3.4.22 \(\int \frac {x \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx\) [322]

3.4.22.1 Optimal result

Integrand size = 26, antiderivative size = 248 \[ \int \frac {x \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {x \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}+\frac {\sqrt {1-c x} \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 c^2 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 c^2 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 c^2 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^2 \sqrt {-1+c x}} \]

output

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

3.4.22.2 Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.88 \[ \int \frac {x \sqrt {1-c^2 x^2}}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {1-c^2 x^2} \left (4 b c x-4 b c^3 x^3+(a+b \text {arccosh}(c x)) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )-3 (a+b \text {arccosh}(c x)) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-a \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-b \text {arccosh}(c x) \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+3 a \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+3 b \text {arccosh}(c x) \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{4 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))} \]

input

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

output

(Sqrt[1 - c^2*x^2]*(4*b*c*x - 4*b*c^3*x^3 + (a + b*ArcCosh[c*x])*CoshInteg 
ral[a/b + ArcCosh[c*x]]*Sinh[a/b] - 3*(a + b*ArcCosh[c*x])*CoshIntegral[3* 
(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - a*Cosh[a/b]*SinhIntegral[a/b + ArcCo 
sh[c*x]] - 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]*Cos 
h[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])]))/(4*b^2*c^2*Sqrt[-1 + c*x 
]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))
 

3.4.22.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.30 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.02, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {6357, 6296, 25, 3042, 26, 3784, 26, 3042, 26, 3779, 3782, 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.

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

\(\Big \downarrow \) 6357

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

\(\Big \downarrow \) 6296

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {1-c x} \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}+\frac {3 c \sqrt {1-c x} \int \frac {x^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \sqrt {1-c x} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}+\frac {3 c \sqrt {1-c x} \int \frac {x^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

\(\displaystyle -\frac {i \sqrt {1-c x} \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}+\frac {3 c \sqrt {1-c x} \int \frac {x^2}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 3782

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

\(\Big \downarrow \) 6302

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

input

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

output

-((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcCosh[c 
*x]))) - (I*Sqrt[1 - c*x]*(I*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b 
] - I*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b]))/(b^2*c^2*Sqrt[-1 + 
c*x]) + (3*Sqrt[1 - c*x]*(-1/4*(CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[ 
a/b]) - (CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/4 + (Cosh 
[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/4 + (Cosh[(3*a)/b]*SinhIntegra 
l[(3*(a + b*ArcCosh[c*x]))/b])/4))/(b^2*c^2*Sqrt[-1 + c*x])
 

3.4.22.3.1 Defintions of rubi rules used

rule 25

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

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

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 6296

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   S 
ubst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, 
b, c, n}, x]
 

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]
 

3.4.22.4 Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.63

method	result	size
default	\(-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-8 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{3} x^{3}-8 b \,c^{4} x^{4}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, b c x +8 b \,c^{2} x^{2}+\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}-3 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+3 \,\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}} b \,\operatorname {arccosh}\left (c x \right )-\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}} b \,\operatorname {arccosh}\left (c x \right )+a \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}-3 a \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+3 \,\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}} a -\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}} a \right )}{8 \left (c x +1\right ) c^{2} \left (c x -1\right ) b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\)	\(405\)

input

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

output

-1/8*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(-8*( 
c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^3*x^3-8*b*c^4*x^4+8*(c*x-1)^(1/2)*(c*x+1)^( 
1/2)*b*c*x+8*b*c^2*x^2+arccosh(c*x)*b*Ei(1,-arccosh(c*x)-a/b)*exp(-(-b*arc 
cosh(c*x)+a)/b)-3*arccosh(c*x)*b*Ei(1,-3*arccosh(c*x)-3*a/b)*exp(-(-b*arcc 
osh(c*x)+3*a)/b)+3*Ei(1,3*arccosh(c*x)+3*a/b)*exp((b*arccosh(c*x)+3*a)/b)* 
b*arccosh(c*x)-Ei(1,arccosh(c*x)+a/b)*exp((a+b*arccosh(c*x))/b)*b*arccosh( 
c*x)+a*Ei(1,-arccosh(c*x)-a/b)*exp(-(-b*arccosh(c*x)+a)/b)-3*a*Ei(1,-3*arc 
cosh(c*x)-3*a/b)*exp(-(-b*arccosh(c*x)+3*a)/b)+3*Ei(1,3*arccosh(c*x)+3*a/b 
)*exp((b*arccosh(c*x)+3*a)/b)*a-Ei(1,arccosh(c*x)+a/b)*exp((a+b*arccosh(c* 
x))/b)*a)/(c*x+1)/c^2/(c*x-1)/b^2/(a+b*arccosh(c*x))
 

3.4.22.5 Fricas [F]

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

input

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

3.4.22.6 Sympy [F]

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

input

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

output

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

3.4.22.7 Maxima [F]

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

input

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

output

-((c^2*x^3 - x)*(c*x + 1)*sqrt(c*x - 1) + (c^3*x^4 - c*x^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*x + 1)^(3/2)*(c*x - 1)* 
c^3*x^3 + (6*c^4*x^4 - 5*c^2*x^2 + 1)*(c*x + 1)*sqrt(c*x - 1) + (3*c^5*x^5 
 - 5*c^3*x^3 + 2*c*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)
 

3.4.22.8 Giac [F]

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

input

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

output

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

3.4.22.9 Mupad [F(-1)]

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

input

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

output

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