\(\int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2} \, dx\) [385]

Optimal result

Integrand size = 24, antiderivative size = 302 \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2} \, dx=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}} \]

[Out]

1/2*x*arccosh(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)-1/5*arccosh(a*x)^(5/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x
+1)^(1/2)+3/128*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)+3/128*erfi(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)+3/16*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-3/8*a*x^2*(-a^2*c*x^2+c)^(1/2)*a
rccosh(a*x)^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5895, 5893, 5884, 5953, 3393, 3388, 2211, 2235, 2236} \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2} \, dx=\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{5/2} \sqrt {c-a^2 c x^2}}{5 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \text {arccosh}(a x)^{3/2} \sqrt {c-a^2 c x^2}-\frac {3 a x^2 \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}}{8 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}}{16 a \sqrt {a x-1} \sqrt {a x+1}} \]

[In]

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

[Out]

(3*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*a*x^2*Sqrt[c - a^2*c*x^2]*
Sqrt[ArcCosh[a*x]])/(8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/2 - (Sqrt[c
- a^2*c*x^2]*ArcCosh[a*x]^(5/2))/(5*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sq
rt[2]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2
]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*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 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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5884

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCosh[c*x])^n/(
m + 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])
), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5895

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*(
(a + b*ArcCosh[c*x])^n/2), x] + (-Dist[(1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])], Int[(a + b*
ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sq
rt[-1 + c*x])], Int[x*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
&& 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}& = \frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\text {arccosh}(a x)^{3/2}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 a \sqrt {c-a^2 c x^2}\right ) \int x \sqrt {\text {arccosh}(a x)} \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 a^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {\text {arccosh}(a x)}} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^2(x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {3 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {3 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {3 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {3 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.45 \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (15 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+15 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-8 \sqrt {\text {arccosh}(a x)} \left (16 \text {arccosh}(a x)^2+15 \cosh (2 \text {arccosh}(a x))-20 \text {arccosh}(a x) \sinh (2 \text {arccosh}(a x))\right )\right )}{640 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \]

[In]

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

[Out]

(Sqrt[c - a^2*c*x^2]*(15*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 15*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[
a*x]]] - 8*Sqrt[ArcCosh[a*x]]*(16*ArcCosh[a*x]^2 + 15*Cosh[2*ArcCosh[a*x]] - 20*ArcCosh[a*x]*Sinh[2*ArcCosh[a*
x]])))/(640*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))

Maple [F]

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

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

[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]

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

[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]

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

[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(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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