\(\int \frac {1}{\text {arcsinh}(a x)^{3/2}} \, dx\) [103]

Optimal result

Integrand size = 8, antiderivative size = 64 \[ \int \frac {1}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5773, 5819, 3389, 2211, 2235, 2236} \[ \int \frac {1}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{a}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{a} \]

[In]

Int[ArcSinh[a*x]^(-3/2),x]

[Out]

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

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 5773

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1
)/(b*c*(n + 1))), x] - Dist[c/(b*(n + 1)), Int[x*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x] /; F
reeQ[{a, b, c}, x] && LtQ[n, -1]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\text {arcsinh}(a x)^{3/2}} \, dx=\frac {-e^{-\text {arcsinh}(a x)}-e^{\text {arcsinh}(a x)}+\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )+\sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )}{a \sqrt {\text {arcsinh}(a x)}} \]

[In]

Integrate[ArcSinh[a*x]^(-3/2),x]

[Out]

(-E^(-ArcSinh[a*x]) - E^ArcSinh[a*x] + Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -ArcSinh[a*x]] + Sqrt[ArcSinh[a*x]]*Gamm
a[1/2, ArcSinh[a*x]])/(a*Sqrt[ArcSinh[a*x]])

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02

method	result	size
default	\(-\frac {\operatorname {arcsinh}\left (a x \right ) \pi \,\operatorname {erf}\left (\sqrt {\operatorname {arcsinh}\left (a x \right )}\right )-\operatorname {arcsinh}\left (a x \right ) \pi \,\operatorname {erfi}\left (\sqrt {\operatorname {arcsinh}\left (a x \right )}\right )+2 \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}}{\sqrt {\pi }\, a \,\operatorname {arcsinh}\left (a x \right )}\)	\(65\)

[In]

int(1/arcsinh(a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\text {arcsinh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/arcsinh(a*x)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {1}{\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(1/asinh(a*x)**(3/2),x)

[Out]

Integral(asinh(a*x)**(-3/2), x)

Maxima [F]

\[ \int \frac {1}{\text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/arcsinh(a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(a*x)^(-3/2), x)

Giac [F]

\[ \int \frac {1}{\text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/arcsinh(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^(-3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}} \,d x \]

[In]

int(1/asinh(a*x)^(3/2),x)

[Out]

int(1/asinh(a*x)^(3/2), x)