3.385 \(\int \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2} \, dx\)

Optimal. Leaf size=302 \[ \frac{3 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{8 \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{16 a \sqrt{a x-1} \sqrt{a x+1}} \]

[Out]

(3*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*a*x^2*Sqrt[c - a^2*c*x^2]*
Sqrt[ArcCosh[a*x]])/(8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/2 - (Sqrt[c
- a^2*c*x^2]*ArcCosh[a*x]^(5/2))/(5*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sq
rt[2]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2
]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

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Rubi [A]  time = 0.61828, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5713, 5683, 5676, 5664, 5781, 3312, 3307, 2180, 2204, 2205} \[ \frac{3 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{8 \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{16 a \sqrt{a x-1} \sqrt{a x+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2),x]

[Out]

(3*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*a*x^2*Sqrt[c - a^2*c*x^2]*
Sqrt[ArcCosh[a*x]])/(8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/2 - (Sqrt[c
- a^2*c*x^2]*ArcCosh[a*x]^(5/2))/(5*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sq
rt[2]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2
]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((-d)^IntPart[p]*(
d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5683

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)], x_Symbol] :
> Simp[(x*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/2, x] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2
*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dis
t[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[x*(a + b*ArcCosh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]

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]

Rule 5664

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcCosh[c*x])^n)/
(m + 1), x] - Dist[(b*c*n)/(m + 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] && GtQ[n, 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 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 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 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2} \, dx}{\sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{\sqrt{c-a^2 c x^2} \int \frac{\cosh ^{-1}(a x)^{3/2}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{2 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 a \sqrt{c-a^2 c x^2}\right ) \int x \sqrt{\cosh ^{-1}(a x)} \, dx}{4 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 a^2 \sqrt{c-a^2 c x^2}\right ) \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx}{16 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cosh (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{16 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{16 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{16 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{32 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{16 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}

Mathematica [A]  time = 0.393647, size = 136, normalized size = 0.45 \[ \frac{\sqrt{c-a^2 c x^2} \left (15 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )+15 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )-8 \sqrt{\cosh ^{-1}(a x)} \left (16 \cosh ^{-1}(a x)^2+15 \cosh \left (2 \cosh ^{-1}(a x)\right )-20 \cosh ^{-1}(a x) \sinh \left (2 \cosh ^{-1}(a x)\right )\right )\right )}{640 a \sqrt{\frac{a x-1}{a x+1}} (a x+1)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2),x]

[Out]

(Sqrt[c - a^2*c*x^2]*(15*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 15*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[
a*x]]] - 8*Sqrt[ArcCosh[a*x]]*(16*ArcCosh[a*x]^2 + 15*Cosh[2*ArcCosh[a*x]] - 20*ArcCosh[a*x]*Sinh[2*ArcCosh[a*
x]])))/(640*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))

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Maple [F]  time = 0.499, size = 0, normalized size = 0. \begin{align*} \int \sqrt{-{a}^{2}c{x}^{2}+c} \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((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(3/2),x)

[Out]

int((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-a^{2} c x^{2} + c} \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((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*c*x^2 + c)*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((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(3/2),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((-a**2*c*x**2+c)**(1/2)*acosh(a*x)**(3/2),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((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(3/2),x, algorithm="giac")

[Out]

sage0*x