\(\int \frac {x^2}{\text {arcsinh}(a x)^{5/2}} \, dx\) [107]

Optimal result

Integrand size = 12, antiderivative size = 161 \[ \int \frac {x^2}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{2 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{2 a^3} \]

[Out]

-1/6*erf(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^3-1/6*erfi(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^3+1/2*erf(3^(1/2)*arcsinh(a*
x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3+1/2*erfi(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3-2/3*x^2*(a^2*x^2+1)^(
1/2)/a/arcsinh(a*x)^(3/2)-8/3*x/a^2/arcsinh(a*x)^(1/2)-4*x^3/arcsinh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5779, 5818, 5780, 5556, 3388, 2211, 2235, 2236, 5774} \[ \int \frac {x^2}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{2 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{2 a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arcsinh}(a x)}} \]

[In]

Int[x^2/ArcSinh[a*x]^(5/2),x]

[Out]

(-2*x^2*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2)) - (8*x)/(3*a^2*Sqrt[ArcSinh[a*x]]) - (4*x^3)/Sqrt[ArcSinh[
a*x]] - (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(6*a^3) + (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(2*a^3) - (S
qrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(6*a^3) + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(2*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 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[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]

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 5774

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cosh[-a/b + x/b], x], x
, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 5779

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSi
nh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n +
1)/Sqrt[1 + c^2*x^2]), x], x] - Dist[m/(b*c*(n + 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] && LtQ[n, -2]

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]

Rule 5818

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Dist[f*(m/
(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x]
/; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {4 \int \frac {x}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}} \, dx}{3 a}+(2 a) \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}} \, dx \\ & = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arcsinh}(a x)}}+12 \int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}} \, dx+\frac {8 \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx}{3 a^2} \\ & = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arcsinh}(a x)}}+\frac {8 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^3}+\frac {12 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arcsinh}(a x)}}+\frac {4 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^3}+\frac {4 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^3}+\frac {12 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {x}}+\frac {\cosh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arcsinh}(a x)}}+\frac {8 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3 a^3}+\frac {8 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3 a^3}-\frac {3 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^3}+\frac {3 \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arcsinh}(a x)}}+\frac {4 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{3 a^3}+\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{3 a^3}+\frac {3 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^3}-\frac {3 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^3}-\frac {3 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^3}+\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^3} \\ & = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arcsinh}(a x)}}+\frac {4 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{3 a^3}+\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{3 a^3}+\frac {3 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a^3}-\frac {3 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a^3}-\frac {3 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a^3}+\frac {3 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{2 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{2 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.38 \[ \int \frac {x^2}{\text {arcsinh}(a x)^{5/2}} \, dx=\frac {-\frac {e^{3 \text {arcsinh}(a x)} (1+6 \text {arcsinh}(a x))+6 \sqrt {3} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-3 \text {arcsinh}(a x)\right )}{12 \text {arcsinh}(a x)^{3/2}}+\frac {e^{\text {arcsinh}(a x)} (1+2 \text {arcsinh}(a x))+2 (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )}{12 \text {arcsinh}(a x)^{3/2}}+\frac {e^{-\text {arcsinh}(a x)} \left (1-2 \text {arcsinh}(a x)+2 e^{\text {arcsinh}(a x)} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )\right )}{12 \text {arcsinh}(a x)^{3/2}}+\frac {1}{12} \left (-\frac {e^{-3 \text {arcsinh}(a x)}}{\text {arcsinh}(a x)^{3/2}}+\frac {6 e^{-3 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}-6 \sqrt {3} \Gamma \left (\frac {1}{2},3 \text {arcsinh}(a x)\right )\right )}{a^3} \]

[In]

Integrate[x^2/ArcSinh[a*x]^(5/2),x]

[Out]

(-1/12*(E^(3*ArcSinh[a*x])*(1 + 6*ArcSinh[a*x]) + 6*Sqrt[3]*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -3*ArcSinh[a*x]])
/ArcSinh[a*x]^(3/2) + (E^ArcSinh[a*x]*(1 + 2*ArcSinh[a*x]) + 2*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -ArcSinh[a*x]]
)/(12*ArcSinh[a*x]^(3/2)) + (1 - 2*ArcSinh[a*x] + 2*E^ArcSinh[a*x]*ArcSinh[a*x]^(3/2)*Gamma[1/2, ArcSinh[a*x]]
)/(12*E^ArcSinh[a*x]*ArcSinh[a*x]^(3/2)) + (-(1/(E^(3*ArcSinh[a*x])*ArcSinh[a*x]^(3/2))) + 6/(E^(3*ArcSinh[a*x
])*Sqrt[ArcSinh[a*x]]) - 6*Sqrt[3]*Gamma[1/2, 3*ArcSinh[a*x]])/12)/a^3

Maple [F]

\[\int \frac {x^{2}}{\operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}}}d x\]

[In]

int(x^2/arcsinh(a*x)^(5/2),x)

[Out]

int(x^2/arcsinh(a*x)^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^2}{\text {arcsinh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^2/arcsinh(a*x)^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x^2}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(x**2/asinh(a*x)**(5/2),x)

[Out]

Integral(x**2/asinh(a*x)**(5/2), x)

Maxima [F]

\[ \int \frac {x^2}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^2/arcsinh(a*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(x^2/arcsinh(a*x)^(5/2), x)

Giac [F]

\[ \int \frac {x^2}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^2/arcsinh(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(x^2/arcsinh(a*x)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \]

[In]

int(x^2/asinh(a*x)^(5/2),x)

[Out]

int(x^2/asinh(a*x)^(5/2), x)