3.4.24 \(\int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx\) [324]

3.4.24.1 Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c x (a+b \text {arccosh}(c x))}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \text {Int}\left (\frac {1}{x^2 (a+b \text {arccosh}(c x))},x\right )}{b c \sqrt {-1+c x}} \]

output

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

3.4.24.2 Mathematica [N/A]

Not integrable

Time = 9.98 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx \]

input

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

output

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

3.4.24.3 Rubi [N/A]

Not integrable

Time = 1.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {6357, 6296, 25, 3042, 26, 3784, 26, 3042, 26, 3779, 3782, 6303}

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.

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

\(\Big \downarrow \) 6357

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

\(\Big \downarrow \) 6296

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

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

\(\Big \downarrow \) 6303

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

input

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

output

$Aborted
 

3.4.24.3.1 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 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 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 6303

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Unintegrable[(d*x)^m*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d, m 
, n}, x]
 

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]
 

3.4.24.4 Maple [N/A] (verified)

Not integrable

Time = 1.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

input

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

output

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

3.4.24.5 Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

input

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

output

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

3.4.24.6 Sympy [N/A]

Not integrable

Time = 2.86 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

input

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

output

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

3.4.24.7 Maxima [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 442, normalized size of antiderivative = 15.79 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

input

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

output

-((c^2*x^2 - 1)*(c*x + 1)*sqrt(c*x - 1) + (c^3*x^3 - c*x)*sqrt(c*x + 1))*s 
qrt(-c*x + 1)/(a*b*c^3*x^3 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x^2 - a*b 
*c*x + (b^2*c^3*x^3 + sqrt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x^2 - b^2*c*x)*l 
og(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) + integrate(((c^3*x^3 + 2*c*x)*(c*x 
 + 1)^(3/2)*(c*x - 1) + (2*c^4*x^4 + c^2*x^2 - 1)*(c*x + 1)*sqrt(c*x - 1) 
+ (c^5*x^5 - c^3*x^3)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^5*x^6 + (c*x + 
1)*(c*x - 1)*a*b*c^3*x^4 - 2*a*b*c^3*x^4 + a*b*c*x^2 + 2*(a*b*c^4*x^5 - a* 
b*c^2*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^6 + (c*x + 1)*(c*x - 1 
)*b^2*c^3*x^4 - 2*b^2*c^3*x^4 + b^2*c*x^2 + 2*(b^2*c^4*x^5 - b^2*c^2*x^3)* 
sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)
 

3.4.24.8 Giac [F(-2)]

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

input

integrate((-c^2*x^2+1)^(1/2)/x/(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
 

3.4.24.9 Mupad [N/A]

Not integrable

Time = 3.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\sqrt {1-c^2\,x^2}}{x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

input

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

output

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