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

Optimal. Leaf size=170 \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{a x-1} \sqrt{a x+1}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \sqrt{c-a^2 c x^2}}{a \sqrt{\cosh ^{-1}(a x)}} \]

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c - a^2*c*x^2])/(a*Sqrt[ArcCosh[a*x]]) - (Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]
*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(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]]])/(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

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Rubi [A]  time = 0.238779, antiderivative size = 176, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5713, 5697, 5670, 5448, 12, 3308, 2180, 2204, 2205} \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2 \sqrt{a x+1} (1-a x) \sqrt{c-a^2 c x^2}}{a \sqrt{a x-1} \sqrt{\cosh ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 - a*x)*Sqrt[1 + a*x]*Sqrt[c - a^2*c*x^2])/(a*Sqrt[-1 + a*x]*Sqrt[ArcCosh[a*x]]) - (Sqrt[Pi/2]*Sqrt[c - a
^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(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]]])/(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 5697

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

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

Mathematica [A]  time = 0.255774, size = 127, normalized size = 0.75 \[ \frac{\sqrt{c-a^2 c x^2} \left (-4 a^2 x^2-\sqrt{2 \pi } \sqrt{\cosh ^{-1}(a x)} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )+\sqrt{2 \pi } \sqrt{\cosh ^{-1}(a x)} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )+4\right )}{2 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[c - a^2*c*x^2]*(4 - 4*a^2*x^2 - Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + Sqrt[2*P
i]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]]))/(2*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCos
h[a*x]])

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Maple [F]  time = 0.487, 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 \frac{\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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )}}{\operatorname{acosh}^{\frac{3}{2}}{\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)**(3/2),x)

[Out]

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

[Out]

sage0*x