3.113 \(\int \frac{1}{\cosh ^{-1}(a x)^{7/2}} \, dx\)

Optimal. Leaf size=122 \[ \frac{4 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}+\frac{4 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{a x-1} \sqrt{a x+1}}{15 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2)) - (4*x)/(15*ArcCosh[a*x]^(3/2)) - (8*Sqrt[-1 + a*x]
*Sqrt[1 + a*x])/(15*a*Sqrt[ArcCosh[a*x]]) + (4*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(15*a) + (4*Sqrt[Pi]*Erfi[Sqr
t[ArcCosh[a*x]]])/(15*a)

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Rubi [A]  time = 0.443552, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {5656, 5775, 5781, 3307, 2180, 2204, 2205} \[ \frac{4 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}+\frac{4 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{a x-1} \sqrt{a x+1}}{15 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^(-7/2),x]

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2)) - (4*x)/(15*ArcCosh[a*x]^(3/2)) - (8*Sqrt[-1 + a*x]
*Sqrt[1 + a*x])/(15*a*Sqrt[ArcCosh[a*x]]) + (4*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(15*a) + (4*Sqrt[Pi]*Erfi[Sqr
t[ArcCosh[a*x]]])/(15*a)

Rule 5656

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

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 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

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{1}{\cosh ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{1}{5} (2 a) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}+\frac{4}{15} \int \frac{1}{\cosh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{1}{15} (8 a) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a}\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{4 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a}+\frac{4 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a}\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}+\frac{8 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}\\ &=-\frac{2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{4 x}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{4 \sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}+\frac{4 \sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{15 a}\\ \end{align*}

Mathematica [A]  time = 0.181454, size = 147, normalized size = 1.2 \[ -\frac{2 e^{-\cosh ^{-1}(a x)} \left (-2 e^{\cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},-\cosh ^{-1}(a x)\right )+2 e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{5/2} \text{Gamma}\left (\frac{1}{2},\cosh ^{-1}(a x)\right )+2 e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^2-2 \cosh ^{-1}(a x)^2+3 \sqrt{\frac{a x-1}{a x+1}} (a x+1) e^{\cosh ^{-1}(a x)}+e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\cosh ^{-1}(a x)\right )}{15 a \cosh ^{-1}(a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^(-7/2),x]

[Out]

(-2*(3*E^ArcCosh[a*x]*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + ArcCosh[a*x] + E^(2*ArcCosh[a*x])*ArcCosh[a*x] -
2*ArcCosh[a*x]^2 + 2*E^(2*ArcCosh[a*x])*ArcCosh[a*x]^2 - 2*E^ArcCosh[a*x]*(-ArcCosh[a*x])^(5/2)*Gamma[1/2, -Ar
cCosh[a*x]] + 2*E^ArcCosh[a*x]*ArcCosh[a*x]^(5/2)*Gamma[1/2, ArcCosh[a*x]]))/(15*a*E^ArcCosh[a*x]*ArcCosh[a*x]
^(5/2))

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Maple [A]  time = 0.101, size = 111, normalized size = 0.9 \begin{align*}{\frac{2}{15\,\sqrt{\pi }a \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}} \left ( -4\,\sqrt{ax-1} \left ({\rm arccosh} \left (ax\right ) \right ) ^{5/2}\sqrt{ax+1}\sqrt{\pi }+2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}\pi \,{\it Erf} \left ( \sqrt{{\rm arccosh} \left (ax\right )} \right ) +2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}\pi \,{\it erfi} \left ( \sqrt{{\rm arccosh} \left (ax\right )} \right ) -2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3/2}\sqrt{\pi }xa-3\,\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccosh(a*x)^(7/2),x)

[Out]

2/15*(-4*(a*x-1)^(1/2)*arccosh(a*x)^(5/2)*(a*x+1)^(1/2)*Pi^(1/2)+2*arccosh(a*x)^3*Pi*erf(arccosh(a*x)^(1/2))+2
*arccosh(a*x)^3*Pi*erfi(arccosh(a*x)^(1/2))-2*arccosh(a*x)^(3/2)*Pi^(1/2)*x*a-3*arccosh(a*x)^(1/2)*Pi^(1/2)*(a
*x+1)^(1/2)*(a*x-1)^(1/2))/Pi^(1/2)/a/arccosh(a*x)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{arcosh}\left (a x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccosh(a*x)^(7/2),x, algorithm="maxima")

[Out]

integrate(arccosh(a*x)^(-7/2), 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(1/arccosh(a*x)^(7/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/acosh(a*x)**(7/2),x)

[Out]

Timed out

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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(1/arccosh(a*x)^(7/2),x, algorithm="giac")

[Out]

sage0*x