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

Optimal. Leaf size=294 \[ -\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{32 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{4 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{32 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{4 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{4 a \sqrt{a x-1} \sqrt{a x+1}} \]

[Out]

(-3*c*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*
x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sq
rt[2]*Sqrt[ArcCosh[a*x]]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[Ar
cCosh[a*x]]])/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCos
h[a*x]]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

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Rubi [A]  time = 0.339293, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {5713, 5701, 3312, 3307, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{32 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{4 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{32 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{4 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{4 a \sqrt{a x-1} \sqrt{a x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*c*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*
x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sq
rt[2]*Sqrt[ArcCosh[a*x]]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[Ar
cCosh[a*x]]])/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCos
h[a*x]]])/(4*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 5701

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Dist[(-(d1*d2))^p/c, Subst[Int[(a + b*x)^n*Sinh[x]^(2*p + 1), x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c
, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && IGtQ[p + 1/2, 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 \frac{\left (c-a^2 c x^2\right )^{3/2}}{\sqrt{\cosh ^{-1}(a x)}} \, dx &=-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \int \frac{(-1+a x)^{3/2} (1+a x)^{3/2}}{\sqrt{\cosh ^{-1}(a x)}} \, dx}{\sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh ^4(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}-\frac{\cosh (2 x)}{2 \sqrt{x}}+\frac{\cosh (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{3 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{4 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (c \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 )}{2 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{3 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{4 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (c \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 )}{4 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (c \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 )}{4 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{3 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{4 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{8 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{2 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{2 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{3 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{4 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{4 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{4 a \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}

Mathematica [A]  time = 0.248487, size = 153, normalized size = 0.52 \[ -\frac{c \sqrt{c-a^2 c x^2} \left (\sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 \cosh ^{-1}(a x)\right )-4 \sqrt{2} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 \cosh ^{-1}(a x)\right )+\sqrt{\cosh ^{-1}(a x)} \left (4 \sqrt{2} \text{Gamma}\left (\frac{1}{2},2 \cosh ^{-1}(a x)\right )-\text{Gamma}\left (\frac{1}{2},4 \cosh ^{-1}(a x)\right )+24 \sqrt{\cosh ^{-1}(a x)}\right )\right )}{32 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c*Sqrt[c - a^2*c*x^2]*(Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -4*ArcCosh[a*x]] - 4*Sqrt[2]*Sqrt[-ArcCosh[a*x]]*Gamma
[1/2, -2*ArcCosh[a*x]] + Sqrt[ArcCosh[a*x]]*(24*Sqrt[ArcCosh[a*x]] + 4*Sqrt[2]*Gamma[1/2, 2*ArcCosh[a*x]] - Ga
mma[1/2, 4*ArcCosh[a*x]])))/(32*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])

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Maple [F]  time = 0.328, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}{\frac{1}{\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)^(3/2)/arccosh(a*x)^(1/2),x)

[Out]

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

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

[Out]

integrate((-a^2*c*x^2 + c)^(3/2)/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)^(3/2)/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 \frac{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}{\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)**(3/2)/acosh(a*x)**(1/2),x)

[Out]

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

[Out]

sage0*x