Integrand size = 20, antiderivative size = 510 \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {d^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {2 d e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}+\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 b^2 c^3}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^5}+\frac {3 d e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {d e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 b^2 c^3}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^5}-\frac {3 d e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5} \]
d^2*Chi((a+b*arccosh(c*x))/b)*cosh(a/b)/b^2/c+1/2*d*e*Chi((a+b*arccosh(c*x ))/b)*cosh(a/b)/b^2/c^3+1/8*e^2*Chi((a+b*arccosh(c*x))/b)*cosh(a/b)/b^2/c^ 5+3/2*d*e*Chi(3*(a+b*arccosh(c*x))/b)*cosh(3*a/b)/b^2/c^3+9/16*e^2*Chi(3*( a+b*arccosh(c*x))/b)*cosh(3*a/b)/b^2/c^5+5/16*e^2*Chi(5*(a+b*arccosh(c*x)) /b)*cosh(5*a/b)/b^2/c^5-d^2*Shi((a+b*arccosh(c*x))/b)*sinh(a/b)/b^2/c-1/2* d*e*Shi((a+b*arccosh(c*x))/b)*sinh(a/b)/b^2/c^3-1/8*e^2*Shi((a+b*arccosh(c *x))/b)*sinh(a/b)/b^2/c^5-3/2*d*e*Shi(3*(a+b*arccosh(c*x))/b)*sinh(3*a/b)/ b^2/c^3-9/16*e^2*Shi(3*(a+b*arccosh(c*x))/b)*sinh(3*a/b)/b^2/c^5-5/16*e^2* Shi(5*(a+b*arccosh(c*x))/b)*sinh(5*a/b)/b^2/c^5-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))
Time = 2.58 (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=-\frac {16 b c^4 d^2 \sqrt {\frac {-1+c x}{1+c x}}+16 b c^5 d^2 x \sqrt {\frac {-1+c x}{1+c x}}+32 b c^4 d e x^2 \sqrt {\frac {-1+c x}{1+c x}}+32 b c^5 d e x^3 \sqrt {\frac {-1+c x}{1+c x}}+16 b c^4 e^2 x^4 \sqrt {\frac {-1+c x}{1+c x}}+16 b c^5 e^2 x^5 \sqrt {\frac {-1+c x}{1+c x}}-2 \left (8 c^4 d^2+4 c^2 d e+e^2\right ) (a+b \text {arccosh}(c x)) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-3 e \left (8 c^2 d+3 e\right ) (a+b \text {arccosh}(c x)) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-5 a e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-5 b e^2 \text {arccosh}(c x) \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+16 a c^4 d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+8 a c^2 d e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+2 a e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+16 b c^4 d^2 \text {arccosh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+8 b c^2 d e \text {arccosh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+2 b e^2 \text {arccosh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+24 a c^2 d e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+9 a e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+24 b c^2 d e \text {arccosh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+9 b e^2 \text {arccosh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+5 a e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+5 b e^2 \text {arccosh}(c x) \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )}{16 b^2 c^5 (a+b \text {arccosh}(c x))} \]
-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]))
Time = 1.19 (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 |
| \(\displaystyle \frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^5}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^5}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^5}+\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 b^2 c^3}+\frac {3 d e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3}-\frac {d e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 b^2 c^3}-\frac {3 d e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{2 b^2 c^3}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {d^2 \sqrt {c x-1} \sqrt {c x+1}}{b c (a+b \text {arccosh}(c x))}-\frac {2 d e x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c (a+b \text {arccosh}(c x))}-\frac {e^2 x^4 \sqrt {c x-1} \sqrt {c x+1}}{b c (a+b \text {arccosh}(c x))}\) |
-((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)
3.6.43.3.1 Defintions of rubi rules used
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])
Leaf count of result is larger than twice the leaf count of optimal. \(1101\) vs. \(2(478)=956\).
Time = 1.98 (sec) , antiderivative size = 1102, normalized size of antiderivative = 2.16
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1102\) |
| default | \(\text {Expression too large to display}\) | \(1102\) |
1/c*(1/32*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+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*x+1)^(1/2)*(c*x-1)^(1 /2)*c^4*x^4+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...
\[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (d + e x^{2}\right )^{2}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
-(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)
\[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]