\(\int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx\) [109]

Optimal result

Integrand size = 8, antiderivative size = 84 \[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {2 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{3 a}+\frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{3 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5773, 5818, 5774, 3388, 2211, 2235, 2236} \[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {2 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{3 a}+\frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{3 a}-\frac {4 x}{3 \sqrt {\text {arcsinh}(a x)}} \]

[In]

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

[Out]

(-2*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2)) - (4*x)/(3*Sqrt[ArcSinh[a*x]]) + (2*Sqrt[Pi]*Erf[Sqrt[ArcSinh[
a*x]]])/(3*a) + (2*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(3*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 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 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 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 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 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {1}{3} (2 a) \int \frac {x}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}} \, dx \\ & = -\frac {2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {4}{3} \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx \\ & = -\frac {2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {4 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a} \\ & = -\frac {2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {2 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a}+\frac {2 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a} \\ & = -\frac {2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {4 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3 a}+\frac {4 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3 a} \\ & = -\frac {2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {2 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{3 a}+\frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{3 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {e^{-\text {arcsinh}(a x)} \left (1+e^{2 \text {arcsinh}(a x)}-2 \text {arcsinh}(a x)+2 e^{2 \text {arcsinh}(a x)} \text {arcsinh}(a x)+2 e^{\text {arcsinh}(a x)} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )+2 e^{\text {arcsinh}(a x)} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )\right )}{3 a \text {arcsinh}(a x)^{3/2}} \]

[In]

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

[Out]

-1/3*(1 + E^(2*ArcSinh[a*x]) - 2*ArcSinh[a*x] + 2*E^(2*ArcSinh[a*x])*ArcSinh[a*x] + 2*E^ArcSinh[a*x]*(-ArcSinh
[a*x])^(3/2)*Gamma[1/2, -ArcSinh[a*x]] + 2*E^ArcSinh[a*x]*ArcSinh[a*x]^(3/2)*Gamma[1/2, ArcSinh[a*x]])/(a*E^Ar
cSinh[a*x]*ArcSinh[a*x]^(3/2))

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {-\frac {4 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a x}{3}+\frac {2 \operatorname {arcsinh}\left (a x \right )^{2} \pi \,\operatorname {erf}\left (\sqrt {\operatorname {arcsinh}\left (a x \right )}\right )}{3}+\frac {2 \operatorname {arcsinh}\left (a x \right )^{2} \pi \,\operatorname {erfi}\left (\sqrt {\operatorname {arcsinh}\left (a x \right )}\right )}{3}-\frac {2 \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}}{3}}{\sqrt {\pi }\, a \operatorname {arcsinh}\left (a x \right )^{2}}\)	\(81\)

[In]

int(1/arcsinh(a*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/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]

\[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {1}{\operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(1/asinh(a*x)**(5/2),x)

[Out]

Integral(asinh(a*x)**(-5/2), x)

Maxima [F]

\[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/arcsinh(a*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(a*x)^(-5/2), x)

Giac [F]

\[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/arcsinh(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^(-5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \]

[In]

int(1/asinh(a*x)^(5/2),x)

[Out]

int(1/asinh(a*x)^(5/2), x)