\(\int \frac {(d-c^2 d x^2)^2}{(a+b \text {arccosh}(c x))^2} \, dx\) [34]

Optimal result

Integrand size = 24, antiderivative size = 240 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c (a+b \text {arccosh}(c x))}+\frac {5 d^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c}-\frac {15 d^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c}+\frac {5 d^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c}-\frac {5 d^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c}+\frac {15 d^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c}-\frac {5 d^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c} \] Output:

-d^2*(c*x-1)^(5/2)*(c*x+1)^(5/2)/b/c/(a+b*arccosh(c*x))+5/8*d^2*cosh(a/b)* 
Chi((a+b*arccosh(c*x))/b)/b^2/c-15/16*d^2*cosh(3*a/b)*Chi(3*(a+b*arccosh(c 
*x))/b)/b^2/c+5/16*d^2*cosh(5*a/b)*Chi(5*(a+b*arccosh(c*x))/b)/b^2/c-5/8*d 
^2*sinh(a/b)*Shi((a+b*arccosh(c*x))/b)/b^2/c+15/16*d^2*sinh(3*a/b)*Shi(3*( 
a+b*arccosh(c*x))/b)/b^2/c-5/16*d^2*sinh(5*a/b)*Shi(5*(a+b*arccosh(c*x))/b 
)/b^2/c
 

Mathematica [A] (warning: unable to verify)

Time = 1.49 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {d^2 \left (-\frac {16 b \left (\frac {-1+c x}{1+c x}\right )^{5/2} (1+c x)^5}{a+b \text {arccosh}(c x)}+20 \left (\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )+5 \left (-2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )\right )}{16 b^2 c} \] Input:

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

Output:

(d^2*((-16*b*((-1 + c*x)/(1 + c*x))^(5/2)*(1 + c*x)^5)/(a + b*ArcCosh[c*x] 
) + 20*(Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] - Cosh[(3*a)/b]*CoshInt 
egral[3*(a/b + ArcCosh[c*x])] - Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] 
 + Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])]) + 5*(-2*Cosh[a/b]*C 
oshIntegral[a/b + ArcCosh[c*x]] + Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcC 
osh[c*x])] + Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcCosh[c*x])] + 2*Sinh[a 
/b]*SinhIntegral[a/b + ArcCosh[c*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b + 
 ArcCosh[c*x])] - Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])])))/(1 
6*b^2*c)
 

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.82, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6319, 6368, 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.

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

\(\Big \downarrow \) 6319

\(\displaystyle \frac {5 c d^2 \int \frac {x (c x-1)^{3/2} (c x+1)^{3/2}}{a+b \text {arccosh}(c x)}dx}{b}-\frac {d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {5 d^2 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c}-\frac {d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {5 d^2 \int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}+\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c}-\frac {d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((d^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(b*c*(a + b*ArcCosh[c*x]))) + (5* 
d^2*((Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/8 - (3*Cosh[(3*a)/b] 
*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/16 + (Cosh[(5*a)/b]*CoshIntegra 
l[(5*(a + b*ArcCosh[c*x]))/b])/16 - (Sinh[a/b]*SinhIntegral[(a + b*ArcCosh 
[c*x])/b])/8 + (3*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/ 
16 - (Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/16))/(b^2*c)
 

Defintions of rubi rules used

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 6368

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x 
_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* 
Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[In 
t[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, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ 
e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(590\) vs. \(2(224)=448\).

Time = 0.22 (sec) , antiderivative size = 591, normalized size of antiderivative = 2.46

method	result	size
derivativedivides	\(\frac {\frac {\left (-16 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+12 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}+16 c^{5} x^{5}-20 c^{3} x^{3}+5 c x \right ) d^{2}}{32 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {5 d^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right )}{32 b^{2}}-\frac {5 \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 ) d^{2}}{32 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {15 d^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{32 b^{2}}+\frac {5 \left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d^{2}}{16 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {5 d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{16 b^{2}}-\frac {5 d^{2} \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{16 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {5 d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{16 b^{2}}+\frac {5 d^{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 )}{32 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {15 d^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{32 b^{2}}-\frac {d^{2} \left (16 c^{5} x^{5}-20 c^{3} x^{3}+16 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+5 c x -12 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right )}{32 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {5 d^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right )}{32 b^{2}}}{c}\)	\(591\)
default	\(\frac {\frac {\left (-16 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+12 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}+16 c^{5} x^{5}-20 c^{3} x^{3}+5 c x \right ) d^{2}}{32 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {5 d^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right )}{32 b^{2}}-\frac {5 \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 ) d^{2}}{32 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {15 d^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{32 b^{2}}+\frac {5 \left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d^{2}}{16 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {5 d^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{16 b^{2}}-\frac {5 d^{2} \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{16 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {5 d^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{16 b^{2}}+\frac {5 d^{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 )}{32 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {15 d^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{32 b^{2}}-\frac {d^{2} \left (16 c^{5} x^{5}-20 c^{3} x^{3}+16 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}+5 c x -12 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right )}{32 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {5 d^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right )}{32 b^{2}}}{c}\)	\(591\)

Input:

int((-c^2*d*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)*d 
^2/b/(a+b*arccosh(c*x))-5/32*d^2/b^2*exp(5*a/b)*Ei(1,5*arccosh(c*x)+5*a/b) 
-5/32*(-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)*d^2/b/(a+b*arccosh(c*x))+15/32*d^2/b^2*exp(3*a/b)*Ei(1,3* 
arccosh(c*x)+3*a/b)+5/16*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x)*d^2/b/(a+b*arc 
cosh(c*x))-5/16*d^2/b^2*exp(a/b)*Ei(1,arccosh(c*x)+a/b)-5/16/b*d^2*(c*x+(c 
*x-1)^(1/2)*(c*x+1)^(1/2))/(a+b*arccosh(c*x))-5/16/b^2*d^2*exp(-a/b)*Ei(1, 
-arccosh(c*x)-a/b)+5/32/b*d^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))+15/32/b^2*d^2*e 
xp(-3*a/b)*Ei(1,-3*arccosh(c*x)-3*a/b)-1/32/b*d^2*(16*c^5*x^5-20*c^3*x^3+1 
6*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*d^2*exp( 
-5*a/b)*Ei(1,-5*arccosh(c*x)-5*a/b))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

d**2*(Integral(-2*c**2*x**2/(a**2 + 2*a*b*acosh(c*x) + b**2*acosh(c*x)**2) 
, x) + Integral(c**4*x**4/(a**2 + 2*a*b*acosh(c*x) + b**2*acosh(c*x)**2), 
x) + Integral(1/(a**2 + 2*a*b*acosh(c*x) + b**2*acosh(c*x)**2), x))
 

Maxima [F]

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

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

Output:

-(c^7*d^2*x^7 - 3*c^5*d^2*x^5 + 3*c^3*d^2*x^3 - c*d^2*x + (c^6*d^2*x^6 - 3 
*c^4*d^2*x^4 + 3*c^2*d^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((5*c^8*d^2*x^8 - 16*c^6*d^2*x^6 + 18*c^4*d^2*x^4 - 8*c 
^2*d^2*x^2 + (5*c^6*d^2*x^6 - 9*c^4*d^2*x^4 + 3*c^2*d^2*x^2 + d^2)*(c*x + 
1)*(c*x - 1) + 5*(2*c^7*d^2*x^7 - 5*c^5*d^2*x^5 + 4*c^3*d^2*x^3 - c*d^2*x) 
*sqrt(c*x + 1)*sqrt(c*x - 1) + d^2)/(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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