\(\int \frac {x^3 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx\) [273]

Optimal result

Integrand size = 28, antiderivative size = 350 \[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}+\frac {\sqrt {1-c x} \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b^2 c^4 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b^2 c^4 \sqrt {-1+c x}}-\frac {5 \sqrt {1-c x} \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b^2 c^4 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^4 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^4 \sqrt {-1+c x}}+\frac {5 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^4 \sqrt {-1+c x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {1-c^2 x^2} \left (16 b c^3 x^3-16 b c^5 x^5+2 (a+b \text {arccosh}(c x)) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )-3 (a+b \text {arccosh}(c x)) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-5 a \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-5 b \text {arccosh}(c x) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-2 a \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-2 b \text {arccosh}(c x) \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+3 a \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+3 b \text {arccosh}(c x) \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+5 a \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+5 b \text {arccosh}(c x) \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{16 b^2 c^4 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))} \] Input:

Integrate[(x^3*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x])^2,x]
 

Output:

(Sqrt[1 - c^2*x^2]*(16*b*c^3*x^3 - 16*b*c^5*x^5 + 2*(a + b*ArcCosh[c*x])*C 
oshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] - 3*(a + b*ArcCosh[c*x])*CoshInt 
egral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - 5*a*CoshIntegral[5*(a/b + Ar 
cCosh[c*x])]*Sinh[(5*a)/b] - 5*b*ArcCosh[c*x]*CoshIntegral[5*(a/b + ArcCos 
h[c*x])]*Sinh[(5*a)/b] - 2*a*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 
2*b*ArcCosh[c*x]*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 3*a*Cosh[(3* 
a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 3*b*ArcCosh[c*x]*Cosh[(3*a)/b 
]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 5*a*Cosh[(5*a)/b]*SinhIntegral[5* 
(a/b + ArcCosh[c*x])] + 5*b*ArcCosh[c*x]*Cosh[(5*a)/b]*SinhIntegral[5*(a/b 
 + ArcCosh[c*x])]))/(16*b^2*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCos 
h[c*x]))
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6357, 6302, 25, 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.

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

\(\Big \downarrow \) 6357

\(\displaystyle \frac {5 c \sqrt {1-c x} \int \frac {x^4}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \int \frac {x^2}{a+b \text {arccosh}(c x)}dx}{b c \sqrt {c x-1}}-\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 6302

\(\displaystyle \frac {5 \sqrt {1-c x} \int -\frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^4 \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \int -\frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^4 \sqrt {c x-1}}-\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {5 \sqrt {1-c x} \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^4 \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^4 \sqrt {c x-1}}-\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

Input:

Int[(x^3*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x])^2,x]
 

Output:

-((x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcCosh 
[c*x]))) - (3*Sqrt[1 - c*x]*(-1/4*(CoshIntegral[(a + b*ArcCosh[c*x])/b]*Si 
nh[a/b]) - (CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/4 + (C 
osh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/4 + (Cosh[(3*a)/b]*SinhInte 
gral[(3*(a + b*ArcCosh[c*x]))/b])/4))/(b^2*c^4*Sqrt[-1 + c*x]) + (5*Sqrt[1 
 - c*x]*(-1/8*(CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b]) - (3*CoshIn 
tegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/16 - (CoshIntegral[(5*(a 
 + b*ArcCosh[c*x]))/b]*Sinh[(5*a)/b])/16 + (Cosh[a/b]*SinhIntegral[(a + b* 
ArcCosh[c*x])/b])/8 + (3*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x] 
))/b])/16 + (Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/16))/ 
(b^2*c^4*Sqrt[-1 + c*x])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 6302

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
1/(b*c^(m + 1))   Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, 
 a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
 

rule 6357

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c* 
x]*(d + e*x^2)^p]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Simp[ 
f*(m/(b*c*(n + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[(f 
*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^( 
n + 1), x], x] - Simp[c*((m + 2*p + 1)/(b*f*(n + 1)))*Simp[(d + e*x^2)^p/(( 
1 + c*x)^p*(-1 + c*x)^p)]   Int[(f*x)^(m + 1)*(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, 
f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 
2*p + 1, 0] && IGtQ[m, -3]
 

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.59

method	result	size
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (32 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{5} x^{5}+32 b \,c^{6} x^{6}-32 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{3} x^{3}-32 b \,c^{4} x^{4}+5 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}+3 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}-2 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}-5 \,\operatorname {expIntegral}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}} b \,\operatorname {arccosh}\left (c x \right )-3 \,\operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+2 \,\operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}} b \,\operatorname {arccosh}\left (c x \right )+5 a \,\operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}+3 a \,\operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}-2 a \,\operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}-5 \,\operatorname {expIntegral}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}} a -3 \,\operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}} a +2 \,\operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}} a \right )}{32 \left (c x +1\right ) c^{4} \left (c x -1\right ) b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\)	\(555\)

Input:

int(x^3*(-c^2*x^2+1)^(1/2)/(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

1/32*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(32*( 
c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^5*x^5+32*b*c^6*x^6-32*(c*x-1)^(1/2)*(c*x+1) 
^(1/2)*b*c^3*x^3-32*b*c^4*x^4+5*arccosh(c*x)*b*Ei(1,-5*arccosh(c*x)-5*a/b) 
*exp(-(-b*arccosh(c*x)+5*a)/b)+3*arccosh(c*x)*b*Ei(1,-3*arccosh(c*x)-3*a/b 
)*exp(-(-b*arccosh(c*x)+3*a)/b)-2*arccosh(c*x)*b*Ei(1,-arccosh(c*x)-a/b)*e 
xp(-(-b*arccosh(c*x)+a)/b)-5*Ei(1,5*arccosh(c*x)+5*a/b)*exp((b*arccosh(c*x 
)+5*a)/b)*b*arccosh(c*x)-3*Ei(1,3*arccosh(c*x)+3*a/b)*exp((b*arccosh(c*x)+ 
3*a)/b)*b*arccosh(c*x)+2*Ei(1,arccosh(c*x)+a/b)*exp((a+b*arccosh(c*x))/b)* 
b*arccosh(c*x)+5*a*Ei(1,-5*arccosh(c*x)-5*a/b)*exp(-(-b*arccosh(c*x)+5*a)/ 
b)+3*a*Ei(1,-3*arccosh(c*x)-3*a/b)*exp(-(-b*arccosh(c*x)+3*a)/b)-2*a*Ei(1, 
-arccosh(c*x)-a/b)*exp(-(-b*arccosh(c*x)+a)/b)-5*Ei(1,5*arccosh(c*x)+5*a/b 
)*exp((b*arccosh(c*x)+5*a)/b)*a-3*Ei(1,3*arccosh(c*x)+3*a/b)*exp((b*arccos 
h(c*x)+3*a)/b)*a+2*Ei(1,arccosh(c*x)+a/b)*exp((a+b*arccosh(c*x))/b)*a)/(c* 
x+1)/c^4/(c*x-1)/b^2/(a+b*arccosh(c*x))
 

Fricas [F]

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

integrate(x^3*(-c^2*x^2+1)^(1/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas 
")
 

Output:

integral(sqrt(-c^2*x^2 + 1)*x^3/(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + 
 a^2), x)
 

Sympy [F]

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

integrate(x**3*(-c**2*x**2+1)**(1/2)/(a+b*acosh(c*x))**2,x)
 

Output:

Integral(x**3*sqrt(-(c*x - 1)*(c*x + 1))/(a + b*acosh(c*x))**2, x)
 

Maxima [F]

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

integrate(x^3*(-c^2*x^2+1)^(1/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima 
")
 

Output:

-((c^2*x^5 - x^3)*(c*x + 1)*sqrt(c*x - 1) + (c^3*x^6 - c*x^4)*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^3*x^5 - 2*c*x^3)*(c*x 
 + 1)^(3/2)*(c*x - 1) + (10*c^4*x^6 - 11*c^2*x^4 + 3*x^2)*(c*x + 1)*sqrt(c 
*x - 1) + (5*c^5*x^7 - 9*c^3*x^5 + 4*c*x^3)*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(-2)]

Exception generated. \[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3*(-c^2*x^2+1)^(1/2)/(a+b*arccosh(c*x))^2,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^3\,\sqrt {1-c^2\,x^2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \] Input:

int((x^3*(1 - c^2*x^2)^(1/2))/(a + b*acosh(c*x))^2,x)
 

Output:

int((x^3*(1 - c^2*x^2)^(1/2))/(a + b*acosh(c*x))^2, x)
 

Reduce [F]

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

int(x^3*(-c^2*x^2+1)^(1/2)/(a+b*acosh(c*x))^2,x)
 

Output:

int((sqrt( - c**2*x**2 + 1)*x**3)/(acosh(c*x)**2*b**2 + 2*acosh(c*x)*a*b + 
 a**2),x)