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

Optimal result

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

[Out]

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)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5785, 5783, 5780, 5556, 12, 3389, 2211, 2235, 2236} \[ \int \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)} \, dx=\frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 c x^2+c} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{16 a \sqrt {a^2 x^2+1}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 c x^2+c} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{16 a \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}}{3 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c} \]

[In]

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

[Out]

(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/2 + (Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/(3*a*Sqrt[1 + a^2*x^2]
) + (Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(16*a*Sqrt[1 + a^2*x^2]) - (Sqrt[Pi/2]*Sq
rt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(16*a*Sqrt[1 + a^2*x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 2235

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 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

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

Rule 5556

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 5780

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[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 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[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 5785

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] + (Dist[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[(a + b*ArcSinh[c*x])^
n/Sqrt[1 + c^2*x^2], x], x] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}+\frac {\sqrt {c+a^2 c x^2} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {1+a^2 x^2}}-\frac {\left (a \sqrt {c+a^2 c x^2}\right ) \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx}{4 \sqrt {1+a^2 x^2}} \\ & = \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 {c+a^2 c x^2} \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{4 a \sqrt {1+a^2 x^2}} \\ & = \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 {c+a^2 c x^2} \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{4 a \sqrt {1+a^2 x^2}} \\ & = \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 {c+a^2 c x^2} \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8 a \sqrt {1+a^2 x^2}} \\ & = \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 {c+a^2 c x^2} \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a \sqrt {1+a^2 x^2}}-\frac {\sqrt {c+a^2 c x^2} \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a \sqrt {1+a^2 x^2}} \\ & = \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 {c+a^2 c x^2} \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a \sqrt {1+a^2 x^2}}-\frac {\sqrt {c+a^2 c x^2} \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a \sqrt {1+a^2 x^2}} \\ & = \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}} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

(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*ArcSinh[a*x]]))/(48*a*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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