\(\int \frac {x^4}{\arcsin (a x)^{3/2}} \, dx\) [123]

Optimal result

Integrand size = 12, antiderivative size = 136 \[ \int \frac {x^4}{\arcsin (a x)^{3/2}} \, dx=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{2 a^5}+\frac {3 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{4 a^5}-\frac {\sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{4 a^5} \] Output:

-2*x^4*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(1/2)-1/4*2^(1/2)*Pi^(1/2)*Fresnel 
S(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^5+3/8*6^(1/2)*Pi^(1/2)*FresnelS(6^ 
(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^5-1/8*10^(1/2)*Pi^(1/2)*FresnelS(10^(1 
/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^5
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.35 \[ \int \frac {x^4}{\arcsin (a x)^{3/2}} \, dx=\frac {-\frac {e^{i \arcsin (a x)}-\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )}{8 \sqrt {\arcsin (a x)}}-\frac {e^{-i \arcsin (a x)}-\sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )}{8 \sqrt {\arcsin (a x)}}+\frac {3 \left (e^{3 i \arcsin (a x)}-\sqrt {3} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-3 i \arcsin (a x)\right )\right )}{16 \sqrt {\arcsin (a x)}}+\frac {3 \left (e^{-3 i \arcsin (a x)}-\sqrt {3} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},3 i \arcsin (a x)\right )\right )}{16 \sqrt {\arcsin (a x)}}-\frac {e^{5 i \arcsin (a x)}-\sqrt {5} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-5 i \arcsin (a x)\right )}{16 \sqrt {\arcsin (a x)}}-\frac {e^{-5 i \arcsin (a x)}-\sqrt {5} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},5 i \arcsin (a x)\right )}{16 \sqrt {\arcsin (a x)}}}{a^5} \] Input:

Integrate[x^4/ArcSin[a*x]^(3/2),x]
 

Output:

(-1/8*(E^(I*ArcSin[a*x]) - Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-I)*ArcSin[a 
*x]])/Sqrt[ArcSin[a*x]] - (E^((-I)*ArcSin[a*x]) - Sqrt[I*ArcSin[a*x]]*Gamm 
a[1/2, I*ArcSin[a*x]])/(8*Sqrt[ArcSin[a*x]]) + (3*(E^((3*I)*ArcSin[a*x]) - 
 Sqrt[3]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-3*I)*ArcSin[a*x]]))/(16*Sqrt[ 
ArcSin[a*x]]) + (3*(E^((-3*I)*ArcSin[a*x]) - Sqrt[3]*Sqrt[I*ArcSin[a*x]]*G 
amma[1/2, (3*I)*ArcSin[a*x]]))/(16*Sqrt[ArcSin[a*x]]) - (E^((5*I)*ArcSin[a 
*x]) - Sqrt[5]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-5*I)*ArcSin[a*x]])/(16* 
Sqrt[ArcSin[a*x]]) - (E^((-5*I)*ArcSin[a*x]) - Sqrt[5]*Sqrt[I*ArcSin[a*x]] 
*Gamma[1/2, (5*I)*ArcSin[a*x]])/(16*Sqrt[ArcSin[a*x]]))/a^5
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5142, 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.

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

\(\Big \downarrow \) 5142

\(\displaystyle \frac {2 \int \left (-\frac {a x}{8 \sqrt {\arcsin (a x)}}+\frac {9 \sin (3 \arcsin (a x))}{16 \sqrt {\arcsin (a x)}}-\frac {5 \sin (5 \arcsin (a x))}{16 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{a^5}-\frac {2 x^4 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )+\frac {3}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {1}{8} \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )\right )}{a^5}-\frac {2 x^4 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\)

Input:

Int[x^4/ArcSin[a*x]^(3/2),x]
 

Output:

(-2*x^4*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) + (2*(-1/4*(Sqrt[Pi/2]*Fr 
esnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]]) + (3*Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/ 
Pi]*Sqrt[ArcSin[a*x]]])/8 - (Sqrt[(5*Pi)/2]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcS 
in[a*x]]])/8))/a^5
 

Defintions of rubi rules used

rule 2009

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

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]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01

method	result	size
default	\(-\frac {\sqrt {5}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-3 \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+2 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+2 \sqrt {-a^{2} x^{2}+1}-3 \cos \left (3 \arcsin \left (a x \right )\right )+\cos \left (5 \arcsin \left (a x \right )\right )}{8 a^{5} \sqrt {\arcsin \left (a x \right )}}\)	\(138\)

Input:

int(x^4/arcsin(a*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/8/a^5*(5^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^( 
1/2)*5^(1/2)*arcsin(a*x)^(1/2))-3*3^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/ 
2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))+2*2^(1/2)*arcsin(a 
*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))+2*(-a^2*x^ 
2+1)^(1/2)-3*cos(3*arcsin(a*x))+cos(5*arcsin(a*x)))/arcsin(a*x)^(1/2)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arcsin (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4/arcsin(a*x)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^4}{\arcsin (a x)^{3/2}} \, dx=\int \frac {x^{4}}{\operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**4/asin(a*x)**(3/2),x)
 

Output:

Integral(x**4/asin(a*x)**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arcsin (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4/arcsin(a*x)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {x^4}{\arcsin (a x)^{3/2}} \, dx=\int { \frac {x^{4}}{\arcsin \left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^4/arcsin(a*x)^(3/2),x, algorithm="giac")
 

Output:

integrate(x^4/arcsin(a*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\arcsin (a x)^{3/2}} \, dx=\int \frac {x^4}{{\mathrm {asin}\left (a\,x\right )}^{3/2}} \,d x \] Input:

int(x^4/asin(a*x)^(3/2),x)
 

Output:

int(x^4/asin(a*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {x^4}{\arcsin (a x)^{3/2}} \, dx=\frac {-\frac {4 \mathit {asin} \left (a x \right ) \left (\int \frac {\sqrt {\mathit {asin} \left (a x \right )}}{\mathit {asin} \left (a x \right )^{2} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{2}}d x \right ) a}{3}+\frac {4 \mathit {asin} \left (a x \right ) \left (\int \frac {\sqrt {\mathit {asin} \left (a x \right )}\, x^{2}}{\mathit {asin} \left (a x \right )^{2} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{2}}d x \right ) a^{3}}{3}+10 \mathit {asin} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x^{5}}{\mathit {asin} \left (a x \right ) a^{2} x^{2}-\mathit {asin} \left (a x \right )}d x \right ) a^{6}-8 \mathit {asin} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x^{3}}{\mathit {asin} \left (a x \right ) a^{2} x^{2}-\mathit {asin} \left (a x \right )}d x \right ) a^{4}-\frac {8 \mathit {asin} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x}{\mathit {asin} \left (a x \right ) a^{2} x^{2}-\mathit {asin} \left (a x \right )}d x \right ) a^{2}}{3}-2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, a^{4} x^{4}+\frac {8 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}}{3}}{\mathit {asin} \left (a x \right ) a^{5}} \] Input:

int(x^4/asin(a*x)^(3/2),x)
 

Output:

(2*( - 2*asin(a*x)*int(sqrt(asin(a*x))/(asin(a*x)**2*a**2*x**2 - asin(a*x) 
**2),x)*a + 2*asin(a*x)*int((sqrt(asin(a*x))*x**2)/(asin(a*x)**2*a**2*x**2 
 - asin(a*x)**2),x)*a**3 + 15*asin(a*x)*int((sqrt( - a**2*x**2 + 1)*sqrt(a 
sin(a*x))*x**5)/(asin(a*x)*a**2*x**2 - asin(a*x)),x)*a**6 - 12*asin(a*x)*i 
nt((sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))*x**3)/(asin(a*x)*a**2*x**2 - as 
in(a*x)),x)*a**4 - 4*asin(a*x)*int((sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x)) 
*x)/(asin(a*x)*a**2*x**2 - asin(a*x)),x)*a**2 - 3*sqrt( - a**2*x**2 + 1)*s 
qrt(asin(a*x))*a**4*x**4 + 4*sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))))/(3*a 
sin(a*x)*a**5)