\(\int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx\) [137]

Optimal result

Integrand size = 12, antiderivative size = 191 \[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {8 x}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^3}{5 \arcsin (a x)^{3/2}}-\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\arcsin (a x)}}+\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\arcsin (a x)}}+\frac {2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{15 a^3}-\frac {6 \sqrt {6 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{5 a^3} \] Output:

-2/5*x^2*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(5/2)-8/15*x/a^2/arcsin(a*x)^(3/ 
2)+4/5*x^3/arcsin(a*x)^(3/2)-16/15*(-a^2*x^2+1)^(1/2)/a^3/arcsin(a*x)^(1/2 
)+24/5*x^2*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(1/2)+2/15*2^(1/2)*Pi^(1/2)*Fr 
esnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^3-6/5*6^(1/2)*Pi^(1/2)*Fresne 
lS(6^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.47 \[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\frac {3 e^{3 i \arcsin (a x)} \left (1+2 i \arcsin (a x)-12 \arcsin (a x)^2\right )+e^{i \arcsin (a x)} \left (-3-2 i \arcsin (a x)+4 \arcsin (a x)^2\right )-4 \sqrt {-i \arcsin (a x)} \arcsin (a x)^2 \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )+e^{-i \arcsin (a x)} \left (-3+2 i \arcsin (a x)+4 \arcsin (a x)^2+4 e^{i \arcsin (a x)} (i \arcsin (a x))^{5/2} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )\right )+36 \sqrt {3} \sqrt {-i \arcsin (a x)} \arcsin (a x)^2 \Gamma \left (\frac {1}{2},-3 i \arcsin (a x)\right )-3 e^{-3 i \arcsin (a x)} \left (-1+2 i \arcsin (a x)+12 \arcsin (a x)^2+12 \sqrt {3} e^{3 i \arcsin (a x)} (i \arcsin (a x))^{5/2} \Gamma \left (\frac {1}{2},3 i \arcsin (a x)\right )\right )}{60 a^3 \arcsin (a x)^{5/2}} \] Input:

Integrate[x^2/ArcSin[a*x]^(7/2),x]
 

Output:

(3*E^((3*I)*ArcSin[a*x])*(1 + (2*I)*ArcSin[a*x] - 12*ArcSin[a*x]^2) + E^(I 
*ArcSin[a*x])*(-3 - (2*I)*ArcSin[a*x] + 4*ArcSin[a*x]^2) - 4*Sqrt[(-I)*Arc 
Sin[a*x]]*ArcSin[a*x]^2*Gamma[1/2, (-I)*ArcSin[a*x]] + (-3 + (2*I)*ArcSin[ 
a*x] + 4*ArcSin[a*x]^2 + 4*E^(I*ArcSin[a*x])*(I*ArcSin[a*x])^(5/2)*Gamma[1 
/2, I*ArcSin[a*x]])/E^(I*ArcSin[a*x]) + 36*Sqrt[3]*Sqrt[(-I)*ArcSin[a*x]]* 
ArcSin[a*x]^2*Gamma[1/2, (-3*I)*ArcSin[a*x]] - (3*(-1 + (2*I)*ArcSin[a*x] 
+ 12*ArcSin[a*x]^2 + 12*Sqrt[3]*E^((3*I)*ArcSin[a*x])*(I*ArcSin[a*x])^(5/2 
)*Gamma[1/2, (3*I)*ArcSin[a*x]]))/E^((3*I)*ArcSin[a*x]))/(60*a^3*ArcSin[a* 
x]^(5/2))
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.32, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5144, 5222, 5132, 5142, 2009, 5224, 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.

\(\displaystyle \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx\)

\(\Big \downarrow \) 5144

\(\displaystyle \frac {4 \int \frac {x}{\sqrt {1-a^2 x^2} \arcsin (a x)^{5/2}}dx}{5 a}-\frac {6}{5} a \int \frac {x^3}{\sqrt {1-a^2 x^2} \arcsin (a x)^{5/2}}dx-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}\)

\(\Big \downarrow \) 5222

\(\displaystyle -\frac {6}{5} a \left (\frac {2 \int \frac {x^2}{\arcsin (a x)^{3/2}}dx}{a}-\frac {2 x^3}{3 a \arcsin (a x)^{3/2}}\right )+\frac {4 \left (\frac {2 \int \frac {1}{\arcsin (a x)^{3/2}}dx}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )}{5 a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}\)

\(\Big \downarrow \) 5132

\(\displaystyle \frac {4 \left (\frac {2 \left (-2 a \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}dx-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )}{5 a}-\frac {6}{5} a \left (\frac {2 \int \frac {x^2}{\arcsin (a x)^{3/2}}dx}{a}-\frac {2 x^3}{3 a \arcsin (a x)^{3/2}}\right )-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}\)

\(\Big \downarrow \) 5142

\(\displaystyle \frac {4 \left (\frac {2 \left (-2 a \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}dx-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )}{5 a}-\frac {6}{5} a \left (\frac {2 \left (\frac {2 \int \left (\frac {3 \sin (3 \arcsin (a x))}{4 \sqrt {\arcsin (a x)}}-\frac {a x}{4 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{a}-\frac {2 x^3}{3 a \arcsin (a x)^{3/2}}\right )-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 \left (\frac {2 \left (-2 a \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}dx-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )}{5 a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {6}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{2} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{a}-\frac {2 x^3}{3 a \arcsin (a x)^{3/2}}\right )\)

\(\Big \downarrow \) 5224

\(\displaystyle \frac {4 \left (\frac {2 \left (-\frac {2 \int \frac {a x}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a}-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )}{5 a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {6}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{2} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{a}-\frac {2 x^3}{3 a \arcsin (a x)^{3/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4 \left (\frac {2 \left (-\frac {2 \int \frac {\sin (\arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a}-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )}{5 a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {6}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{2} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{a}-\frac {2 x^3}{3 a \arcsin (a x)^{3/2}}\right )\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {4 \left (\frac {2 \left (-\frac {4 \int a xd\sqrt {\arcsin (a x)}}{a}-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )}{5 a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {6}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{2} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{a}-\frac {2 x^3}{3 a \arcsin (a x)^{3/2}}\right )\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {4 \left (\frac {2 \left (-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}-\frac {2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a}\right )}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )}{5 a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {6}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{2} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{a}-\frac {2 x^3}{3 a \arcsin (a x)^{3/2}}\right )\)

Input:

Int[x^2/ArcSin[a*x]^(7/2),x]
 

Output:

(-2*x^2*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) + (4*((-2*x)/(3*a*ArcSi 
n[a*x]^(3/2)) + (2*((-2*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) - (2*Sqrt 
[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a))/(3*a)))/(5*a) - (6*a*(( 
-2*x^3)/(3*a*ArcSin[a*x]^(3/2)) + (2*((-2*x^2*Sqrt[1 - a^2*x^2])/(a*Sqrt[A 
rcSin[a*x]]) + (2*(-1/2*(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]] 
) + (Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/2))/a^3))/a))/ 
5
 

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 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 5132

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

rule 5142

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

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 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]
 

rule 5224

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x 
^2)^p]   Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, 
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] 
 && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.81

method	result	size
default	\(-\frac {36 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-4 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-4 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+36 \arcsin \left (a x \right )^{2} \cos \left (3 \arcsin \left (a x \right )\right )-2 \arcsin \left (a x \right ) a x +6 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )+3 \sqrt {-a^{2} x^{2}+1}-3 \cos \left (3 \arcsin \left (a x \right )\right )}{30 a^{3} \arcsin \left (a x \right )^{\frac {5}{2}}}\)	\(154\)

Input:

int(x^2/arcsin(a*x)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/30/a^3*(36*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*a 
rcsin(a*x)^(1/2))*arcsin(a*x)^(5/2)-4*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi 
^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(5/2)-4*arcsin(a*x)^2*(-a^2*x^2+1)^( 
1/2)+36*arcsin(a*x)^2*cos(3*arcsin(a*x))-2*arcsin(a*x)*a*x+6*arcsin(a*x)*s 
in(3*arcsin(a*x))+3*(-a^2*x^2+1)^(1/2)-3*cos(3*arcsin(a*x)))/arcsin(a*x)^( 
5/2)
                                                                                    
                                                                                    
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^2/arcsin(a*x)^(7/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\int \frac {x^{2}}{\operatorname {asin}^{\frac {7}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**2/asin(a*x)**(7/2),x)
 

Output:

Integral(x**2/asin(a*x)**(7/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^2/arcsin(a*x)^(7/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\int { \frac {x^{2}}{\arcsin \left (a x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(x^2/arcsin(a*x)^(7/2),x, algorithm="giac")
 

Output:

integrate(x^2/arcsin(a*x)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\int \frac {x^2}{{\mathrm {asin}\left (a\,x\right )}^{7/2}} \,d x \] Input:

int(x^2/asin(a*x)^(7/2),x)
 

Output:

int(x^2/asin(a*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {x^2}{\arcsin (a x)^{7/2}} \, dx=\frac {\frac {6 \mathit {asin} \left (a x \right )^{3} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x^{3}}{\mathit {asin} \left (a x \right )^{3} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{3}}d x \right ) a^{2}}{5}-\frac {4 \mathit {asin} \left (a x \right )^{3} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x}{\mathit {asin} \left (a x \right )^{3} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{3}}d x \right )}{5}-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x^{2}}{5}}{\mathit {asin} \left (a x \right )^{3} a} \] Input:

int(x^2/asin(a*x)^(7/2),x)
 

Output:

(2*(3*asin(a*x)**3*int((sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))*x**3)/(asin 
(a*x)**3*a**2*x**2 - asin(a*x)**3),x)*a**2 - 2*asin(a*x)**3*int((sqrt( - a 
**2*x**2 + 1)*sqrt(asin(a*x))*x)/(asin(a*x)**3*a**2*x**2 - asin(a*x)**3),x 
) - sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))*x**2))/(5*asin(a*x)**3*a)