\(\int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx\) [235]

Optimal result

Integrand size = 25, antiderivative size = 139 \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{2 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {-1+c x}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.30 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {-((-1+c x) (1+c x))} \left (\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\log (a+b \text {arccosh}(c x))-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{2 b c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \] Input:

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

Output:

(Sqrt[-((-1 + c*x)*(1 + c*x))]*(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCos 
h[c*x])] - Log[a + b*ArcCosh[c*x]] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + A 
rcCosh[c*x])]))/(2*b*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.65, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6321, 3042, 25, 3793, 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.

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

\(\Big \downarrow \) 6321

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Sqrt[1 - c*x]*((Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/2 
 - Log[a + b*ArcCosh[c*x]]/2 - (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCo 
sh[c*x]))/b])/2))/(b*c*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 3042

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

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6321

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] 
  Subst[Int[x^n*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] 
/; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0]
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.19

method	result	size
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+2 \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c x +\operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}+\operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}\right )}{4 \left (c x -1\right ) \left (c x +1\right ) c b}\)	\(165\)

Input:

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

Output:

1/4*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(2*(c* 
x-1)^(1/2)*(c*x+1)^(1/2)*ln(a+b*arccosh(c*x))+2*ln(a+b*arccosh(c*x))*c*x+E 
i(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)+Ei(1,-2*arccosh(c*x) 
-2*a/b)*exp(-(-b*arccosh(c*x)+2*a)/b))/(c*x-1)/(c*x+1)/c/b
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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