3.76 \(\int \sqrt {\cosh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=53 \[ -\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{4 a}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{4 a}+x \sqrt {\cosh ^{-1}(a x)} \]

[Out]

-1/4*erf(arccosh(a*x)^(1/2))*Pi^(1/2)/a-1/4*erfi(arccosh(a*x)^(1/2))*Pi^(1/2)/a+x*arccosh(a*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.22, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5654, 5781, 3307, 2180, 2204, 2205} \[ -\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{4 a}-\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{4 a}+x \sqrt {\cosh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[ArcCosh[a*x]],x]

[Out]

x*Sqrt[ArcCosh[a*x]] - (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(4*a) - (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(4*a)

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]

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 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && 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])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 45, normalized size = 0.85 \[ \frac {\frac {\sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-\cosh ^{-1}(a x)\right )}{\sqrt {-\cosh ^{-1}(a x)}}+\Gamma \left (\frac {3}{2},\cosh ^{-1}(a x)\right )}{2 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[ArcCosh[a*x]],x]

[Out]

((Sqrt[ArcCosh[a*x]]*Gamma[3/2, -ArcCosh[a*x]])/Sqrt[-ArcCosh[a*x]] + Gamma[3/2, ArcCosh[a*x]])/(2*a)

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\operatorname {arcosh}\left (a x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(arccosh(a*x)), x)

________________________________________________________________________________________

maple [A]  time = 0.13, size = 43, normalized size = 0.81 \[ \frac {4 \sqrt {\mathrm {arccosh}\left (a x \right )}\, \sqrt {\pi }\, x a -\pi \erf \left (\sqrt {\mathrm {arccosh}\left (a x \right )}\right )-\pi \erfi \left (\sqrt {\mathrm {arccosh}\left (a x \right )}\right )}{4 \sqrt {\pi }\, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*arccosh(a*x)^(1/2)*Pi^(1/2)*x*a-Pi*erf(arccosh(a*x)^(1/2))-Pi*erfi(arccosh(a*x)^(1/2)))/Pi^(1/2)/a

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\operatorname {arcosh}\left (a x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(arccosh(a*x)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {\mathrm {acosh}\left (a\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acosh(a*x)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\operatorname {acosh}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**(1/2),x)

[Out]

Integral(sqrt(acosh(a*x)), x)

________________________________________________________________________________________