\(\int \frac {1}{\sqrt {a-b \text {arccosh}(c+d x)}} \, dx\) [10]

Optimal result

Integrand size = 15, antiderivative size = 94 \[ \int \frac {1}{\sqrt {a-b \text {arccosh}(c+d x)}} \, dx=-\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a-b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a-b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d} \] Output:

-1/2*exp(a/b)*Pi^(1/2)*erf((a-b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(1/2)/d+1 
/2*Pi^(1/2)*erfi((a-b*arccosh(d*x+c))^(1/2)/b^(1/2))/b^(1/2)/d/exp(a/b)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt {a-b \text {arccosh}(c+d x)}} \, dx=\frac {e^{-\frac {a}{b}} \left (e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}-\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}-\text {arccosh}(c+d x)\right )+\sqrt {-\frac {a}{b}+\text {arccosh}(c+d x)} \Gamma \left (\frac {1}{2},-\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )}{2 d \sqrt {a-b \text {arccosh}(c+d x)}} \] Input:

Integrate[1/Sqrt[a - b*ArcCosh[c + d*x]],x]
 

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.49 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {6410, 6296, 3042, 26, 3789, 2611, 2633, 2634}

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.

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

\(\Big \downarrow \) 6410

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

\(\Big \downarrow \) 6296

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3789

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

Int[1/Sqrt[a - b*ArcCosh[c + d*x]],x]
 

Output:

(I*((I/2)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a - b*ArcCosh[c + d*x]]/Sqrt[b 
]] - ((I/2)*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a - b*ArcCosh[c + d*x]]/Sqrt[b]])/E 
^(a/b)))/(b*d)
 

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 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

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

rule 3789

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
/2   Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2   Int[(c + d*x)^m*E 
^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
 

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 6410

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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