\(\int \frac {x^3}{\text {arcsinh}(a x)^{3/2}} \, dx\) [100]

Optimal result

Integrand size = 12, antiderivative size = 138 \[ \int \frac {x^3}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{2 a^4}+\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{2 a^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5778, 3388, 2211, 2235, 2236} \[ \int \frac {x^3}{\text {arcsinh}(a x)^{3/2}} \, dx=\frac {\sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{2 a^4}+\frac {\sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{2 a^4}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}} \]

[In]

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

[Out]

(-2*x^3*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]]) + (Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[a*x]]])/(4*a^4) - (Sqrt[Pi/2]
*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(2*a^4) + (Sqrt[Pi]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(4*a^4) - (Sqrt[Pi/2]*Erfi[S
qrt[2]*Sqrt[ArcSinh[a*x]]])/(2*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 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 5778

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[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Si
nh[-a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}
, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{\text {arcsinh}(a x)^{3/2}} \, dx=\frac {\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-4 \text {arcsinh}(a x)\right )-\sqrt {2} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )+\sqrt {2} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )-\sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},4 \text {arcsinh}(a x)\right )+2 \sinh (2 \text {arcsinh}(a x))-\sinh (4 \text {arcsinh}(a x))}{4 a^4 \sqrt {\text {arcsinh}(a x)}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {x^{3}}{\operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x^3/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]

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

[In]

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

[Out]

Integral(x**3/asinh(a*x)**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^3/arcsinh(a*x)^(3/2), x)

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

int(x^3/asinh(a*x)^(3/2), x)