\(\int \frac {x^2}{\text {arcsinh}(a x)^{5/2}} \, dx\) [138]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.38 \[ \int \frac {x^2}{\text {arcsinh}(a x)^{5/2}} \, dx=\frac {-\frac {e^{3 \text {arcsinh}(a x)} (1+6 \text {arcsinh}(a x))+6 \sqrt {3} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-3 \text {arcsinh}(a x)\right )}{12 \text {arcsinh}(a x)^{3/2}}+\frac {e^{\text {arcsinh}(a x)} (1+2 \text {arcsinh}(a x))+2 (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )}{12 \text {arcsinh}(a x)^{3/2}}+\frac {e^{-\text {arcsinh}(a x)} \left (1-2 \text {arcsinh}(a x)+2 e^{\text {arcsinh}(a x)} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )\right )}{12 \text {arcsinh}(a x)^{3/2}}+\frac {1}{12} \left (-\frac {e^{-3 \text {arcsinh}(a x)}}{\text {arcsinh}(a x)^{3/2}}+\frac {6 e^{-3 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}-6 \sqrt {3} \Gamma \left (\frac {1}{2},3 \text {arcsinh}(a x)\right )\right )}{a^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.33, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6194, 6233, 6189, 3042, 3788, 26, 2611, 2633, 2634, 6195, 5971, 2009}

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.

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

\(\Big \downarrow \) 6194

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

\(\Big \downarrow \) 6233

\(\displaystyle 2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )+\frac {4 \left (\frac {2 \int \frac {1}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6189

\(\displaystyle \frac {4 \left (\frac {2 \int \frac {\sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^2}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}+2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4 \left (-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \int \frac {\sin \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^2}\right )}{3 a}+2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\)

\(\Big \downarrow \) 3788

\(\displaystyle \frac {4 \left (-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \left (\frac {1}{2} i \int -\frac {i e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {i e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{a^2}\right )}{3 a}+2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {4 \left (\frac {2 \left (\frac {1}{2} \int \frac {e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)+\frac {1}{2} \int \frac {e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{a^2}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}+2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {4 \left (\frac {2 \left (\int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\int e^{\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{a^2}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}+2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {4 \left (\frac {2 \left (\int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )\right )}{a^2}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}+2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\)

\(\Big \downarrow \) 2634

\(\displaystyle 2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )+\frac {4 \left (\frac {2 \left (\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )\right )}{a^2}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6195

\(\displaystyle 2 a \left (\frac {6 \int \frac {a^2 x^2 \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^4}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )+\frac {4 \left (\frac {2 \left (\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )\right )}{a^2}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\)

\(\Big \downarrow \) 5971

\(\displaystyle 2 a \left (\frac {6 \int \left (\frac {\cosh (3 \text {arcsinh}(a x))}{4 \sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {a^2 x^2+1}}{4 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{a^4}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )+\frac {4 \left (\frac {2 \left (\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )\right )}{a^2}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 a \left (\frac {6 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^4}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )+\frac {4 \left (\frac {2 \left (\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )\right )}{a^2}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\)

Input:

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

Output:

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

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[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 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 6189

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

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 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2}{\text {arcsinh}(a x)^{5/2}} \, dx=\frac {2 \mathit {asinh} \left (a x \right )^{2} \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}\, x^{3}}{\mathit {asinh} \left (a x \right )^{2} a^{2} x^{2}+\mathit {asinh} \left (a x \right )^{2}}d x \right ) a^{2}+\frac {4 \mathit {asinh} \left (a x \right )^{2} \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}\, x}{\mathit {asinh} \left (a x \right )^{2} a^{2} x^{2}+\mathit {asinh} \left (a x \right )^{2}}d x \right )}{3}-\frac {2 \sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}\, x^{2}}{3}}{\mathit {asinh} \left (a x \right )^{2} a} \] Input:

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

Output:

(2*(3*asinh(a*x)**2*int((sqrt(a**2*x**2 + 1)*sqrt(asinh(a*x))*x**3)/(asinh 
(a*x)**2*a**2*x**2 + asinh(a*x)**2),x)*a**2 + 2*asinh(a*x)**2*int((sqrt(a* 
*2*x**2 + 1)*sqrt(asinh(a*x))*x)/(asinh(a*x)**2*a**2*x**2 + asinh(a*x)**2) 
,x) - sqrt(a**2*x**2 + 1)*sqrt(asinh(a*x))*x**2))/(3*asinh(a*x)**2*a)