3.97 \(\int \frac{x^4}{\cosh ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=193 \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^5}-\frac{2 x^4 \sqrt{a x-1} \sqrt{a x+1}}{a \sqrt{\cosh ^{-1}(a x)}} \]

[Out]

(-2*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]]) + (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(8*a^5) + (3
*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(16*a^5) + (Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(16*a^5)
+ (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(8*a^5) + (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(16*a^5) + (Sq
rt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(16*a^5)

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Rubi [A]  time = 0.184238, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5666, 3307, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^5}-\frac{2 x^4 \sqrt{a x-1} \sqrt{a x+1}}{a \sqrt{\cosh ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcCosh[a*x]^(3/2),x]

[Out]

(-2*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]]) + (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(8*a^5) + (3
*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(16*a^5) + (Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(16*a^5)
+ (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(8*a^5) + (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(16*a^5) + (Sq
rt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(16*a^5)

Rule 5666

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[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a +
b*x)^(n + 1)*Cosh[x]^(m - 1)*(m - (m + 1)*Cosh[x]^2), x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{x^4}{\cosh ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{8 \sqrt{x}}-\frac{9 \cosh (3 x)}{16 \sqrt{x}}-\frac{5 \cosh (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{a \sqrt{\cosh ^{-1}(a x)}}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}+\frac{9 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{a \sqrt{\cosh ^{-1}(a x)}}+\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}+\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{5 \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{9 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{9 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{a \sqrt{\cosh ^{-1}(a x)}}+\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{4 a^5}+\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{4 a^5}+\frac{5 \operatorname{Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac{5 \operatorname{Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac{9 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac{9 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{a \sqrt{\cosh ^{-1}(a x)}}+\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac{3 \sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{5 \pi } \text{erf}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac{3 \sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{5 \pi } \text{erfi}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a^5}\\ \end{align*}

Mathematica [A]  time = 0.29186, size = 201, normalized size = 1.04 \[ -\frac{-\sqrt{5} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-5 \cosh ^{-1}(a x)\right )-3 \sqrt{3} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 \cosh ^{-1}(a x)\right )-2 \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-\cosh ^{-1}(a x)\right )+2 \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},\cosh ^{-1}(a x)\right )+3 \sqrt{3} \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 \cosh ^{-1}(a x)\right )+\sqrt{5} \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},5 \cosh ^{-1}(a x)\right )+4 \sqrt{\frac{a x-1}{a x+1}} (a x+1)+6 \sinh \left (3 \cosh ^{-1}(a x)\right )+2 \sinh \left (5 \cosh ^{-1}(a x)\right )}{16 a^5 \sqrt{\cosh ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/ArcCosh[a*x]^(3/2),x]

[Out]

-(4*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) - Sqrt[5]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -5*ArcCosh[a*x]] - 3*Sqrt[3]
*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -3*ArcCosh[a*x]] - 2*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -ArcCosh[a*x]] + 2*Sqrt[Ar
cCosh[a*x]]*Gamma[1/2, ArcCosh[a*x]] + 3*Sqrt[3]*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 3*ArcCosh[a*x]] + Sqrt[5]*Sqrt[
ArcCosh[a*x]]*Gamma[1/2, 5*ArcCosh[a*x]] + 6*Sinh[3*ArcCosh[a*x]] + 2*Sinh[5*ArcCosh[a*x]])/(16*a^5*Sqrt[ArcCo
sh[a*x]])

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Maple [F]  time = 0.183, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ({\rm arccosh} \left (ax\right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arccosh(a*x)^(3/2),x)

[Out]

int(x^4/arccosh(a*x)^(3/2),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 )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccosh(a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^4/arccosh(a*x)^(3/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccosh(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{acosh}^{\frac{3}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/acosh(a*x)**(3/2),x)

[Out]

Integral(x**4/acosh(a*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccosh(a*x)^(3/2),x, algorithm="giac")

[Out]

sage0*x