\(\int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx\) [93]

Optimal result

Integrand size = 10, antiderivative size = 63 \[ \int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a^2}+\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a^2} \]

[Out]

-1/8*erf(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^2+1/8*erfi(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2
)/a^2

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5887, 5556, 12, 3389, 2211, 2235, 2236} \[ \int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx=\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a^2}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a^2} \]

[In]

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

[Out]

-1/4*(Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/a^2 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(4*a^2)

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]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.78 \[ \int \frac {x}{\sqrt {\text {arccosh}(a x)}} \, dx=\frac {\sqrt {\frac {\pi }{2}} \left (-\text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{4 a^2} \]

[In]

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

[Out]

(Sqrt[Pi/2]*(-Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]]))/(4*a^2)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.59

method	result	size
default	\(-\frac {\sqrt {\pi }\, \sqrt {2}\, \left (\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )-\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )\right )}{8 a^{2}}\)	\(37\)

[In]

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

[Out]

-1/8*Pi^(1/2)*2^(1/2)*(erf(2^(1/2)*arccosh(a*x)^(1/2))-erfi(2^(1/2)*arccosh(a*x)^(1/2)))/a^2

Fricas [F(-2)]

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

[In]

integrate(x/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]

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

[In]

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

[Out]

Integral(x/sqrt(acosh(a*x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x/sqrt(arccosh(a*x)), x)

Giac [F]

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

[In]

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

[Out]

integrate(x/sqrt(arccosh(a*x)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x/acosh(a*x)^(1/2), x)