 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{16 a^5}+\frac{\text{Shi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0713324, size = 31, normalized size = 0.76 \[ \frac{2 \text{Shi}\left (\cosh ^{-1}(a x)\right )+3 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )+\text{Shi}\left (5 \cosh ^{-1}(a x)\right )}{16 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*SinhIntegral[ArcCosh[a*x]] + 3*SinhIntegral[3*ArcCosh[a*x]] + SinhIntegral[5*ArcCosh[a*x]])/(16*a^5)

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Maple [A]  time = 0.026, size = 31, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{{\it Shi} \left ({\rm arccosh} \left (ax\right ) \right ) }{8}}+{\frac{3\,{\it Shi} \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{16}}+{\frac{{\it Shi} \left ( 5\,{\rm arccosh} \left (ax\right ) \right ) }{16}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^5*(1/8*Shi(arccosh(a*x))+3/16*Shi(3*arccosh(a*x))+1/16*Shi(5*arccosh(a*x)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/arccosh(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{arcosh}\left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^4/arccosh(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**4/acosh(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/arccosh(a*x), x)