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

Optimal. Leaf size=205 \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{3 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.374543, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5713, 5683, 5676, 5670, 5448, 12, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{3 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]],x]

[Out]

(x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/2 - (Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(3*a*Sqrt[-1 + a*x]*Sq
rt[1 + a*x]) + (Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 +
a*x]) - (Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(16*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 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3308

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

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

Mathematica [A]  time = 0.132218, size = 117, normalized size = 0.57 \[ -\frac{\sqrt{-c (a x-1) (a x+1)} \left (3 \sqrt{2} \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},2 \cosh ^{-1}(a x)\right )+3 \sqrt{2} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-2 \cosh ^{-1}(a x)\right )+16 \cosh ^{-1}(a x)^2\right )}{48 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]],x]

[Out]

-(Sqrt[-(c*(-1 + a*x)*(1 + a*x))]*(16*ArcCosh[a*x]^2 + 3*Sqrt[2]*Sqrt[-ArcCosh[a*x]]*Gamma[3/2, -2*ArcCosh[a*x
]] + 3*Sqrt[2]*Sqrt[ArcCosh[a*x]]*Gamma[3/2, 2*ArcCosh[a*x]]))/(48*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt
[ArcCosh[a*x]])

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Maple [F]  time = 0.526, size = 0, normalized size = 0. \begin{align*} \int \sqrt{-{a}^{2}c{x}^{2}+c}\sqrt{{\rm arccosh} \left (ax\right )}\, 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)^(1/2),x)

[Out]

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

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} \sqrt{\operatorname{acosh}{\left (a x \right )}}\, dx \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)**(1/2),x)

[Out]

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

[Out]

sage0*x