\(\int \frac {\sqrt {d-c^2 d x^2}}{(a+b \text {arccosh}(c x))^2} \, dx\) [78]

Optimal result

Integrand size = 26, antiderivative size = 175 \[ \int \frac {\sqrt {d-c^2 d x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x} \sqrt {d-c^2 d x^2}}{b c (a+b \text {arccosh}(c x))}-\frac {\sqrt {d-c^2 d x^2} \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{b^2 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {d-c^2 d x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b^2 c \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/b/c/(a+b*arccosh(c*x))-( 
-c^2*d*x^2+d)^(1/2)*Chi(2*(a+b*arccosh(c*x))/b)*sinh(2*a/b)/b^2/c/(c*x-1)^ 
(1/2)/(c*x+1)^(1/2)+(-c^2*d*x^2+d)^(1/2)*cosh(2*a/b)*Shi(2*(a+b*arccosh(c* 
x))/b)/b^2/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {d-c^2 d x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {d-c^2 d x^2} \left (b \left (-1+c^2 x^2\right )+(a+b \text {arccosh}(c x)) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-(a+b \text {arccosh}(c x)) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{b^2 c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))} \] Input:

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

Output:

-((Sqrt[d - c^2*d*x^2]*(b*(-1 + c^2*x^2) + (a + b*ArcCosh[c*x])*CoshIntegr 
al[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] - (a + b*ArcCosh[c*x])*Cosh[(2*a) 
/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])]))/(b^2*c*Sqrt[-1 + c*x]*Sqrt[1 + 
c*x]*(a + b*ArcCosh[c*x])))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.83, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6319, 6302, 25, 5971, 27, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}

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.

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

\(\Big \downarrow \) 6319

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

\(\Big \downarrow \) 6302

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

\(\displaystyle -\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {d-c^2 d x^2}}{b c (a+b \text {arccosh}(c x))}+\frac {i \sqrt {d-c^2 d x^2} \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 3782

\(\displaystyle -\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {d-c^2 d x^2}}{b c (a+b \text {arccosh}(c x))}+\frac {i \sqrt {d-c^2 d x^2} \left (i \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[d - c^2*d*x^2])/(b*c*(a + b*ArcCosh[c 
*x]))) + (I*Sqrt[d - c^2*d*x^2]*(I*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b 
]*Sinh[(2*a)/b] - I*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b] 
))/(b^2*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 3042

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

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

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 [A] (verified)

Time = 0.16 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.55

method	result	size
default	\(\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{2} x^{2}+2 b \,c^{3} x^{3}+\operatorname {arccosh}\left (c x \right ) b \,\operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-\operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} b \,\operatorname {arccosh}\left (c x \right )-2 \sqrt {c x -1}\, \sqrt {c x +1}\, b +a \,\operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-\operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} a -2 b c x \right )}{2 \left (c x -1\right ) \left (c x +1\right ) c \,b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\)	\(271\)

Input:

int((-c^2*d*x^2+d)^(1/2)/(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

1/2*(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(2 
*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^2*x^2+2*b*c^3*x^3+arccosh(c*x)*b*Ei(1,-2* 
arccosh(c*x)-2*a/b)*exp(-(-b*arccosh(c*x)+2*a)/b)-Ei(1,2*arccosh(c*x)+2*a/ 
b)*exp((b*arccosh(c*x)+2*a)/b)*b*arccosh(c*x)-2*(c*x-1)^(1/2)*(c*x+1)^(1/2 
)*b+a*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-(-b*arccosh(c*x)+2*a)/b)-Ei(1,2*arc 
cosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)*a-2*b*c*x)/(c*x-1)/(c*x+1)/c/ 
b^2/(a+b*arccosh(c*x))
 

Fricas [F]

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

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

Output:

integral(sqrt(-c^2*d*x^2 + d)/(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a 
^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-((c^2*sqrt(d)*x^2 - sqrt(d))*(c*x + 1)*sqrt(c*x - 1) + (c^3*sqrt(d)*x^3 - 
 c*sqrt(d)*x)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^3*x^2 + sqrt(c*x + 1)*s 
qrt(c*x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1 
)*b^2*c^2*x - b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) + integrate(( 
(2*c^2*sqrt(d)*x^2 + sqrt(d))*(c*x + 1)^(3/2)*(c*x - 1) + 2*(2*c^3*sqrt(d) 
*x^3 - c*sqrt(d)*x)*(c*x + 1)*sqrt(c*x - 1) + (2*c^4*sqrt(d)*x^4 - 3*c^2*s 
qrt(d)*x^2 + sqrt(d))*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^4*x^4 + (c*x + 
1)*(c*x - 1)*a*b*c^2*x^2 - 2*a*b*c^2*x^2 + 2*(a*b*c^3*x^3 - a*b*c*x)*sqrt( 
c*x + 1)*sqrt(c*x - 1) + a*b + (b^2*c^4*x^4 + (c*x + 1)*(c*x - 1)*b^2*c^2* 
x^2 - 2*b^2*c^2*x^2 + 2*(b^2*c^3*x^3 - b^2*c*x)*sqrt(c*x + 1)*sqrt(c*x - 1 
) + b^2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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