\(\int \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)} \, dx\) [83]

Optimal result

Integrand size = 24, antiderivative size = 205 \[ \int \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)} \, dx=\frac {1}{2} x \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}} \] Output:

1/2*x*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(1/2)-1/3*(-a^2*c*x^2+c)^(1/2)*arc 
cosh(a*x)^(3/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+1/32*2^(1/2)*Pi^(1/2)*(-a^2* 
c*x^2+c)^(1/2)*erf(2^(1/2)*arccosh(a*x)^(1/2))/a/(a*x-1)^(1/2)/(a*x+1)^(1/ 
2)-1/32*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erfi(2^(1/2)*arccosh(a*x)^(1 
/2))/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.57 \[ \int \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)} \, dx=-\frac {\sqrt {-c (-1+a x) (1+a x)} \left (16 \text {arccosh}(a x)^2+3 \sqrt {2} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},-2 \text {arccosh}(a x)\right )+3 \sqrt {2} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {3}{2},2 \text {arccosh}(a x)\right )\right )}{48 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\text {arccosh}(a x)}} \] Input:

Integrate[Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]],x]
 

Output:

-1/48*(Sqrt[-(c*(-1 + a*x)*(1 + a*x))]*(16*ArcCosh[a*x]^2 + 3*Sqrt[2]*Sqrt 
[-ArcCosh[a*x]]*Gamma[3/2, -2*ArcCosh[a*x]] + 3*Sqrt[2]*Sqrt[ArcCosh[a*x]] 
*Gamma[3/2, 2*ArcCosh[a*x]]))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt 
[ArcCosh[a*x]])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.97 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.88, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6310, 6302, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634, 6308}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2} \, dx\)

\(\Big \downarrow \) 6310

\(\displaystyle -\frac {a \sqrt {c-a^2 c x^2} \int \frac {x}{\sqrt {\text {arccosh}(a x)}}dx}{4 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 6302

\(\displaystyle -\frac {\sqrt {c-a^2 c x^2} \int \frac {a x \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{4 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\sinh (2 \text {arccosh}(a x))}{2 \sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{4 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\sinh (2 \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \int -\frac {i \sin (2 i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {i \sqrt {c-a^2 c x^2} \int \frac {\sin (2 i \text {arccosh}(a x))}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 3789

\(\displaystyle \frac {i \sqrt {c-a^2 c x^2} \left (\frac {1}{2} i \int \frac {e^{2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \int \frac {e^{-2 \text {arccosh}(a x)}}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {i \sqrt {c-a^2 c x^2} \left (i \int e^{2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}-i \int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {i \sqrt {c-a^2 c x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-i \int e^{-2 \text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {\sqrt {c-a^2 c x^2} \int \frac {\sqrt {\text {arccosh}(a x)}}{\sqrt {a x-1} \sqrt {a x+1}}dx}{2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {i \sqrt {c-a^2 c x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 6308

\(\displaystyle \frac {i \sqrt {c-a^2 c x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\text {arccosh}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{3 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}\)

Input:

Int[Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]],x]
 

Output:

(x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/2 - (Sqrt[c - a^2*c*x^2]*ArcCos 
h[a*x]^(3/2))/(3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + ((I/8)*Sqrt[c - a^2*c*x 
^2]*((-1/2*I)*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + (I/2)*Sqrt[Pi/2 
]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]]))/(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[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 2633

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 2634

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] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3789

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
/2   Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2   Int[(c + d*x)^m*E 
^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
 

rule 5971

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 6302

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
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 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(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*ArcCosh[ 
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 6310

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] + (-Simp[( 
1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[(a + b*ArcC 
osh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sq 
rt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)} \, dx=\sqrt {c}\, \left (\int \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acosh} \left (a x \right )}d x \right ) \] Input:

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

Output:

sqrt(c)*int(sqrt( - a**2*x**2 + 1)*sqrt(acosh(a*x)),x)