\(\int \frac {(d-c^2 d x^2)^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx\) [77]

Optimal result

Integrand size = 26, antiderivative size = 307 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (d-c^2 d x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}-\frac {d \sqrt {d-c^2 d x^2} \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{b^2 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d \sqrt {d-c^2 d x^2} \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{2 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d \sqrt {d-c^2 d x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b^2 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d \sqrt {d-c^2 d x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-c^2*d*x^2+d)^(3/2)/b/c/(a+b*arccosh(c*x))-d 
*(-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)+1/2*d*(-c^2*d*x^2+d)^(1/2)*Chi(4*(a+b*arccosh(c*x))/ 
b)*sinh(4*a/b)/b^2/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+d*(-c^2*d*x^2+d)^(1/2)*co 
sh(2*a/b)*Shi(2*(a+b*arccosh(c*x))/b)/b^2/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/ 
2*d*(-c^2*d*x^2+d)^(1/2)*cosh(4*a/b)*Shi(4*(a+b*arccosh(c*x))/b)/b^2/c/(c* 
x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.55 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {d^3 \sqrt {-1+c x} \sqrt {1+c x} \left (\frac {2 b \left (-1+c^2 x^2\right )^3}{a+b \text {arccosh}(c x)}+(-1+c x) (1+c x) \left (-2 \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )+\text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )\right )}{2 b^2 c \left (d-c^2 d x^2\right )^{3/2}} \] Input:

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

Output:

(d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((2*b*(-1 + c^2*x^2)^3)/(a + b*ArcCosh[c 
*x]) + (-1 + c*x)*(1 + c*x)*(-2*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[ 
(2*a)/b] + CoshIntegral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] + 2*Cosh[(2* 
a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - Cosh[(4*a)/b]*SinhIntegral[4* 
(a/b + ArcCosh[c*x])])))/(2*b^2*c*(d - c^2*d*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.64, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {6319, 25, 6327, 6367, 25, 5971, 2009}

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.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx\)

\(\Big \downarrow \) 6319

\(\displaystyle -\frac {4 c d \sqrt {d-c^2 d x^2} \int -\frac {x (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (d-c^2 d x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 c d \sqrt {d-c^2 d x^2} \int \frac {x (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (d-c^2 d x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 6327

\(\displaystyle \frac {4 c d \sqrt {d-c^2 d x^2} \int \frac {x \left (1-c^2 x^2\right )}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (d-c^2 d x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 6367

\(\displaystyle -\frac {4 d \sqrt {d-c^2 d x^2} \int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (d-c^2 d x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 d \sqrt {d-c^2 d x^2} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (d-c^2 d x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {4 d \sqrt {d-c^2 d x^2} \int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 (a+b \text {arccosh}(c x))}-\frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{4 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (d-c^2 d x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 d \sqrt {d-c^2 d x^2} \left (\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (d-c^2 d x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

Input:

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

Output:

-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d - c^2*d*x^2)^(3/2))/(b*c*(a + b*ArcCosh 
[c*x]))) - (4*d*Sqrt[d - c^2*d*x^2]*((CoshIntegral[(2*(a + b*ArcCosh[c*x]) 
)/b]*Sinh[(2*a)/b])/4 - (CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b]*Sinh[(4* 
a)/b])/8 - (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/4 + (C 
osh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/8))/(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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 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]
 

rule 6327

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( 
e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 
*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 
, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
 

rule 6367

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x 
)^p*(-1 + c*x)^p)]   Subst[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, d, e, n}, x] && Eq 
Q[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.46

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 (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{4} x^{4}-4 b \,c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{2} x^{2}+8 b \,c^{3} x^{3}+2 \,\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 {arccosh}\left (c x \right ) b \,\operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}+\operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} b \,\operatorname {arccosh}\left (c x \right )-2 \,\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 )-4 \sqrt {c x -1}\, \sqrt {c x +1}\, b +2 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}}-a \,\operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}+\operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} a -2 \,\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 -4 b c x \right ) d}{4 \left (c x -1\right ) \left (c x +1\right ) c \,b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\)	\(448\)

Input:

int((-c^2*d*x^2+d)^(3/2)/(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

1/4*(-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)*(- 
4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^4*x^4-4*b*c^5*x^5+8*(c*x-1)^(1/2)*(c*x+1 
)^(1/2)*b*c^2*x^2+8*b*c^3*x^3+2*arccosh(c*x)*b*Ei(1,-2*arccosh(c*x)-2*a/b) 
*exp(-(-b*arccosh(c*x)+2*a)/b)-arccosh(c*x)*b*Ei(1,-4*arccosh(c*x)-4*a/b)* 
exp(-(-b*arccosh(c*x)+4*a)/b)+Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c* 
x)+4*a)/b)*b*arccosh(c*x)-2*Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x) 
+2*a)/b)*b*arccosh(c*x)-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b+2*a*Ei(1,-2*arccos 
h(c*x)-2*a/b)*exp(-(-b*arccosh(c*x)+2*a)/b)-a*Ei(1,-4*arccosh(c*x)-4*a/b)* 
exp(-(-b*arccosh(c*x)+4*a)/b)+Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c* 
x)+4*a)/b)*a-2*Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)*a-4* 
b*c*x)*d/(c*x-1)/(c*x+1)/c/b^2/(a+b*arccosh(c*x))
 

Fricas [F]

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

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

Output:

integral((-c^2*d*x^2 + d)^(3/2)/(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + 
 a^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

((c^4*d^(3/2)*x^4 - 2*c^2*d^(3/2)*x^2 + d^(3/2))*(c*x + 1)*sqrt(c*x - 1) + 
 (c^5*d^(3/2)*x^5 - 2*c^3*d^(3/2)*x^3 + c*d^(3/2)*x)*sqrt(c*x + 1))*sqrt(- 
c*x + 1)/(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt(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(((4*c^4*d^(3/2)*x^4 - 3*c^2*d^(3/2)* 
x^2 - d^(3/2))*(c*x + 1)^(3/2)*(c*x - 1) + 4*(2*c^5*d^(3/2)*x^5 - 3*c^3*d^ 
(3/2)*x^3 + c*d^(3/2)*x)*(c*x + 1)*sqrt(c*x - 1) + (4*c^6*d^(3/2)*x^6 - 9* 
c^4*d^(3/2)*x^4 + 6*c^2*d^(3/2)*x^2 - d^(3/2))*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)*s 
qrt(c*x + 1)*sqrt(c*x - 1) + b^2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), 
 x)
                                                                                    
                                                                                    
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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