\(\int \sqrt {\text {arccosh}(a x)} \, dx\) [76]

Optimal result

Integrand size = 8, antiderivative size = 53 \[ \int \sqrt {\text {arccosh}(a x)} \, dx=x \sqrt {\text {arccosh}(a x)}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{4 a}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{4 a} \]

[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] (verified)

Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5879, 5953, 3388, 2211, 2235, 2236} \[ \int \sqrt {\text {arccosh}(a x)} \, dx=-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{4 a}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{4 a}+x \sqrt {\text {arccosh}(a x)} \]

[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 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 3388

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 5879

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 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.85 \[ \int \sqrt {\text {arccosh}(a x)} \, dx=\frac {\frac {\sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},-\text {arccosh}(a x)\right )}{\sqrt {-\text {arccosh}(a x)}}+\Gamma \left (\frac {3}{2},\text {arccosh}(a x)\right )}{2 a} \]

[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)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77

method	result	size
default	\(-\frac {-4 \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a x +\pi \,\operatorname {erf}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )+\pi \,\operatorname {erfi}\left (\sqrt {\operatorname {arccosh}\left (a x \right )}\right )}{4 \sqrt {\pi }\, a}\)	\(41\)

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \sqrt {\text {arccosh}(a x)} \, dx=\text {Exception raised: TypeError} \]

[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)

Sympy [F]

\[ \int \sqrt {\text {arccosh}(a x)} \, dx=\int \sqrt {\operatorname {acosh}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

Integral(sqrt(acosh(a*x)), x)

Maxima [F]

\[ \int \sqrt {\text {arccosh}(a x)} \, dx=\int { \sqrt {\operatorname {arcosh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

integrate(sqrt(arccosh(a*x)), x)

Giac [F]

\[ \int \sqrt {\text {arccosh}(a x)} \, dx=\int { \sqrt {\operatorname {arcosh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

integrate(sqrt(arccosh(a*x)), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {\text {arccosh}(a x)} \, dx=\int \sqrt {\mathrm {acosh}\left (a\,x\right )} \,d x \]

[In]

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

[Out]

int(acosh(a*x)^(1/2), x)