\(\int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx\) [410]

Optimal result

Integrand size = 24, antiderivative size = 170 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {arccosh}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}} \]

[Out]

-1/2*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)+1/2*e
rfi(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)-2*(a*x-1)^
(1/2)*(a*x+1)^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/arccosh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5904, 5887, 5556, 12, 3389, 2211, 2235, 2236} \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{a \sqrt {a x-1} \sqrt {a x+1}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \sqrt {c-a^2 c x^2}}{a \sqrt {\text {arccosh}(a x)}} \]

[In]

Int[Sqrt[c - a^2*c*x^2]/ArcCosh[a*x]^(3/2),x]

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c - a^2*c*x^2])/(a*Sqrt[ArcCosh[a*x]]) - (Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]
*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(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]]])/(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 5904

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

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (4-4 a^2 x^2-\sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{2 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\text {arccosh}(a x)}} \]

[In]

Integrate[Sqrt[c - a^2*c*x^2]/ArcCosh[a*x]^(3/2),x]

[Out]

(Sqrt[c - a^2*c*x^2]*(4 - 4*a^2*x^2 - Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + Sqrt[2*P
i]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]]))/(2*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCos
h[a*x]])

Maple [F]

\[\int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\operatorname {arccosh}\left (a x \right )^{\frac {3}{2}}}d x\]

[In]

int((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(3/2),x)

[Out]

int((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(3/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate((-a**2*c*x**2+c)**(1/2)/acosh(a*x)**(3/2),x)

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))/acosh(a*x)**(3/2), x)

Maxima [F]

\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*c*x^2 + c)/arccosh(a*x)^(3/2), x)

Giac [F]

\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(-a^2*c*x^2 + c)/arccosh(a*x)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,d x \]

[In]

int((c - a^2*c*x^2)^(1/2)/acosh(a*x)^(3/2),x)

[Out]

int((c - a^2*c*x^2)^(1/2)/acosh(a*x)^(3/2), x)