3.5.73 \(\int \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)} \, dx\) [473]

3.5.73.1 Optimal result

Integrand size = 23, antiderivative size = 186 \[ \int \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)} \, dx=\frac {1}{2} x \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}+\frac {\sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}}{3 a \sqrt {1+a^2 x^2}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}} \]

output

1/3*arcsinh(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+1/32*erf(2^ 
(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+ 
1)^(1/2)-1/32*erfi(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*c*x^2 
+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+1/2*x*(a^2*c*x^2+c)^(1/2)*arcsinh(a*x)^(1/2)
 

3.5.73.2 Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.56 \[ \int \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)} \, dx=\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (16 \text {arcsinh}(a x)^2-3 \sqrt {2} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {3}{2},-2 \text {arcsinh}(a x)\right )-3 \sqrt {2} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {3}{2},2 \text {arcsinh}(a x)\right )\right )}{48 a \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \]

input

Integrate[Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]],x]
 

output

(Sqrt[c*(1 + a^2*x^2)]*(16*ArcSinh[a*x]^2 - 3*Sqrt[2]*Sqrt[-ArcSinh[a*x]]* 
Gamma[3/2, -2*ArcSinh[a*x]] - 3*Sqrt[2]*Sqrt[ArcSinh[a*x]]*Gamma[3/2, 2*Ar 
cSinh[a*x]]))/(48*a*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])
 

3.5.73.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.01 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.90, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {6200, 6195, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634, 6198}

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.

\(\displaystyle \int \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c} \, dx\)

\(\Big \downarrow \) 6200

\(\displaystyle -\frac {a \sqrt {a^2 c x^2+c} \int \frac {x}{\sqrt {\text {arcsinh}(a x)}}dx}{4 \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c}\)

\(\Big \downarrow \) 6195

\(\displaystyle -\frac {\sqrt {a^2 c x^2+c} \int \frac {a x \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {\sqrt {a^2 c x^2+c} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}-\frac {\sqrt {a^2 c x^2+c} \int \frac {\sinh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2 c x^2+c} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}-\frac {\sqrt {a^2 c x^2+c} \int \frac {\sinh (2 \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {a^2 c x^2+c} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}-\frac {\sqrt {a^2 c x^2+c} \int -\frac {i \sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\sqrt {a^2 c x^2+c} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {i \sqrt {a^2 c x^2+c} \int \frac {\sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c}\)

\(\Big \downarrow \) 3789

\(\displaystyle \frac {i \sqrt {a^2 c x^2+c} \left (\frac {1}{2} i \int \frac {e^{2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {e^{-2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{8 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {i \sqrt {a^2 c x^2+c} \left (i \int e^{2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{8 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {i \sqrt {a^2 c x^2+c} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{8 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {\sqrt {a^2 c x^2+c} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {i \sqrt {a^2 c x^2+c} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c}\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {i \sqrt {a^2 c x^2+c} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}}{3 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c}\)

input

Int[Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]],x]
 

output

(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/2 + (Sqrt[c + a^2*c*x^2]*ArcSin 
h[a*x]^(3/2))/(3*a*Sqrt[1 + a^2*x^2]) + ((I/8)*Sqrt[c + a^2*c*x^2]*((-1/2* 
I)*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] + (I/2)*Sqrt[Pi/2]*Erfi[Sqrt 
[2]*Sqrt[ArcSinh[a*x]]]))/(a*Sqrt[1 + a^2*x^2])
 

3.5.73.3.1 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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6195

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

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -1]
 

rule 6200

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 
/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[(a + b*ArcSinh[c*x])^n/Sq 
rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* 
x^2]]   Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
 

3.5.73.4 Maple [F]

\[\int \sqrt {a^{2} c \,x^{2}+c}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}d x\]

input

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

output

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

3.5.73.5 Fricas [F(-2)]

Exception generated. \[ \int \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)} \, dx=\text {Exception raised: TypeError} \]

input

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

output

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

3.5.73.6 Sympy [F]

\[ \int \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)} \, dx=\int \sqrt {c \left (a^{2} x^{2} + 1\right )} \sqrt {\operatorname {asinh}{\left (a x \right )}}\, dx \]

input

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

output

Integral(sqrt(c*(a**2*x**2 + 1))*sqrt(asinh(a*x)), x)
 

3.5.73.7 Maxima [F]

\[ \int \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)} \, dx=\int { \sqrt {a^{2} c x^{2} + c} \sqrt {\operatorname {arsinh}\left (a x\right )} \,d x } \]

input

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

output

integrate(sqrt(a^2*c*x^2 + c)*sqrt(arcsinh(a*x)), x)
 

3.5.73.8 Giac [F(-2)]

Exception generated. \[ \int \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

3.5.73.9 Mupad [F(-1)]

Timed out. \[ \int \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)} \, dx=\int \sqrt {\mathrm {asinh}\left (a\,x\right )}\,\sqrt {c\,a^2\,x^2+c} \,d x \]

input

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

output

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