3.2.8 \(\int \frac {x}{\text {arcsinh}(a x)^{5/2}} \, dx\) [108]

3.2.8.1 Optimal result

Integrand size = 10, antiderivative size = 118 \[ \int \frac {x}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 x \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {8 x^2}{3 \sqrt {\text {arcsinh}(a x)}}-\frac {2 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^2}+\frac {2 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^2} \]

output

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

3.2.8.2 Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.83 \[ \int \frac {x}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 \text {arcsinh}(a x) \left (e^{-2 \text {arcsinh}(a x)}+e^{2 \text {arcsinh}(a x)}-\sqrt {2} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )-\sqrt {2} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )\right )+\sinh (2 \text {arcsinh}(a x))}{3 a^2 \text {arcsinh}(a x)^{3/2}} \]

input

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

output

-1/3*(2*ArcSinh[a*x]*(E^(-2*ArcSinh[a*x]) + E^(2*ArcSinh[a*x]) - Sqrt[2]*S 
qrt[-ArcSinh[a*x]]*Gamma[1/2, -2*ArcSinh[a*x]] - Sqrt[2]*Sqrt[ArcSinh[a*x] 
]*Gamma[1/2, 2*ArcSinh[a*x]]) + Sinh[2*ArcSinh[a*x]])/(a^2*ArcSinh[a*x]^(3 
/2))
 

3.2.8.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.85 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6194, 6198, 6233, 6195, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6194

\(\displaystyle \frac {2 \int \frac {1}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}dx}{3 a}+\frac {4}{3} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}dx-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {4}{3} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}dx-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}\)

\(\Big \downarrow \) 6233

\(\displaystyle \frac {4}{3} a \left (\frac {4 \int \frac {x}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}\)

\(\Big \downarrow \) 6195

\(\displaystyle \frac {4}{3} a \left (\frac {4 \int \frac {a x \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^3}-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {4}{3} a \left (\frac {4 \int \frac {\sinh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^3}-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{3} a \left (\frac {2 \int \frac {\sinh (2 \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^3}-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4}{3} a \left (-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \int -\frac {i \sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^3}\right )-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {4}{3} a \left (-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \int \frac {\sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^3}\right )-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}\)

\(\Big \downarrow \) 3789

\(\displaystyle \frac {4}{3} a \left (-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \int \frac {e^{2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {e^{-2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{a^3}\right )-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {4}{3} a \left (-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \left (i \int e^{2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{a^3}\right )-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {4}{3} a \left (-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{a^3}\right )-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {4}{3} a \left (-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^3}\right )-\frac {2 x \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\text {arcsinh}(a x)}}\)

input

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

output

(-2*x*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2)) - 4/(3*a^2*Sqrt[ArcSinh[ 
a*x]]) + (4*a*((-2*x^2)/(a*Sqrt[ArcSinh[a*x]]) - ((2*I)*((-1/2*I)*Sqrt[Pi/ 
2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] + (I/2)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[Ar 
cSinh[a*x]]]))/a^3))/3
 

3.2.8.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[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 2633

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 2634

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] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3789

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
/2   Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2   Int[(c + d*x)^m*E 
^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
 

rule 5971

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 6194

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

rule 6195

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
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 6198

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

rule 6233

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] - Simp[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]
 

3.2.8.4 Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.01

method	result	size
default	\(-\frac {\sqrt {2}\, \left (4 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, a x +2 \operatorname {arcsinh}\left (a x \right )^{2} \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )-2 \operatorname {arcsinh}\left (a x \right )^{2} \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )+2 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\right )}{3 \sqrt {\pi }\, a^{2} \operatorname {arcsinh}\left (a x \right )^{2}}\)	\(119\)

input

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

output

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

3.2.8.5 Fricas [F(-2)]

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

input

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

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

3.2.8.6 Sympy [F]

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

input

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

output

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

3.2.8.7 Maxima [F]

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

input

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

output

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

3.2.8.8 Giac [F]

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

input

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

output

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

3.2.8.9 Mupad [F(-1)]

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

input

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

output

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