\(\int \frac {1}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx\) [93]

Optimal result

Integrand size = 12, antiderivative size = 88 \[ \int \frac {1}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {e^{-\frac {a}{b}} \left (-e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )\right )}{2 c \sqrt {a+b \text {arcsinh}(c x)}} \] Input:

Integrate[1/Sqrt[a + b*ArcSinh[c*x]],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6189, 3042, 3788, 26, 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.

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

\(\Big \downarrow \) 6189

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

\(\displaystyle \frac {\int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}}d\sqrt {a+b \text {arcsinh}(c x)}+\int e^{\frac {a+b \text {arcsinh}(c x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c x)}}{b c}\)

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

Int[1/Sqrt[a + b*ArcSinh[c*x]],x]
 

Output:

((Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/2 + (Sqr 
t[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*E^(a/b)))/(b*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 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 3788

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

rule 6189

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(1/sqrt(b*arcsinh(c*x) + a), x)
 

Giac [F]

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

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

Output:

integrate(1/sqrt(b*arcsinh(c*x) + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(asinh(c*x)*b + a)/(asinh(c*x)*b + a),x)