Integrand size = 8, antiderivative size = 43 \[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a} \] Output:
1/2*Pi^(1/2)*erf(arcsinh(a*x)^(1/2))/a+1/2*Pi^(1/2)*erfi(arcsinh(a*x)^(1/2 ))/a
Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {\frac {\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}-\Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )}{2 a} \] Input:
Integrate[1/Sqrt[ArcSinh[a*x]],x]
Output:
((Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -ArcSinh[a*x]])/Sqrt[ArcSinh[a*x]] - Gamm a[1/2, ArcSinh[a*x]])/(2*a)
Time = 0.52 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, 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.
\(\displaystyle \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx\) |
\(\Big \downarrow \) 6189 |
\(\displaystyle \frac {\int \frac {\sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sin \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a}\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle \frac {\frac {1}{2} i \int -\frac {i e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {i e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)+\frac {1}{2} \int \frac {e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {\int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\int e^{\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}}{a}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {\int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{a}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{a}\) |
Input:
Int[1/Sqrt[ArcSinh[a*x]],x]
Output:
((Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/2 + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]] )/2)/a
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]
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]
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]
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]
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]
Time = 0.84 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.56
method | result | size |
default | \(\frac {\sqrt {\pi }\, \left (\operatorname {erf}\left (\sqrt {\operatorname {arcsinh}\left (x a \right )}\right )+\operatorname {erfi}\left (\sqrt {\operatorname {arcsinh}\left (x a \right )}\right )\right )}{2 a}\) | \(24\) |
Input:
int(1/arcsinh(x*a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*Pi^(1/2)*(erf(arcsinh(x*a)^(1/2))+erfi(arcsinh(x*a)^(1/2)))/a
Exception generated. \[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/arcsinh(a*x)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {1}{\sqrt {\operatorname {asinh}{\left (a x \right )}}}\, dx \] Input:
integrate(1/asinh(a*x)**(1/2),x)
Output:
Integral(1/sqrt(asinh(a*x)), x)
\[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {1}{\sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \] Input:
integrate(1/arcsinh(a*x)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(arcsinh(a*x)), x)
\[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {1}{\sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \] Input:
integrate(1/arcsinh(a*x)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(arcsinh(a*x)), x)
Timed out. \[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {1}{\sqrt {\mathrm {asinh}\left (a\,x\right )}} \,d x \] Input:
int(1/asinh(a*x)^(1/2),x)
Output:
int(1/asinh(a*x)^(1/2), x)
\[ \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {2 \sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}-2 \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}\, x}{a^{2} x^{2}+1}d x \right ) a^{2}}{a} \] Input:
int(1/asinh(a*x)^(1/2),x)
Output:
(2*(sqrt(a**2*x**2 + 1)*sqrt(asinh(a*x)) - int((sqrt(a**2*x**2 + 1)*sqrt(a sinh(a*x))*x)/(a**2*x**2 + 1),x)*a**2))/a