\(\int \frac {x}{\arcsin (a x)^{5/2}} \, dx\) [132]

Optimal result

Integrand size = 10, antiderivative size = 89 \[ \int \frac {x}{\arcsin (a x)^{5/2}} \, dx=-\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {8 x^2}{3 \sqrt {\arcsin (a x)}}-\frac {8 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{3 a^2} \] Output:

-2/3*x*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(3/2)-4/3/a^2/arcsin(a*x)^(1/2)+8/ 
3*x^2/arcsin(a*x)^(1/2)-8/3*Pi^(1/2)*FresnelS(2*arcsin(a*x)^(1/2)/Pi^(1/2) 
)/a^2
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.26 \[ \int \frac {x}{\arcsin (a x)^{5/2}} \, dx=-\frac {2 \arcsin (a x) \left (e^{-2 i \arcsin (a x)}+e^{2 i \arcsin (a x)}-\sqrt {2} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-2 i \arcsin (a x)\right )-\sqrt {2} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},2 i \arcsin (a x)\right )\right )+\sin (2 \arcsin (a x))}{3 a^2 \arcsin (a x)^{3/2}} \] Input:

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

Output:

-1/3*(2*ArcSin[a*x]*(E^((-2*I)*ArcSin[a*x]) + E^((2*I)*ArcSin[a*x]) - Sqrt 
[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-2*I)*ArcSin[a*x]] - Sqrt[2]*Sqrt[I 
*ArcSin[a*x]]*Gamma[1/2, (2*I)*ArcSin[a*x]]) + Sin[2*ArcSin[a*x]])/(a^2*Ar 
cSin[a*x]^(3/2))
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5144, 5152, 5222, 5146, 4906, 27, 3042, 3786, 3832}

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.

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

\(\Big \downarrow \) 5144

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

\(\Big \downarrow \) 5152

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5146

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3832

\(\displaystyle -\frac {4}{3} a \left (\frac {2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{a^3}-\frac {2 x^2}{a \sqrt {\arcsin (a x)}}\right )-\frac {2 x \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {4}{3 a^2 \sqrt {\arcsin (a x)}}\)

Input:

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

Output:

(-2*x*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2)) - 4/(3*a^2*Sqrt[ArcSin[a* 
x]]) - (4*a*((-2*x^2)/(a*Sqrt[ArcSin[a*x]]) + (2*Sqrt[Pi]*FresnelS[(2*Sqrt 
[ArcSin[a*x]])/Sqrt[Pi]])/a^3))/3
 

Defintions of rubi rules used

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 3042

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

rule 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d 
   Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f 
}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
 

rule 3832

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 5144

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

rule 5146

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[1 
/(b*c^(m + 1))   Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b], x], x, a 
+ b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
 

rule 5152

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a 
 + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d 
+ e, 0] && NeQ[n, -1]
 

rule 5222

Int[(((a_.) + ArcSin[(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*ArcSin[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* 
ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2* 
d + e, 0] && LtQ[n, -1]
 

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.63

method	result	size
default	\(-\frac {8 \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}+4 \arcsin \left (a x \right ) \cos \left (2 \arcsin \left (a x \right )\right )+\sin \left (2 \arcsin \left (a x \right )\right )}{3 a^{2} \arcsin \left (a x \right )^{\frac {3}{2}}}\)	\(56\)

Input:

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

Output:

-1/3/a^2*(8*Pi^(1/2)*FresnelS(2*arcsin(a*x)^(1/2)/Pi^(1/2))*arcsin(a*x)^(3 
/2)+4*arcsin(a*x)*cos(2*arcsin(a*x))+sin(2*arcsin(a*x)))/arcsin(a*x)^(3/2)
                                                                                    
                                                                                    
 

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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