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

Optimal. Leaf size=157 \[ \frac{8 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac{8 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{32 x \sqrt{a x-1} \sqrt{a x+1}}{15 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]

[Out]

(-2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2)) + 4/(15*a^2*ArcCosh[a*x]^(3/2)) - (8*x^2)/(15*Arc
Cosh[a*x]^(3/2)) - (32*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(15*a*Sqrt[ArcCosh[a*x]]) + (8*Sqrt[2*Pi]*Erf[Sqrt[2]*S
qrt[ArcCosh[a*x]]])/(15*a^2) + (8*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(15*a^2)

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Rubi [A]  time = 0.495435, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {5668, 5775, 5666, 3307, 2180, 2204, 2205, 5676} \[ \frac{8 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac{8 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{32 x \sqrt{a x-1} \sqrt{a x+1}}{15 a \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1}}{5 a \cosh ^{-1}(a x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2)) + 4/(15*a^2*ArcCosh[a*x]^(3/2)) - (8*x^2)/(15*Arc
Cosh[a*x]^(3/2)) - (32*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(15*a*Sqrt[ArcCosh[a*x]]) + (8*Sqrt[2*Pi]*Erf[Sqrt[2]*S
qrt[ArcCosh[a*x]]])/(15*a^2) + (8*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(15*a^2)

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 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]

Rule 5676

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

Rubi steps

\begin{align*} \int \frac{x}{\cosh ^{-1}(a x)^{7/2}} \, dx &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}-\frac{2 \int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac{1}{5} (4 a) \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}+\frac{16}{15} \int \frac{x}{\cosh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{32 x \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{32 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^2}\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{32 x \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^2}+\frac{16 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{15 a^2}\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{32 x \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{32 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac{32 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}\\ &=-\frac{2 x \sqrt{-1+a x} \sqrt{1+a x}}{5 a \cosh ^{-1}(a x)^{5/2}}+\frac{4}{15 a^2 \cosh ^{-1}(a x)^{3/2}}-\frac{8 x^2}{15 \cosh ^{-1}(a x)^{3/2}}-\frac{32 x \sqrt{-1+a x} \sqrt{1+a x}}{15 a \sqrt{\cosh ^{-1}(a x)}}+\frac{8 \sqrt{2 \pi } \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}+\frac{8 \sqrt{2 \pi } \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{15 a^2}\\ \end{align*}

Mathematica [A]  time = 0.285803, size = 91, normalized size = 0.58 \[ -\frac{-8 \sqrt{2 \pi } \left (\text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )+\text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )\right )+\frac{4 \cosh \left (2 \cosh ^{-1}(a x)\right )}{\cosh ^{-1}(a x)^{3/2}}+\frac{\left (16 \cosh ^{-1}(a x)^2+3\right ) \sinh \left (2 \cosh ^{-1}(a x)\right )}{\cosh ^{-1}(a x)^{5/2}}}{15 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((4*Cosh[2*ArcCosh[a*x]])/ArcCosh[a*x]^(3/2) - 8*Sqrt[2*Pi]*(Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + Erfi[Sqrt[2]*S
qrt[ArcCosh[a*x]]]) + ((3 + 16*ArcCosh[a*x]^2)*Sinh[2*ArcCosh[a*x]])/ArcCosh[a*x]^(5/2))/(15*a^2)

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Maple [A]  time = 0.118, size = 153, normalized size = 1. \begin{align*}{\frac{\sqrt{2}}{15\,\sqrt{\pi }{a}^{2} \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}} \left ( -16\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{5/2}\sqrt{2}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1}xa-4\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }{x}^{2}{a}^{2}-3\,\sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1}xa+8\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}\pi \,{\it Erf} \left ( \sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )} \right ) +8\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}\pi \,{\it erfi} \left ( \sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )} \right ) +2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*2^(1/2)*(-16*arccosh(a*x)^(5/2)*2^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x*a-4*arccosh(a*x)^(3/2)*2^(
1/2)*Pi^(1/2)*x^2*a^2-3*2^(1/2)*arccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x*a+8*arccosh(a*x)^3*P
i*erf(2^(1/2)*arccosh(a*x)^(1/2))+8*arccosh(a*x)^3*Pi*erfi(2^(1/2)*arccosh(a*x)^(1/2))+2*arccosh(a*x)^(3/2)*2^
(1/2)*Pi^(1/2))/Pi^(1/2)/a^2/arccosh(a*x)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\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(x/arccosh(a*x)^(7/2),x, algorithm="maxima")

[Out]

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

[Out]

sage0*x