3.4.80 \(\int \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)} \, dx\) [380]

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

[Out]

-1/3*arccosh(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+1/32*erf(2^(1/2)*arccosh(a*x)^(1/2)
)*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-1/32*erfi(2^(1/2)*arccosh(a*x)^(1/2))*2^
(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+1/2*x*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(1/2
)

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Rubi [A]
time = 0.17, antiderivative size = 205, normalized size of antiderivative = 1.00, 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 = {5895, 5893, 5887, 5556, 12, 3389, 2211, 2235, 2236} \begin {gather*} \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)} \end {gather*}

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 12

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

Rule 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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 2236

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 3389

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 5556

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 5887

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Cosh
[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5893

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

Rule 5895

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*(
(a + b*ArcCosh[c*x])^n/2), x] + (-Dist[(1/2)*Simp[Sqrt[d + e*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] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sq
rt[-1 + c*x])], Int[x*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
&& GtQ[n, 0]

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} \text {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} \text {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} \text {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} \text {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} \text {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} \text {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} \text {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*}

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Mathematica [A]
time = 0.10, size = 117, normalized size = 0.57 \begin {gather*} -\frac {\sqrt {-c (-1+a x) (1+a x)} \left (16 \cosh ^{-1}(a x)^2+3 \sqrt {2} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-2 \cosh ^{-1}(a x)\right )+3 \sqrt {2} \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},2 \cosh ^{-1}(a x)\right )\right )}{48 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\cosh ^{-1}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \sqrt {-a^{2} c \,x^{2}+c}\, \sqrt {\mathrm {arccosh}\left (a x \right )}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \sqrt {\operatorname {acosh}{\left (a x \right )}}\, dx \end {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {\mathrm {acosh}\left (a\,x\right )}\,\sqrt {c-a^2\,c\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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