3.58 \(\int \frac{x^4}{\cosh ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=102 \[ \frac{\text{Shi}\left (\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{27 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{25 \text{Shi}\left (5 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac{5 x^5}{2 \cosh ^{-1}(a x)}-\frac{x^4 \sqrt{a x-1} \sqrt{a x+1}}{2 a \cosh ^{-1}(a x)^2} \]

[Out]

-(x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]^2) + (2*x^3)/(a^2*ArcCosh[a*x]) - (5*x^5)/(2*ArcCosh[a*x
]) + SinhIntegral[ArcCosh[a*x]]/(16*a^5) + (27*SinhIntegral[3*ArcCosh[a*x]])/(32*a^5) + (25*SinhIntegral[5*Arc
Cosh[a*x]])/(32*a^5)

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Rubi [A]  time = 0.644703, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5668, 5775, 5670, 5448, 3298} \[ \frac{\text{Shi}\left (\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{27 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{25 \text{Shi}\left (5 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac{5 x^5}{2 \cosh ^{-1}(a x)}-\frac{x^4 \sqrt{a x-1} \sqrt{a x+1}}{2 a \cosh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcCosh[a*x]^3,x]

[Out]

-(x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]^2) + (2*x^3)/(a^2*ArcCosh[a*x]) - (5*x^5)/(2*ArcCosh[a*x
]) + SinhIntegral[ArcCosh[a*x]]/(16*a^5) + (27*SinhIntegral[3*ArcCosh[a*x]])/(32*a^5) + (25*SinhIntegral[5*Arc
Cosh[a*x]])/(32*a^5)

Rule 5668

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(
a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcCosh
[c*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] + Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcCosh[c
*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5775

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[((f*x)^m*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1
, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 0]

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)^3} \, dx &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{2 a \cosh ^{-1}(a x)^2}-\frac{2 \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2} \, dx}{a}+\frac{1}{2} (5 a) \int \frac{x^5}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2} \, dx\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac{2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac{5 x^5}{2 \cosh ^{-1}(a x)}+\frac{25}{2} \int \frac{x^4}{\cosh ^{-1}(a x)} \, dx-\frac{6 \int \frac{x^2}{\cosh ^{-1}(a x)} \, dx}{a^2}\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac{2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac{5 x^5}{2 \cosh ^{-1}(a x)}-\frac{6 \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\cosh ^4(x) \sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac{2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac{5 x^5}{2 \cosh ^{-1}(a x)}-\frac{6 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{4 x}+\frac{\sinh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac{25 \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 )}{2 a^5}\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac{2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac{5 x^5}{2 \cosh ^{-1}(a x)}+\frac{25 \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{75 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac{2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac{5 x^5}{2 \cosh ^{-1}(a x)}+\frac{\text{Shi}\left (\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{27 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{25 \text{Shi}\left (5 \cosh ^{-1}(a x)\right )}{32 a^5}\\ \end{align*}

Mathematica [A]  time = 0.134561, size = 107, normalized size = 1.05 \[ \frac{-16 a^4 x^4 \sqrt{a x-1} \sqrt{a x+1}-80 a^5 x^5 \cosh ^{-1}(a x)+64 a^3 x^3 \cosh ^{-1}(a x)+2 \cosh ^{-1}(a x)^2 \text{Shi}\left (\cosh ^{-1}(a x)\right )+27 \cosh ^{-1}(a x)^2 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )+25 \cosh ^{-1}(a x)^2 \text{Shi}\left (5 \cosh ^{-1}(a x)\right )}{32 a^5 \cosh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/ArcCosh[a*x]^3,x]

[Out]

(-16*a^4*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x] + 64*a^3*x^3*ArcCosh[a*x] - 80*a^5*x^5*ArcCosh[a*x] + 2*ArcCosh[a*x]
^2*SinhIntegral[ArcCosh[a*x]] + 27*ArcCosh[a*x]^2*SinhIntegral[3*ArcCosh[a*x]] + 25*ArcCosh[a*x]^2*SinhIntegra
l[5*ArcCosh[a*x]])/(32*a^5*ArcCosh[a*x]^2)

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Maple [A]  time = 0.042, size = 123, normalized size = 1.2 \begin{align*}{\frac{1}{{a}^{5}} \left ( -{\frac{1}{16\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{ax}{16\,{\rm arccosh} \left (ax\right )}}+{\frac{{\it Shi} \left ({\rm arccosh} \left (ax\right ) \right ) }{16}}-{\frac{3\,\sinh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{32\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}-{\frac{9\,\cosh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{32\,{\rm arccosh} \left (ax\right )}}+{\frac{27\,{\it Shi} \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{32}}-{\frac{\sinh \left ( 5\,{\rm arccosh} \left (ax\right ) \right ) }{32\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}-{\frac{5\,\cosh \left ( 5\,{\rm arccosh} \left (ax\right ) \right ) }{32\,{\rm arccosh} \left (ax\right )}}+{\frac{25\,{\it Shi} \left ( 5\,{\rm arccosh} \left (ax\right ) \right ) }{32}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arccosh(a*x)^3,x)

[Out]

1/a^5*(-1/16/arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/16*a*x/arccosh(a*x)+1/16*Shi(arccosh(a*x))-3/32/arcc
osh(a*x)^2*sinh(3*arccosh(a*x))-9/32/arccosh(a*x)*cosh(3*arccosh(a*x))+27/32*Shi(3*arccosh(a*x))-1/32/arccosh(
a*x)^2*sinh(5*arccosh(a*x))-5/32/arccosh(a*x)*cosh(5*arccosh(a*x))+25/32*Shi(5*arccosh(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccosh(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a^8*x^11 - 3*a^6*x^9 + 3*a^4*x^7 - a^2*x^5 + (a^5*x^8 - a^3*x^6)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + (3*a^
6*x^9 - 5*a^4*x^7 + 2*a^2*x^5)*(a*x + 1)*(a*x - 1) + (3*a^7*x^10 - 7*a^5*x^8 + 5*a^3*x^6 - a*x^4)*sqrt(a*x + 1
)*sqrt(a*x - 1) + (5*a^8*x^11 - 15*a^6*x^9 + 15*a^4*x^7 - 5*a^2*x^5 + (5*a^5*x^8 - 8*a^3*x^6 + 3*a*x^4)*(a*x +
1)^(3/2)*(a*x - 1)^(3/2) + (15*a^6*x^9 - 31*a^4*x^7 + 20*a^2*x^5 - 4*x^3)*(a*x + 1)*(a*x - 1) + (15*a^7*x^10
- 38*a^5*x^8 + 32*a^3*x^6 - 9*a*x^4)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)))/((a^
8*x^6 + (a*x + 1)^(3/2)*(a*x - 1)^(3/2)*a^5*x^3 - 3*a^6*x^4 + 3*a^4*x^2 + 3*(a^6*x^4 - a^4*x^2)*(a*x + 1)*(a*x
- 1) + 3*(a^7*x^5 - 2*a^5*x^3 + a^3*x)*sqrt(a*x + 1)*sqrt(a*x - 1) - a^2)*log(a*x + sqrt(a*x + 1)*sqrt(a*x -
1))^2) + integrate(1/2*(25*a^10*x^12 - 100*a^8*x^10 + 150*a^6*x^8 - 100*a^4*x^6 + 25*a^2*x^4 + (25*a^6*x^8 - 2
4*a^4*x^6 + 3*a^2*x^4)*(a*x + 1)^2*(a*x - 1)^2 + (100*a^7*x^9 - 172*a^5*x^7 + 87*a^3*x^5 - 12*a*x^3)*(a*x + 1)
^(3/2)*(a*x - 1)^(3/2) + 3*(50*a^8*x^10 - 124*a^6*x^8 + 105*a^4*x^6 - 35*a^2*x^4 + 4*x^2)*(a*x + 1)*(a*x - 1)
+ (100*a^9*x^11 - 324*a^7*x^9 + 381*a^5*x^7 - 193*a^3*x^5 + 36*a*x^3)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^10*x^8
+ (a*x + 1)^2*(a*x - 1)^2*a^6*x^4 - 4*a^8*x^6 + 6*a^6*x^4 - 4*a^4*x^2 + 4*(a^7*x^5 - a^5*x^3)*(a*x + 1)^(3/2)*
(a*x - 1)^(3/2) + 6*(a^8*x^6 - 2*a^6*x^4 + a^4*x^2)*(a*x + 1)*(a*x - 1) + 4*(a^9*x^7 - 3*a^7*x^5 + 3*a^5*x^3 -
a^3*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + a^2)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), 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 )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccosh(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^4/arccosh(a*x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{acosh}^{3}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/acosh(a*x)**3,x)

[Out]

Integral(x**4/acosh(a*x)**3, 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 )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccosh(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^4/arccosh(a*x)^3, x)