3.563 \(\int \frac {1}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx\)

Optimal. Leaf size=88 \[ \frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {\sqrt {\pi } e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5658, 3308, 2180, 2205, 2204} \[ \frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}-\frac {\sqrt {\pi } e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*ArcCosh[c*x]],x]

[Out]

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

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 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 5658

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 100, normalized size = 1.14 \[ \frac {e^{-\frac {a}{b}} \left (e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right )+\sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right )\right )}{2 c \sqrt {a+b \cosh ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[a + b*ArcCosh[c*x]],x]

[Out]

(E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] + Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1
/2, -((a + b*ArcCosh[c*x])/b)])/(2*c*E^(a/b)*Sqrt[a + b*ArcCosh[c*x]])

________________________________________________________________________________________

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(1/(a+b*arccosh(c*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 \[ \int \frac {1}{\sqrt {b \operatorname {arcosh}\left (c x\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*arccosh(c*x) + a), x)

________________________________________________________________________________________

maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a +b \,\mathrm {arccosh}\left (c x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \operatorname {arcosh}\left (c x\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*arccosh(c*x) + a), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+b\,\mathrm {acosh}\left (c\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*acosh(c*x))^(1/2),x)

[Out]

int(1/(a + b*acosh(c*x))^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \operatorname {acosh}{\left (c x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*acosh(c*x))**(1/2),x)

[Out]

Integral(1/sqrt(a + b*acosh(c*x)), x)

________________________________________________________________________________________