\(\int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx\) [494]

Optimal result

Integrand size = 17, antiderivative size = 33 \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=-\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(x)}\right ) \]

[Out]

-1/2*erf(arcsinh(x)^(1/2))*Pi^(1/2)+1/2*erfi(arcsinh(x)^(1/2))*Pi^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {5819, 3389, 2211, 2235, 2236} \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(x)}\right )-\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(x)}\right ) \]

[In]

Int[x/(Sqrt[1 + x^2]*Sqrt[ArcSinh[x]]),x]

[Out]

-1/2*(Sqrt[Pi]*Erf[Sqrt[ArcSinh[x]]]) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[x]]])/2

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 5819

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(x)\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(x)\right )\right )+\frac {1}{2} \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(x)\right ) \\ & = -\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(x)}\right )+\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(x)}\right ) \\ & = -\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=\frac {1}{2} \left (\frac {\sqrt {-\text {arcsinh}(x)} \Gamma \left (\frac {1}{2},-\text {arcsinh}(x)\right )}{\sqrt {\text {arcsinh}(x)}}+\Gamma \left (\frac {1}{2},\text {arcsinh}(x)\right )\right ) \]

[In]

Integrate[x/(Sqrt[1 + x^2]*Sqrt[ArcSinh[x]]),x]

[Out]

((Sqrt[-ArcSinh[x]]*Gamma[1/2, -ArcSinh[x]])/Sqrt[ArcSinh[x]] + Gamma[1/2, ArcSinh[x]])/2

Maple [F]

\[\int \frac {x}{\sqrt {x^{2}+1}\, \sqrt {\operatorname {arcsinh}\left (x \right )}}d x\]

[In]

int(x/(x^2+1)^(1/2)/arcsinh(x)^(1/2),x)

[Out]

int(x/(x^2+1)^(1/2)/arcsinh(x)^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x/(sqrt(x**2 + 1)*sqrt(asinh(x))), x)

Maxima [F]

\[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=\int { \frac {x}{\sqrt {x^{2} + 1} \sqrt {\operatorname {arsinh}\left (x\right )}} \,d x } \]

[In]

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

[Out]

integrate(x/(sqrt(x^2 + 1)*sqrt(arcsinh(x))), x)

Giac [F]

\[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=\int { \frac {x}{\sqrt {x^{2} + 1} \sqrt {\operatorname {arsinh}\left (x\right )}} \,d x } \]

[In]

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

[Out]

integrate(x/(sqrt(x^2 + 1)*sqrt(arcsinh(x))), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=\int \frac {x}{\sqrt {\mathrm {asinh}\left (x\right )}\,\sqrt {x^2+1}} \,d x \]

[In]

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

[Out]

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