\(\int \frac {(d+e x)^2}{(a+b \text {arccosh}(c x))^2} \, dx\) [32]

Optimal result

Integrand size = 18, antiderivative size = 374 \[ \int \frac {(d+e x)^2}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {d^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {2 d e x \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}-\frac {e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^3}+\frac {2 d e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b^2 c^2}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^3}-\frac {2 d e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b^2 c^2}-\frac {3 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 1.37 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.42 \[ \int \frac {(d+e x)^2}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {4 b c^2 d^2 \sqrt {\frac {-1+c x}{1+c x}}+4 b c^3 d^2 x \sqrt {\frac {-1+c x}{1+c x}}+8 b c^2 d e x \sqrt {\frac {-1+c x}{1+c x}}+8 b c^3 d e x^2 \sqrt {\frac {-1+c x}{1+c x}}+4 b c^2 e^2 x^2 \sqrt {\frac {-1+c x}{1+c x}}+4 b c^3 e^2 x^3 \sqrt {\frac {-1+c x}{1+c x}}-\left (4 c^2 d^2+e^2\right ) (a+b \text {arccosh}(c x)) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-8 c d e (a+b \text {arccosh}(c x)) \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-3 a e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-3 b e^2 \text {arccosh}(c x) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+4 a c^2 d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+a e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+4 b c^2 d^2 \text {arccosh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+b e^2 \text {arccosh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+8 a c d e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+8 b c d e \text {arccosh}(c x) \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+3 a e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+3 b e^2 \text {arccosh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )}{4 b^2 c^3 (a+b \text {arccosh}(c x))} \] Input:

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

Output:

-1/4*(4*b*c^2*d^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 4*b*c^3*d^2*x*Sqrt[(-1 + c* 
x)/(1 + c*x)] + 8*b*c^2*d*e*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 8*b*c^3*d*e*x^2 
*Sqrt[(-1 + c*x)/(1 + c*x)] + 4*b*c^2*e^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 
 4*b*c^3*e^2*x^3*Sqrt[(-1 + c*x)/(1 + c*x)] - (4*c^2*d^2 + e^2)*(a + b*Arc 
Cosh[c*x])*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] - 8*c*d*e*(a + b*Arc 
Cosh[c*x])*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x])] - 3*a*e^2*Co 
sh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] - 3*b*e^2*ArcCosh[c*x]*Co 
sh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] + 4*a*c^2*d^2*Sinh[a/b]*S 
inhIntegral[a/b + ArcCosh[c*x]] + a*e^2*Sinh[a/b]*SinhIntegral[a/b + ArcCo 
sh[c*x]] + 4*b*c^2*d^2*ArcCosh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c 
*x]] + b*e^2*ArcCosh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 8*a 
*c*d*e*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + 8*b*c*d*e*ArcC 
osh[c*x]*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + 3*a*e^2*Sinh 
[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 3*b*e^2*ArcCosh[c*x]*Sinh 
[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])])/(b^2*c^3*(a + b*ArcCosh[c* 
x]))
 

Rubi [A] (verified)

Time = 1.64 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6379, 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.

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

\(\Big \downarrow \) 6379

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

\(\Big \downarrow \) 2009

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

Input:

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

Defintions of rubi rules used

rule 2009

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

rule 6379

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_S 
ymbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcCosh[c*x])^n, x], x] /; 
 FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
 

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 649, normalized size of antiderivative = 1.74

method	result	size
derivativedivides	\(\frac {\frac {\left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}+4 c^{3} x^{3}-3 c x \right ) e^{2}}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b^{2}}-\frac {e^{2} \left (4 c^{3} x^{3}-3 c x +4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 b \,c^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2} c^{2}}+\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d^{2}}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}+\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) e^{2}}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{8 c^{2} b^{2}}-\frac {d^{2} \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}-\frac {e^{2} \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{8 c^{2} b^{2}}+\frac {\left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 c^{2} x^{2}-1\right ) d e}{2 c b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {e d \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right )}{c \,b^{2}}-\frac {e d \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 c^{2} x^{2}-1\right )}{2 b c \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {e d \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right )}{b^{2} c}}{c}\)	\(649\)
default	\(\frac {\frac {\left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}+4 c^{3} x^{3}-3 c x \right ) e^{2}}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b^{2}}-\frac {e^{2} \left (4 c^{3} x^{3}-3 c x +4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 b \,c^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2} c^{2}}+\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d^{2}}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}+\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) e^{2}}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{8 c^{2} b^{2}}-\frac {d^{2} \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}-\frac {e^{2} \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 c^{2} b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{8 c^{2} b^{2}}+\frac {\left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 c^{2} x^{2}-1\right ) d e}{2 c b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {e d \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right )}{c \,b^{2}}-\frac {e d \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 c^{2} x^{2}-1\right )}{2 b c \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {e d \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right )}{b^{2} c}}{c}\)	\(649\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(c^3*e^2*x^5 + 2*c^3*d*e*x^4 - 2*c*d*e*x^2 - c*d^2*x + (c^3*d^2 - c*e^2)* 
x^3 + (c^2*e^2*x^4 + 2*c^2*d*e*x^3 - 2*d*e*x + (c^2*d^2 - e^2)*x^2 - d^2)* 
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((3*c^5*e^2*x^6 + 
 4*c^5*d*e*x^5 - 8*c^3*d*e*x^3 + (c^5*d^2 - 6*c^3*e^2)*x^4 + 4*c*d*e*x + ( 
3*c^3*e^2*x^4 + 4*c^3*d*e*x^3 + c*d^2 + (c^3*d^2 - c*e^2)*x^2)*(c*x + 1)*( 
c*x - 1) + c*d^2 - (2*c^3*d^2 - 3*c*e^2)*x^2 + (6*c^4*e^2*x^5 + 8*c^4*d*e* 
x^4 - 8*c^2*d*e*x^2 + (2*c^4*d^2 - 7*c^2*e^2)*x^3 + 2*d*e - (c^2*d^2 - 2*e 
^2)*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)*sqrt(c*x - 1)) 
*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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