\(\int \frac {x^2}{\arcsin (a x)^{5/2}} \, dx\) [131]

Optimal result

Integrand size = 12, antiderivative size = 125 \[ \int \frac {x^2}{\arcsin (a x)^{5/2}} \, dx=-\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {4 x^3}{\sqrt {\arcsin (a x)}}-\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^3}+\frac {\sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^3} \] Output:

-2/3*x^2*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(3/2)-8/3*x/a^2/arcsin(a*x)^(1/2 
)+4*x^3/arcsin(a*x)^(1/2)-1/3*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*a 
rcsin(a*x)^(1/2))/a^3+6^(1/2)*Pi^(1/2)*FresnelC(6^(1/2)/Pi^(1/2)*arcsin(a* 
x)^(1/2))/a^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.22 \[ \int \frac {x^2}{\arcsin (a x)^{5/2}} \, dx=\frac {\frac {i e^{i \arcsin (a x)} (i-2 \arcsin (a x))-2 (-i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )}{12 \arcsin (a x)^{3/2}}-\frac {e^{-i \arcsin (a x)} \left (1-2 i \arcsin (a x)+2 e^{i \arcsin (a x)} (i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )\right )}{12 \arcsin (a x)^{3/2}}-\frac {i e^{3 i \arcsin (a x)} (i-6 \arcsin (a x))-6 \sqrt {3} (-i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},-3 i \arcsin (a x)\right )}{12 \arcsin (a x)^{3/2}}+\frac {e^{-3 i \arcsin (a x)} \left (1-6 i \arcsin (a x)+6 \sqrt {3} e^{3 i \arcsin (a x)} (i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},3 i \arcsin (a x)\right )\right )}{12 \arcsin (a x)^{3/2}}}{a^3} \] Input:

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

Output:

((I*E^(I*ArcSin[a*x])*(I - 2*ArcSin[a*x]) - 2*((-I)*ArcSin[a*x])^(3/2)*Gam 
ma[1/2, (-I)*ArcSin[a*x]])/(12*ArcSin[a*x]^(3/2)) - (1 - (2*I)*ArcSin[a*x] 
 + 2*E^(I*ArcSin[a*x])*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, I*ArcSin[a*x]])/(1 
2*E^(I*ArcSin[a*x])*ArcSin[a*x]^(3/2)) - (I*E^((3*I)*ArcSin[a*x])*(I - 6*A 
rcSin[a*x]) - 6*Sqrt[3]*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-3*I)*ArcSin[ 
a*x]])/(12*ArcSin[a*x]^(3/2)) + (1 - (6*I)*ArcSin[a*x] + 6*Sqrt[3]*E^((3*I 
)*ArcSin[a*x])*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, (3*I)*ArcSin[a*x]])/(12*E^ 
((3*I)*ArcSin[a*x])*ArcSin[a*x]^(3/2)))/a^3
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.41, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5144, 5222, 5134, 3042, 3785, 3833, 5146, 4906, 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.

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

\(\Big \downarrow \) 5144

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3833

\(\displaystyle -2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\arcsin (a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\arcsin (a x)}}\right )+\frac {4 \left (\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^2}-\frac {2 x}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 5146

\(\displaystyle -2 a \left (\frac {6 \int \frac {a^2 x^2 \sqrt {1-a^2 x^2}}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a^4}-\frac {2 x^3}{a \sqrt {\arcsin (a x)}}\right )+\frac {4 \left (\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^2}-\frac {2 x}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 4906

\(\displaystyle -2 a \left (\frac {6 \int \left (\frac {\sqrt {1-a^2 x^2}}{4 \sqrt {\arcsin (a x)}}-\frac {\cos (3 \arcsin (a x))}{4 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{a^4}-\frac {2 x^3}{a \sqrt {\arcsin (a x)}}\right )+\frac {4 \left (\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^2}-\frac {2 x}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 a \left (\frac {6 \left (\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )\right )}{a^4}-\frac {2 x^3}{a \sqrt {\arcsin (a x)}}\right )+\frac {4 \left (\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^2}-\frac {2 x}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}\)

Input:

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

Output:

(-2*x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2)) + (4*((-2*x)/(a*Sqrt[Ar 
cSin[a*x]]) + (2*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a^2))/ 
(3*a) - 2*a*((-2*x^3)/(a*Sqrt[ArcSin[a*x]]) + (6*((Sqrt[Pi/2]*FresnelC[Sqr 
t[2/Pi]*Sqrt[ArcSin[a*x]]])/2 - (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSi 
n[a*x]]])/2))/a^4)
 

Defintions of rubi rules used

rule 2009

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

rule 3042

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

rule 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[2/d   Subst[Int[Cos[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 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[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 5134

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   Su 
bst[Int[x^n*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, 
c, n}, x]
 

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 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 = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {-6 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}+2 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}-2 \arcsin \left (a x \right ) a x +6 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )+\sqrt {-a^{2} x^{2}+1}-\cos \left (3 \arcsin \left (a x \right )\right )}{6 a^{3} \arcsin \left (a x \right )^{\frac {3}{2}}}\)	\(117\)

Input:

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

Output:

-1/6/a^3*(-6*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*ar 
csin(a*x)^(1/2))*arcsin(a*x)^(3/2)+2*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^ 
(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(3/2)-2*arcsin(a*x)*a*x+6*arcsin(a*x) 
*sin(3*arcsin(a*x))+(-a^2*x^2+1)^(1/2)-cos(3*arcsin(a*x)))/arcsin(a*x)^(3/ 
2)
                                                                                    
                                                                                    
 

Fricas [F(-2)]

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

integrate(x^2/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]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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