\(\int \frac {(d-c^2 d x^2)^3}{(a+b \text {arccosh}(c x))^2} \, dx\) [33]

Optimal result

Integrand size = 24, antiderivative size = 307 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {d^3 (-1+c x)^{7/2} (1+c x)^{7/2}}{b c (a+b \text {arccosh}(c x))}+\frac {35 d^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 b^2 c}-\frac {63 d^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 b^2 c}+\frac {35 d^3 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 b^2 c}-\frac {7 d^3 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 b^2 c}-\frac {35 d^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 b^2 c}+\frac {63 d^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 b^2 c}-\frac {35 d^3 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 b^2 c}+\frac {7 d^3 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 b^2 c} \] Output:

d^3*(c*x-1)^(7/2)*(c*x+1)^(7/2)/b/c/(a+b*arccosh(c*x))+35/64*d^3*cosh(a/b) 
*Chi((a+b*arccosh(c*x))/b)/b^2/c-63/64*d^3*cosh(3*a/b)*Chi(3*(a+b*arccosh( 
c*x))/b)/b^2/c+35/64*d^3*cosh(5*a/b)*Chi(5*(a+b*arccosh(c*x))/b)/b^2/c-7/6 
4*d^3*cosh(7*a/b)*Chi(7*(a+b*arccosh(c*x))/b)/b^2/c-35/64*d^3*sinh(a/b)*Sh 
i((a+b*arccosh(c*x))/b)/b^2/c+63/64*d^3*sinh(3*a/b)*Shi(3*(a+b*arccosh(c*x 
))/b)/b^2/c-35/64*d^3*sinh(5*a/b)*Shi(5*(a+b*arccosh(c*x))/b)/b^2/c+7/64*d 
^3*sinh(7*a/b)*Shi(7*(a+b*arccosh(c*x))/b)/b^2/c
 

Mathematica [A] (warning: unable to verify)

Time = 2.06 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.87 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{(a+b \text {arccosh}(c x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(d^3*(-64*b*Sqrt[(-1 + c*x)/(1 + c*x)] - 64*b*c*x*Sqrt[(-1 + c*x)/(1 + c*x 
)] + 192*b*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 192*b*c^3*x^3*Sqrt[(-1 + c 
*x)/(1 + c*x)] - 192*b*c^4*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - 192*b*c^5*x^5* 
Sqrt[(-1 + c*x)/(1 + c*x)] + 64*b*c^6*x^6*Sqrt[(-1 + c*x)/(1 + c*x)] + 64* 
b*c^7*x^7*Sqrt[(-1 + c*x)/(1 + c*x)] + 35*(a + b*ArcCosh[c*x])*Cosh[a/b]*C 
oshIntegral[a/b + ArcCosh[c*x]] - 63*(a + b*ArcCosh[c*x])*Cosh[(3*a)/b]*Co 
shIntegral[3*(a/b + ArcCosh[c*x])] + 35*a*Cosh[(5*a)/b]*CoshIntegral[5*(a/ 
b + ArcCosh[c*x])] + 35*b*ArcCosh[c*x]*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + 
 ArcCosh[c*x])] - 7*a*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcCosh[c*x])] - 
 7*b*ArcCosh[c*x]*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcCosh[c*x])] - 35* 
a*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 35*b*ArcCosh[c*x]*Sinh[a/b] 
*SinhIntegral[a/b + ArcCosh[c*x]] + 63*a*Sinh[(3*a)/b]*SinhIntegral[3*(a/b 
 + ArcCosh[c*x])] + 63*b*ArcCosh[c*x]*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + 
ArcCosh[c*x])] - 35*a*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] - 
 35*b*ArcCosh[c*x]*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] + 7* 
a*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcCosh[c*x])] + 7*b*ArcCosh[c*x]*Si 
nh[(7*a)/b]*SinhIntegral[7*(a/b + ArcCosh[c*x])]))/(64*b^2*c*(a + b*ArcCos 
h[c*x]))
 

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.80, 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.

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

\(\Big \downarrow \) 6319

\(\displaystyle \frac {d^3 (c x-1)^{7/2} (c x+1)^{7/2}}{b c (a+b \text {arccosh}(c x))}-\frac {7 c d^3 \int \frac {x (c x-1)^{5/2} (c x+1)^{5/2}}{a+b \text {arccosh}(c x)}dx}{b}\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {d^3 (c x-1)^{7/2} (c x+1)^{7/2}}{b c (a+b \text {arccosh}(c x))}-\frac {7 d^3 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^6\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {d^3 (c x-1)^{7/2} (c x+1)^{7/2}}{b c (a+b \text {arccosh}(c x))}-\frac {7 d^3 \int \left (\frac {\cosh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arccosh}(c x))}{b}\right )}{64 (a+b \text {arccosh}(c x))}-\frac {5 \cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{64 (a+b \text {arccosh}(c x))}+\frac {9 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{64 (a+b \text {arccosh}(c x))}-\frac {5 \cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{64 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^3 (c x-1)^{7/2} (c x+1)^{7/2}}{b c (a+b \text {arccosh}(c x))}-\frac {7 d^3 \left (-\frac {5}{64} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {9}{64} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {5}{64} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{64} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )+\frac {5}{64} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {9}{64} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {5}{64} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{64} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c}\)

Input:

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

Output:

(d^3*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2))/(b*c*(a + b*ArcCosh[c*x])) - (7*d^3 
*((-5*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/64 + (9*Cosh[(3*a)/b 
]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/64 - (5*Cosh[(5*a)/b]*CoshInte 
gral[(5*(a + b*ArcCosh[c*x]))/b])/64 + (Cosh[(7*a)/b]*CoshIntegral[(7*(a + 
 b*ArcCosh[c*x]))/b])/64 + (5*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/ 
b])/64 - (9*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/64 + ( 
5*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/64 - (Sinh[(7*a) 
/b]*SinhIntegral[(7*(a + b*ArcCosh[c*x]))/b])/64))/(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. \(907\) vs. \(2(287)=574\).

Time = 0.37 (sec) , antiderivative size = 908, normalized size of antiderivative = 2.96

method	result	size
derivativedivides	\(\frac {-\frac {\left (-64 c^{6} x^{6} \sqrt {c x -1}\, \sqrt {c x +1}+80 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}-24 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}+64 c^{7} x^{7}-112 c^{5} x^{5}+56 c^{3} x^{3}-7 c x \right ) d^{3}}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {7 d^{3} {\mathrm e}^{\frac {7 a}{b}} \operatorname {expIntegral}_{1}\left (7 \,\operatorname {arccosh}\left (c x \right )+\frac {7 a}{b}\right )}{128 b^{2}}+\frac {7 \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^{3}}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {35 d^{3} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right )}{128 b^{2}}-\frac {21 \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^{3}}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {63 d^{3} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{128 b^{2}}+\frac {35 \left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d^{3}}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {35 d^{3} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{128 b^{2}}-\frac {35 d^{3} \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {35 d^{3} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{128 b^{2}}+\frac {21 d^{3} \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 )}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {63 d^{3} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{128 b^{2}}-\frac {7 d^{3} \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 )}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {35 d^{3} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right )}{128 b^{2}}+\frac {d^{3} \left (64 c^{7} x^{7}-112 c^{5} x^{5}+64 c^{6} x^{6} \sqrt {c x -1}\, \sqrt {c x +1}+56 c^{3} x^{3}-80 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}-7 c x +24 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right )}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {7 d^{3} {\mathrm e}^{-\frac {7 a}{b}} \operatorname {expIntegral}_{1}\left (-7 \,\operatorname {arccosh}\left (c x \right )-\frac {7 a}{b}\right )}{128 b^{2}}}{c}\)	\(908\)
default	\(\frac {-\frac {\left (-64 c^{6} x^{6} \sqrt {c x -1}\, \sqrt {c x +1}+80 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}-24 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}+64 c^{7} x^{7}-112 c^{5} x^{5}+56 c^{3} x^{3}-7 c x \right ) d^{3}}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {7 d^{3} {\mathrm e}^{\frac {7 a}{b}} \operatorname {expIntegral}_{1}\left (7 \,\operatorname {arccosh}\left (c x \right )+\frac {7 a}{b}\right )}{128 b^{2}}+\frac {7 \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^{3}}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {35 d^{3} {\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right )}{128 b^{2}}-\frac {21 \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^{3}}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {63 d^{3} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{128 b^{2}}+\frac {35 \left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d^{3}}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {35 d^{3} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{128 b^{2}}-\frac {35 d^{3} \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {35 d^{3} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{128 b^{2}}+\frac {21 d^{3} \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 )}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {63 d^{3} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{128 b^{2}}-\frac {7 d^{3} \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 )}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {35 d^{3} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right )}{128 b^{2}}+\frac {d^{3} \left (64 c^{7} x^{7}-112 c^{5} x^{5}+64 c^{6} x^{6} \sqrt {c x -1}\, \sqrt {c x +1}+56 c^{3} x^{3}-80 c^{4} x^{4} \sqrt {c x -1}\, \sqrt {c x +1}-7 c x +24 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right )}{128 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}+\frac {7 d^{3} {\mathrm e}^{-\frac {7 a}{b}} \operatorname {expIntegral}_{1}\left (-7 \,\operatorname {arccosh}\left (c x \right )-\frac {7 a}{b}\right )}{128 b^{2}}}{c}\)	\(908\)

Input:

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

Output:

1/c*(-1/128*(-64*c^6*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)+80*c^4*x^4*(c*x-1)^(1 
/2)*(c*x+1)^(1/2)-24*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c* 
x+1)^(1/2)+64*c^7*x^7-112*c^5*x^5+56*c^3*x^3-7*c*x)*d^3/b/(a+b*arccosh(c*x 
))+7/128*d^3/b^2*exp(7*a/b)*Ei(1,7*arccosh(c*x)+7*a/b)+7/128*(-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^3/b/(a+b*arccosh(c*x)) 
-35/128*d^3/b^2*exp(5*a/b)*Ei(1,5*arccosh(c*x)+5*a/b)-21/128*(-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^ 
3/b/(a+b*arccosh(c*x))+63/128*d^3/b^2*exp(3*a/b)*Ei(1,3*arccosh(c*x)+3*a/b 
)+35/128*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x)*d^3/b/(a+b*arccosh(c*x))-35/12 
8*d^3/b^2*exp(a/b)*Ei(1,arccosh(c*x)+a/b)-35/128/b*d^3*(c*x+(c*x-1)^(1/2)* 
(c*x+1)^(1/2))/(a+b*arccosh(c*x))-35/128/b^2*d^3*exp(-a/b)*Ei(1,-arccosh(c 
*x)-a/b)+21/128/b*d^3*(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))+63/128/b^2*d^3*exp(-3*a 
/b)*Ei(1,-3*arccosh(c*x)-3*a/b)-7/128/b*d^3*(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))-35/128/b^2*d^3*exp(-5*a 
/b)*Ei(1,-5*arccosh(c*x)-5*a/b)+1/128/b*d^3*(64*c^7*x^7-112*c^5*x^5+64*c^6 
*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)+56*c^3*x^3-80*c^4*x^4*(c*x-1)^(1/2)*(c*x+ 
1)^(1/2)-7*c*x+24*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*...
 

Fricas [F]

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

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

Output:

integral(-(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3)/(b^2*arccosh 
(c*x)^2 + 2*a*b*arccosh(c*x) + a^2), x)
 

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3}{(a+b \text {arccosh}(c x))^2} \, dx=- d^{3} \left (\int \frac {3 c^{2} x^{2}}{a^{2} + 2 a b \operatorname {acosh}{\left (c x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c x \right )}}\, dx + \int \left (- \frac {3 c^{4} x^{4}}{a^{2} + 2 a b \operatorname {acosh}{\left (c x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c x \right )}}\right )\, dx + \int \frac {c^{6} x^{6}}{a^{2} + 2 a b \operatorname {acosh}{\left (c x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c x \right )}}\, dx + \int \left (- \frac {1}{a^{2} + 2 a b \operatorname {acosh}{\left (c x \right )} + b^{2} \operatorname {acosh}^{2}{\left (c x \right )}}\right )\, dx\right ) \] Input:

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

Output:

-d**3*(Integral(3*c**2*x**2/(a**2 + 2*a*b*acosh(c*x) + b**2*acosh(c*x)**2) 
, x) + Integral(-3*c**4*x**4/(a**2 + 2*a*b*acosh(c*x) + b**2*acosh(c*x)**2 
), x) + Integral(c**6*x**6/(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]

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

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

Output:

(c^9*d^3*x^9 - 4*c^7*d^3*x^7 + 6*c^5*d^3*x^5 - 4*c^3*d^3*x^3 + c*d^3*x + ( 
c^8*d^3*x^8 - 4*c^6*d^3*x^6 + 6*c^4*d^3*x^4 - 4*c^2*d^3*x^2 + d^3)*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)*l 
og(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) - integrate((7*c^10*d^3*x^10 - 29*c 
^8*d^3*x^8 + 46*c^6*d^3*x^6 - 34*c^4*d^3*x^4 + 11*c^2*d^3*x^2 + (7*c^8*d^3 
*x^8 - 20*c^6*d^3*x^6 + 18*c^4*d^3*x^4 - 4*c^2*d^3*x^2 - d^3)*(c*x + 1)*(c 
*x - 1) - d^3 + 7*(2*c^9*d^3*x^9 - 7*c^7*d^3*x^7 + 9*c^5*d^3*x^5 - 5*c^3*d 
^3*x^3 + c*d^3*x)*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)*sqrt(c*x + 1)*sqrt(c*x - 1) + b 
^2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3}{(a+b \text {arccosh}(c x))^2} \, dx=d^{3} \left (-\left (\int \frac {x^{6}}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) c^{6}+3 \left (\int \frac {x^{4}}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}-3 \left (\int \frac {x^{2}}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {1}{\mathit {acosh} \left (c x \right )^{2} b^{2}+2 \mathit {acosh} \left (c x \right ) a b +a^{2}}d x \right ) \] Input:

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

Output:

d**3*( - int(x**6/(acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b + a**2),x)*c**6 + 
 3*int(x**4/(acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b + a**2),x)*c**4 - 3*int 
(x**2/(acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b + a**2),x)*c**2 + int(1/(acos 
h(c*x)**2*b**2 + 2*acosh(c*x)*a*b + a**2),x))