\(\int \sqrt {a+b \text {arcsinh}(c+d x)} \, dx\) [122]

Optimal result

Integrand size = 14, antiderivative size = 115 \[ \int \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d}+\frac {\sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d}-\frac {\sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d} \] Output:

(d*x+c)*(a+b*arcsinh(d*x+c))^(1/2)/d+1/4*b^(1/2)*exp(a/b)*Pi^(1/2)*erf((a+ 
b*arcsinh(d*x+c))^(1/2)/b^(1/2))/d-1/4*b^(1/2)*Pi^(1/2)*erfi((a+b*arcsinh( 
d*x+c))^(1/2)/b^(1/2))/d/exp(a/b)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97 \[ \int \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {e^{-\frac {a}{b}} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )}{\sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}}}\right )}{2 d} \] Input:

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

Output:

(Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c + 
 d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + d* 
x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/(2*d*E^(a/b))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6273, 6187, 6234, 25, 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.

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

\(\Big \downarrow \) 6273

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

\(\Big \downarrow \) 6187

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

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3789

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

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

Output:

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

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 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 6187

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A 
rcSinh[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

rule 6234

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

rule 6273

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((a+b*arcsinh(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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