\(\int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx\) [112]

Optimal result

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

-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)-1 
/2*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)+1/2*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erf 
i(2^(1/2)*arccosh(a*x)^(1/2))/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.27 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{3/2}} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (4-4 a^2 x^2-\sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{2 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\text {arccosh}(a x)}} \] Input:

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

Output:

(Sqrt[c - a^2*c*x^2]*(4 - 4*a^2*x^2 - Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erf[Sq 
rt[2]*Sqrt[ArcCosh[a*x]]] + Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqr 
t[ArcCosh[a*x]]]))/(2*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 = 0.66 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.87, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6319, 6302, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634}

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.

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

\(\Big \downarrow \) 6319

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

\(\Big \downarrow \) 6302

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3789

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

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

Output:

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c - a^2*c*x^2])/(a*Sqrt[ArcCosh[a*x] 
]) - ((2*I)*Sqrt[c - a^2*c*x^2]*((-1/2*I)*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcC 
osh[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 6319

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*A 
rcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Si 
mp[(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*ArcCosh[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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(c)*(2*acosh(a*x)*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a**2*x** 
2 + 1)*sqrt(acosh(a*x))*x)/(acosh(a*x)*a**2*x**2 - acosh(a*x)),x)*a**2 - s 
qrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a**2*x**2 + 1)*sqrt(acosh(a*x))))/(acos 
h(a*x)*a)