\(\int \frac {x^3}{\sqrt {\text {arcsinh}(a x)}} \, dx\) [93]

Optimal result

Integrand size = 12, antiderivative size = 109 \[ \int \frac {x^3}{\sqrt {\text {arcsinh}(a x)}} \, dx=-\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{32 a^4}+\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^4}+\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{32 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^4} \]

[Out]

1/16*erf(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4-1/16*erfi(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/
2)/a^4-1/32*erf(2*arcsinh(a*x)^(1/2))*Pi^(1/2)/a^4+1/32*erfi(2*arcsinh(a*x)^(1/2))*Pi^(1/2)/a^4

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5780, 5556, 3389, 2211, 2235, 2236} \[ \int \frac {x^3}{\sqrt {\text {arcsinh}(a x)}} \, dx=-\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{32 a^4}+\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^4}+\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{32 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^4} \]

[In]

Int[x^3/Sqrt[ArcSinh[a*x]],x]

[Out]

-1/32*(Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[a*x]]])/a^4 + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(8*a^4) + (Sqrt[
Pi]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(32*a^4) - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(8*a^4)

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh ^3(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^4} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^4} \\ & = \frac {\text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^4}-\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{4 a^4} \\ & = -\frac {\text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^4}+\frac {\text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^4}+\frac {\text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^4}-\frac {\text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^4} \\ & = -\frac {\text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a^4}+\frac {\text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a^4}+\frac {\text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{4 a^4}-\frac {\text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{4 a^4} \\ & = -\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{32 a^4}+\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^4}+\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{32 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {\frac {\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-4 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\frac {2 \sqrt {2} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}-2 \sqrt {2} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )+\Gamma \left (\frac {1}{2},4 \text {arcsinh}(a x)\right )}{32 a^4} \]

[In]

Integrate[x^3/Sqrt[ArcSinh[a*x]],x]

[Out]

((Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -4*ArcSinh[a*x]])/Sqrt[ArcSinh[a*x]] + (2*Sqrt[2]*Sqrt[ArcSinh[a*x]]*Gamma[1/
2, -2*ArcSinh[a*x]])/Sqrt[-ArcSinh[a*x]] - 2*Sqrt[2]*Gamma[1/2, 2*ArcSinh[a*x]] + Gamma[1/2, 4*ArcSinh[a*x]])/
(32*a^4)

Maple [F]

\[\int \frac {x^{3}}{\sqrt {\operatorname {arcsinh}\left (a x \right )}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x^3/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]

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

[In]

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

[Out]

Integral(x**3/sqrt(asinh(a*x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^3/sqrt(arcsinh(a*x)), x)

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

int(x^3/asinh(a*x)^(1/2), x)