\(\int x^2 \sqrt {\text {arcsinh}(a x)} \, dx\) [76]

Optimal result

Integrand size = 12, antiderivative size = 120 \[ \int x^2 \sqrt {\text {arcsinh}(a x)} \, dx=\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{48 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{48 a^3} \]

[Out]

1/144*erf(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3-1/144*erfi(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(
1/2)/a^3-1/16*erf(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^3+1/16*erfi(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^3+1/3*x^3*arcsinh(
a*x)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5777, 5819, 3393, 3389, 2211, 2235, 2236} \[ \int x^2 \sqrt {\text {arcsinh}(a x)} \, dx=-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{48 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{48 a^3}+\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)} \]

[In]

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

[Out]

(x^3*Sqrt[ArcSinh[a*x]])/3 - (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(16*a^3) + (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSin
h[a*x]]])/(48*a^3) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(16*a^3) - (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]
]])/(48*a^3)

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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5777

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcSinh[c*x])^n/(
m + 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /;
FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

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 {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {1}{6} a \int \frac {x^3}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx \\ & = \frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {\text {Subst}\left (\int \frac {\sinh ^3(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{6 a^3} \\ & = \frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {i \text {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 \sqrt {x}}-\frac {i \sinh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{6 a^3} \\ & = \frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {\text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{24 a^3}+\frac {\text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^3} \\ & = \frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}+\frac {\text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{48 a^3}-\frac {\text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{48 a^3}-\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^3}+\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^3} \\ & = \frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}+\frac {\text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{24 a^3}-\frac {\text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{24 a^3}-\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a^3}+\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a^3} \\ & = \frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{48 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{48 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.82 \[ \int x^2 \sqrt {\text {arcsinh}(a x)} \, dx=\frac {\frac {\sqrt {3} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {3}{2},-3 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+\frac {9 \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {3}{2},-\text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+9 \Gamma \left (\frac {3}{2},\text {arcsinh}(a x)\right )-\sqrt {3} \Gamma \left (\frac {3}{2},3 \text {arcsinh}(a x)\right )}{72 a^3} \]

[In]

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

[Out]

((Sqrt[3]*Sqrt[ArcSinh[a*x]]*Gamma[3/2, -3*ArcSinh[a*x]])/Sqrt[-ArcSinh[a*x]] + (9*Sqrt[-ArcSinh[a*x]]*Gamma[3
/2, -ArcSinh[a*x]])/Sqrt[ArcSinh[a*x]] + 9*Gamma[3/2, ArcSinh[a*x]] - Sqrt[3]*Gamma[3/2, 3*ArcSinh[a*x]])/(72*
a^3)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x^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]

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

[In]

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

[Out]

Integral(x**2*sqrt(asinh(a*x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^2*sqrt(arcsinh(a*x)), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^2*sqrt(arcsinh(a*x)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2*asinh(a*x)^(1/2), x)