\(\int \frac {x^4}{\arcsin (a x)^{5/2}} \, dx\) [129]

Optimal result

Integrand size = 12, antiderivative size = 171 \[ \int \frac {x^4}{\arcsin (a x)^{5/2}} \, dx=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\arcsin (a x)}}+\frac {20 x^5}{3 \sqrt {\arcsin (a x)}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^5}+\frac {3 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{2 a^5}-\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{6 a^5} \] Output:

-2/3*x^4*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(3/2)-16/3*x^3/a^2/arcsin(a*x)^( 
1/2)+20/3*x^5/arcsin(a*x)^(1/2)-1/6*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^( 
1/2)*arcsin(a*x)^(1/2))/a^5+3/4*6^(1/2)*Pi^(1/2)*FresnelC(6^(1/2)/Pi^(1/2) 
*arcsin(a*x)^(1/2))/a^5-5/12*10^(1/2)*Pi^(1/2)*FresnelC(10^(1/2)/Pi^(1/2)* 
arcsin(a*x)^(1/2))/a^5
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.44 \[ \int \frac {x^4}{\arcsin (a x)^{5/2}} \, dx=\frac {\frac {i e^{i \arcsin (a x)} (i-2 \arcsin (a x))-2 (-i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )}{24 \arcsin (a x)^{3/2}}-\frac {e^{-i \arcsin (a x)} \left (1-2 i \arcsin (a x)+2 e^{i \arcsin (a x)} (i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )\right )}{24 \arcsin (a x)^{3/2}}-\frac {i e^{3 i \arcsin (a x)} (i-6 \arcsin (a x))-6 \sqrt {3} (-i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},-3 i \arcsin (a x)\right )}{16 \arcsin (a x)^{3/2}}+\frac {e^{-3 i \arcsin (a x)} \left (1-6 i \arcsin (a x)+6 \sqrt {3} e^{3 i \arcsin (a x)} (i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},3 i \arcsin (a x)\right )\right )}{16 \arcsin (a x)^{3/2}}+\frac {i e^{5 i \arcsin (a x)} (i-10 \arcsin (a x))-10 \sqrt {5} (-i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},-5 i \arcsin (a x)\right )}{48 \arcsin (a x)^{3/2}}-\frac {e^{-5 i \arcsin (a x)} \left (1-10 i \arcsin (a x)+10 \sqrt {5} e^{5 i \arcsin (a x)} (i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},5 i \arcsin (a x)\right )\right )}{48 \arcsin (a x)^{3/2}}}{a^5} \] Input:

Integrate[x^4/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]])/(24*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]])/(2 
4*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]])/(16*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]])/(16*E^ 
((3*I)*ArcSin[a*x])*ArcSin[a*x]^(3/2)) + (I*E^((5*I)*ArcSin[a*x])*(I - 10* 
ArcSin[a*x]) - 10*Sqrt[5]*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-5*I)*ArcSi 
n[a*x]])/(48*ArcSin[a*x]^(3/2)) - (1 - (10*I)*ArcSin[a*x] + 10*Sqrt[5]*E^( 
(5*I)*ArcSin[a*x])*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, (5*I)*ArcSin[a*x]])/(4 
8*E^((5*I)*ArcSin[a*x])*ArcSin[a*x]^(3/2)))/a^5
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.47, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5144, 5222, 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.

\(\displaystyle \int \frac {x^4}{\arcsin (a x)^{5/2}} \, dx\)

\(\Big \downarrow \) 5144

\(\displaystyle -\frac {10}{3} a \int \frac {x^5}{\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}dx+\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}dx}{3 a}-\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5146

\(\displaystyle \frac {8 \left (\frac {6 \int \frac {a^2 x^2 \sqrt {1-a^2 x^2}}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a^4}-\frac {2 x^3}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {10}{3} a \left (\frac {10 \int \frac {a^4 x^4 \sqrt {1-a^2 x^2}}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a^6}-\frac {2 x^5}{a \sqrt {\arcsin (a x)}}\right )-\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {10}{3} a \left (\frac {10 \int \left (-\frac {3 \cos (3 \arcsin (a x))}{16 \sqrt {\arcsin (a x)}}+\frac {\cos (5 \arcsin (a x))}{16 \sqrt {\arcsin (a x)}}+\frac {\sqrt {1-a^2 x^2}}{8 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{a^6}-\frac {2 x^5}{a \sqrt {\arcsin (a x)}}\right )+\frac {8 \left (\frac {6 \int \left (\frac {\sqrt {1-a^2 x^2}}{4 \sqrt {\arcsin (a x)}}-\frac {\cos (3 \arcsin (a x))}{4 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{a^4}-\frac {2 x^3}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 2009

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

Input:

Int[x^4/ArcSin[a*x]^(5/2),x]
 

Output:

(-2*x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2)) + (8*((-2*x^3)/(a*Sqrt[ 
ArcSin[a*x]]) + (6*((Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/2 
- (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/2))/a^4))/(3*a) - (1 
0*a*((-2*x^5)/(a*Sqrt[ArcSin[a*x]]) + (10*((Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi] 
*Sqrt[ArcSin[a*x]]])/4 - (Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a 
*x]]])/8 + (Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/8))/a^6)) 
/3
 

Defintions of rubi rules used

rule 2009

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

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 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 = 0.12 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.01

method	result	size
default	\(-\frac {-18 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}+10 \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}+4 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}-4 \arcsin \left (a x \right ) a x +18 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )-10 \arcsin \left (a x \right ) \sin \left (5 \arcsin \left (a x \right )\right )+2 \sqrt {-a^{2} x^{2}+1}-3 \cos \left (3 \arcsin \left (a x \right )\right )+\cos \left (5 \arcsin \left (a x \right )\right )}{24 a^{5} \arcsin \left (a x \right )^{\frac {3}{2}}}\)	\(173\)

Input:

int(x^4/arcsin(a*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/24/a^5*(-18*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)* 
arcsin(a*x)^(1/2))*arcsin(a*x)^(3/2)+10*2^(1/2)*Pi^(1/2)*5^(1/2)*FresnelC( 
2^(1/2)/Pi^(1/2)*5^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(3/2)+4*2^(1/2)*Pi 
^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(3/2)-4*ar 
csin(a*x)*a*x+18*arcsin(a*x)*sin(3*arcsin(a*x))-10*arcsin(a*x)*sin(5*arcsi 
n(a*x))+2*(-a^2*x^2+1)^(1/2)-3*cos(3*arcsin(a*x))+cos(5*arcsin(a*x)))/arcs 
in(a*x)^(3/2)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arcsin (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4/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]

\[ \int \frac {x^4}{\arcsin (a x)^{5/2}} \, dx=\int \frac {x^{4}}{\operatorname {asin}^{\frac {5}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**4/asin(a*x)**(5/2),x)
 

Output:

Integral(x**4/asin(a*x)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arcsin (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4/arcsin(a*x)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {x^4}{\arcsin (a x)^{5/2}} \, dx=\int { \frac {x^{4}}{\arcsin \left (a x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4/arcsin(a*x)^(5/2),x, algorithm="giac")
 

Output:

integrate(x^4/arcsin(a*x)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\arcsin (a x)^{5/2}} \, dx=\int \frac {x^4}{{\mathrm {asin}\left (a\,x\right )}^{5/2}} \,d x \] Input:

int(x^4/asin(a*x)^(5/2),x)
 

Output:

int(x^4/asin(a*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^4}{\arcsin (a x)^{5/2}} \, dx=\frac {-\frac {4 \mathit {asin} \left (a x \right )^{2} \left (\int \frac {\sqrt {\mathit {asin} \left (a x \right )}}{\mathit {asin} \left (a x \right )^{3} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{3}}d x \right ) a}{3}+\frac {4 \mathit {asin} \left (a x \right )^{2} \left (\int \frac {\sqrt {\mathit {asin} \left (a x \right )}\, x^{2}}{\mathit {asin} \left (a x \right )^{3} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{3}}d x \right ) a^{3}}{3}+\frac {10 \mathit {asin} \left (a x \right )^{2} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x^{5}}{\mathit {asin} \left (a x \right )^{2} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{2}}d x \right ) a^{6}}{3}-\frac {8 \mathit {asin} \left (a x \right )^{2} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x^{3}}{\mathit {asin} \left (a x \right )^{2} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{2}}d x \right ) a^{4}}{3}-\frac {8 \mathit {asin} \left (a x \right )^{2} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x}{\mathit {asin} \left (a x \right )^{2} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{2}}d x \right ) a^{2}}{9}-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, a^{4} x^{4}}{3}+\frac {8 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}}{9}}{\mathit {asin} \left (a x \right )^{2} a^{5}} \] Input:

int(x^4/asin(a*x)^(5/2),x)
 

Output:

(2*( - 6*asin(a*x)**2*int(sqrt(asin(a*x))/(asin(a*x)**3*a**2*x**2 - asin(a 
*x)**3),x)*a + 6*asin(a*x)**2*int((sqrt(asin(a*x))*x**2)/(asin(a*x)**3*a** 
2*x**2 - asin(a*x)**3),x)*a**3 + 15*asin(a*x)**2*int((sqrt( - a**2*x**2 + 
1)*sqrt(asin(a*x))*x**5)/(asin(a*x)**2*a**2*x**2 - asin(a*x)**2),x)*a**6 - 
 12*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**4 - 4*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)* 
a**2 - 3*sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))*a**4*x**4 + 4*sqrt( - a**2 
*x**2 + 1)*sqrt(asin(a*x))))/(9*asin(a*x)**2*a**5)