\(\int \frac {1}{(a+b \text {arccosh}(c x))^2} \, dx\) [140]

Optimal result

Integrand size = 10, antiderivative size = 90 \[ \int \frac {1}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x}}{b c (a+b \text {arccosh}(c x))}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b^2 c} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {-\frac {b \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}{a+b \text {arccosh}(c x)}+\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )}{b^2 c} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6295, 6368, 3042, 3784, 26, 3042, 26, 3779, 3782}

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.

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

\(\Big \downarrow \) 6295

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

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

Input:

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

Output:

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

Defintions of rubi rules used

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 6295

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

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 [A] (verified)

Time = 0.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.39

method	result	size
derivativedivides	\(\frac {\frac {-\sqrt {c x -1}\, \sqrt {c x +1}+c x}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {c x +\sqrt {c x -1}\, \sqrt {c x +1}}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}}{c}\)	\(125\)
default	\(\frac {\frac {-\sqrt {c x -1}\, \sqrt {c x +1}+c x}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {c x +\sqrt {c x -1}\, \sqrt {c x +1}}{2 b \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}}{c}\)	\(125\)

Input:

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

Output:

1/c*(1/2*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x)/b/(a+b*arccosh(c*x))-1/2/b^2*e 
xp(a/b)*Ei(1,arccosh(c*x)+a/b)-1/2/b*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(a+ 
b*arccosh(c*x))-1/2/b^2*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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