\(\int \frac {(d+e x^2)^2}{(a+b \text {arccosh}(c x))^2} \, dx\) [138]

Optimal result

Integrand size = 20, antiderivative size = 510 \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {d^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {2 d e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}+\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 b^2 c^3}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^5}+\frac {3 d e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {d e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 b^2 c^3}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^5}-\frac {3 d e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5} \] Output:

-d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))-2*d*e*x^2*(c*x-1)^ 
(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))-e^2*x^4*(c*x-1)^(1/2)*(c*x+1)^( 
1/2)/b/c/(a+b*arccosh(c*x))+d^2*cosh(a/b)*Chi((a+b*arccosh(c*x))/b)/b^2/c+ 
1/2*d*e*cosh(a/b)*Chi((a+b*arccosh(c*x))/b)/b^2/c^3+1/8*e^2*cosh(a/b)*Chi( 
(a+b*arccosh(c*x))/b)/b^2/c^5+3/2*d*e*cosh(3*a/b)*Chi(3*(a+b*arccosh(c*x)) 
/b)/b^2/c^3+9/16*e^2*cosh(3*a/b)*Chi(3*(a+b*arccosh(c*x))/b)/b^2/c^5+5/16* 
e^2*cosh(5*a/b)*Chi(5*(a+b*arccosh(c*x))/b)/b^2/c^5-d^2*sinh(a/b)*Shi((a+b 
*arccosh(c*x))/b)/b^2/c-1/2*d*e*sinh(a/b)*Shi((a+b*arccosh(c*x))/b)/b^2/c^ 
3-1/8*e^2*sinh(a/b)*Shi((a+b*arccosh(c*x))/b)/b^2/c^5-3/2*d*e*sinh(3*a/b)* 
Shi(3*(a+b*arccosh(c*x))/b)/b^2/c^3-9/16*e^2*sinh(3*a/b)*Shi(3*(a+b*arccos 
h(c*x))/b)/b^2/c^5-5/16*e^2*sinh(5*a/b)*Shi(5*(a+b*arccosh(c*x))/b)/b^2/c^ 
5
 

Mathematica [A] (warning: unable to verify)

Time = 2.34 (sec) , antiderivative size = 663, normalized size of antiderivative = 1.30 \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/16*(16*b*c^4*d^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 16*b*c^5*d^2*x*Sqrt[(-1 + 
 c*x)/(1 + c*x)] + 32*b*c^4*d*e*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 32*b*c^5* 
d*e*x^3*Sqrt[(-1 + c*x)/(1 + c*x)] + 16*b*c^4*e^2*x^4*Sqrt[(-1 + c*x)/(1 + 
 c*x)] + 16*b*c^5*e^2*x^5*Sqrt[(-1 + c*x)/(1 + c*x)] - 2*(8*c^4*d^2 + 4*c^ 
2*d*e + e^2)*(a + b*ArcCosh[c*x])*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x 
]] - 3*e*(8*c^2*d + 3*e)*(a + b*ArcCosh[c*x])*Cosh[(3*a)/b]*CoshIntegral[3 
*(a/b + ArcCosh[c*x])] - 5*a*e^2*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcCo 
sh[c*x])] - 5*b*e^2*ArcCosh[c*x]*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcCo 
sh[c*x])] + 16*a*c^4*d^2*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 8*a* 
c^2*d*e*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 2*a*e^2*Sinh[a/b]*Sin 
hIntegral[a/b + ArcCosh[c*x]] + 16*b*c^4*d^2*ArcCosh[c*x]*Sinh[a/b]*SinhIn 
tegral[a/b + ArcCosh[c*x]] + 8*b*c^2*d*e*ArcCosh[c*x]*Sinh[a/b]*SinhIntegr 
al[a/b + ArcCosh[c*x]] + 2*b*e^2*ArcCosh[c*x]*Sinh[a/b]*SinhIntegral[a/b + 
 ArcCosh[c*x]] + 24*a*c^2*d*e*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[ 
c*x])] + 9*a*e^2*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 24*b 
*c^2*d*e*ArcCosh[c*x]*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 
 9*b*e^2*ArcCosh[c*x]*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 
 5*a*e^2*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] + 5*b*e^2*ArcC 
osh[c*x]*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])])/(b^2*c^5*(a + 
 b*ArcCosh[c*x]))
 

Rubi [A] (verified)

Time = 1.65 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6324, 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.

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

\(\Big \downarrow \) 6324

\(\displaystyle \int \left (\frac {d^2}{(a+b \text {arccosh}(c x))^2}+\frac {2 d e x^2}{(a+b \text {arccosh}(c x))^2}+\frac {e^2 x^4}{(a+b \text {arccosh}(c x))^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^5}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^5}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 b^2 c^3}+\frac {3 d e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3}-\frac {d e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 b^2 c^3}-\frac {3 d e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {d^2 \sqrt {c x-1} \sqrt {c x+1}}{b c (a+b \text {arccosh}(c x))}-\frac {2 d e x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c (a+b \text {arccosh}(c x))}-\frac {e^2 x^4 \sqrt {c x-1} \sqrt {c x+1}}{b c (a+b \text {arccosh}(c x))}\)

Input:

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

Output:

-((d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x]))) - (2*d*e* 
x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x])) - (e^2*x^4*Sq 
rt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x])) + (d^2*Cosh[a/b]*Co 
shIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c) + (d*e*Cosh[a/b]*CoshIntegral[ 
(a + b*ArcCosh[c*x])/b])/(2*b^2*c^3) + (e^2*Cosh[a/b]*CoshIntegral[(a + b* 
ArcCosh[c*x])/b])/(8*b^2*c^5) + (3*d*e*Cosh[(3*a)/b]*CoshIntegral[(3*(a + 
b*ArcCosh[c*x]))/b])/(2*b^2*c^3) + (9*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*(a 
 + b*ArcCosh[c*x]))/b])/(16*b^2*c^5) + (5*e^2*Cosh[(5*a)/b]*CoshIntegral[( 
5*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^5) - (d^2*Sinh[a/b]*SinhIntegral[(a 
+ b*ArcCosh[c*x])/b])/(b^2*c) - (d*e*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh 
[c*x])/b])/(2*b^2*c^3) - (e^2*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/ 
b])/(8*b^2*c^5) - (3*d*e*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x] 
))/b])/(2*b^2*c^3) - (9*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c 
*x]))/b])/(16*b^2*c^5) - (5*e^2*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCo 
sh[c*x]))/b])/(16*b^2*c^5)
 

Defintions of rubi rules used

rule 2009

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

rule 6324

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && 
(p > 0 || IGtQ[n, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1101\) vs. \(2(478)=956\).

Time = 0.57 (sec) , antiderivative size = 1102, normalized size of antiderivative = 2.16

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1102\)
default	\(\text {Expression too large to display}\)	\(1102\)

Input:

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

Output:

1/c*(1/32*(-16*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)+12*(c*x-1)^(1/2)*(c*x+1 
)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2)+16*c^5*x^5-20*c^3*x^3+5*c*x)*e 
^2/c^4/b/(a+b*arccosh(c*x))-5/32*e^2/c^4/b^2*exp(5*a/b)*Ei(1,5*arccosh(c*x 
)+5*a/b)-1/32/b*e^2/c^4*(16*c^5*x^5-20*c^3*x^3+16*c^4*x^4*(c*x-1)^(1/2)*(c 
*x+1)^(1/2)+5*c*x-12*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c* 
x+1)^(1/2))/(a+b*arccosh(c*x))-5/32/b^2*e^2/c^4*exp(-5*a/b)*Ei(1,-5*arccos 
h(c*x)-5*a/b)+1/2*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x)*d^2/b/(a+b*arccosh(c* 
x))-1/2*d^2/b^2*exp(a/b)*Ei(1,arccosh(c*x)+a/b)+1/4*(-(c*x-1)^(1/2)*(c*x+1 
)^(1/2)+c*x)*d*e/c^2/b/(a+b*arccosh(c*x))-1/4/c^2*d*e/b^2*exp(a/b)*Ei(1,ar 
ccosh(c*x)+a/b)+1/16*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x)*e^2/c^4/b/(a+b*arc 
cosh(c*x))-1/16/c^4*e^2/b^2*exp(a/b)*Ei(1,arccosh(c*x)+a/b)-1/2/b*d^2*(c*x 
+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(a+b*arccosh(c*x))-1/2/b^2*d^2*exp(-a/b)*Ei( 
1,-arccosh(c*x)-a/b)-1/4/c^2/b*d*e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(a+b* 
arccosh(c*x))-1/4/c^2/b^2*d*e*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)-1/16/c^4/b 
*e^2*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(a+b*arccosh(c*x))-1/16/c^4/b^2*e^2 
*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)+1/4*(-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2 
*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c^3*x^3-3*c*x)*d*e/c^2/b/(a+b*arccosh(c 
*x))+3/32*(-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1 
/2)+4*c^3*x^3-3*c*x)*e^2/c^4/b/(a+b*arccosh(c*x))-3/4*e/c^2/b^2*exp(3*a/b) 
*Ei(1,3*arccosh(c*x)+3*a/b)*d-9/32*e^2/c^4/b^2*exp(3*a/b)*Ei(1,3*arccos...
 

Fricas [F]

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

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

Output:

integral((e^2*x^4 + 2*d*e*x^2 + d^2)/(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c 
*x) + a^2), x)
 

Sympy [F]

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

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

Output:

Integral((d + e*x**2)**2/(a + b*acosh(c*x))**2, x)
 

Maxima [F]

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

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

Output:

-(c^3*e^2*x^7 + (2*c^3*d*e - c*e^2)*x^5 - c*d^2*x + (c^3*d^2 - 2*c*d*e)*x^ 
3 + (c^2*e^2*x^6 + (2*c^2*d*e - e^2)*x^4 + (c^2*d^2 - 2*d*e)*x^2 - d^2)*sq 
rt(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((5*c^5*e^2*x^8 + 2 
*(3*c^5*d*e - 5*c^3*e^2)*x^6 + (c^5*d^2 - 12*c^3*d*e + 5*c*e^2)*x^4 + (5*c 
^3*e^2*x^6 + 3*(2*c^3*d*e - c*e^2)*x^4 + c*d^2 + (c^3*d^2 - 2*c*d*e)*x^2)* 
(c*x + 1)*(c*x - 1) + c*d^2 - 2*(c^3*d^2 - 3*c*d*e)*x^2 + (10*c^4*e^2*x^7 
+ (12*c^4*d*e - 13*c^2*e^2)*x^5 + 2*(c^4*d^2 - 7*c^2*d*e + 2*e^2)*x^3 - (c 
^2*d^2 - 4*d*e)*x)*sqrt(c*x + 1)*sqrt(c*x - 1))/(a*b*c^5*x^4 + (c*x + 1)*( 
c*x - 1)*a*b*c^3*x^2 - 2*a*b*c^3*x^2 + a*b*c + 2*(a*b*c^4*x^3 - a*b*c^2*x) 
*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^4 + (c*x + 1)*(c*x - 1)*b^2*c^3* 
x^2 - 2*b^2*c^3*x^2 + b^2*c + 2*(b^2*c^4*x^3 - b^2*c^2*x)*sqrt(c*x + 1)*sq 
rt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx=\left (\int \frac {x^{4}}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) e^{2}+2 \left (\int \frac {x^{2}}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) d e +\left (\int \frac {1}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) d^{2} \] Input:

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

Output:

int(x**4/(acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b + a**2),x)*e**2 + 2*int(x* 
*2/(acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b + a**2),x)*d*e + int(1/(acosh(c* 
x)**2*b**2 + 2*acosh(c*x)*a*b + a**2),x)*d**2