\(\int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx\) [121]

Optimal result

Integrand size = 12, antiderivative size = 171 \[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=-\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}-\frac {5 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7}+\frac {9 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7}-\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7}+\frac {\sqrt {\frac {7 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {14}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^7} \] Output:

-2*x^6*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(1/2)-5/32*2^(1/2)*Pi^(1/2)*Fresne 
lS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^7+9/32*6^(1/2)*Pi^(1/2)*FresnelS( 
6^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^7-5/32*10^(1/2)*Pi^(1/2)*FresnelS(10 
^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^7+1/32*14^(1/2)*Pi^(1/2)*FresnelS(14^ 
(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^7
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.16 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.50 \[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=\frac {-\frac {5 \left (e^{i \arcsin (a x)}-\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )\right )}{64 \sqrt {\arcsin (a x)}}-\frac {5 \left (e^{-i \arcsin (a x)}-\sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )\right )}{64 \sqrt {\arcsin (a x)}}+\frac {9 \left (e^{3 i \arcsin (a x)}-\sqrt {3} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-3 i \arcsin (a x)\right )\right )}{64 \sqrt {\arcsin (a x)}}+\frac {9 \left (e^{-3 i \arcsin (a x)}-\sqrt {3} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},3 i \arcsin (a x)\right )\right )}{64 \sqrt {\arcsin (a x)}}-\frac {5 \left (e^{5 i \arcsin (a x)}-\sqrt {5} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-5 i \arcsin (a x)\right )\right )}{64 \sqrt {\arcsin (a x)}}-\frac {5 \left (e^{-5 i \arcsin (a x)}-\sqrt {5} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},5 i \arcsin (a x)\right )\right )}{64 \sqrt {\arcsin (a x)}}+\frac {e^{7 i \arcsin (a x)}-\sqrt {7} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-7 i \arcsin (a x)\right )}{64 \sqrt {\arcsin (a x)}}+\frac {e^{-7 i \arcsin (a x)}-\sqrt {7} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},7 i \arcsin (a x)\right )}{64 \sqrt {\arcsin (a x)}}}{a^7} \] Input:

Integrate[x^6/ArcSin[a*x]^(3/2),x]
 

Output:

((-5*(E^(I*ArcSin[a*x]) - Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-I)*ArcSin[a* 
x]]))/(64*Sqrt[ArcSin[a*x]]) - (5*(E^((-I)*ArcSin[a*x]) - Sqrt[I*ArcSin[a* 
x]]*Gamma[1/2, I*ArcSin[a*x]]))/(64*Sqrt[ArcSin[a*x]]) + (9*(E^((3*I)*ArcS 
in[a*x]) - Sqrt[3]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-3*I)*ArcSin[a*x]])) 
/(64*Sqrt[ArcSin[a*x]]) + (9*(E^((-3*I)*ArcSin[a*x]) - Sqrt[3]*Sqrt[I*ArcS 
in[a*x]]*Gamma[1/2, (3*I)*ArcSin[a*x]]))/(64*Sqrt[ArcSin[a*x]]) - (5*(E^(( 
5*I)*ArcSin[a*x]) - Sqrt[5]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-5*I)*ArcSi 
n[a*x]]))/(64*Sqrt[ArcSin[a*x]]) - (5*(E^((-5*I)*ArcSin[a*x]) - Sqrt[5]*Sq 
rt[I*ArcSin[a*x]]*Gamma[1/2, (5*I)*ArcSin[a*x]]))/(64*Sqrt[ArcSin[a*x]]) + 
 (E^((7*I)*ArcSin[a*x]) - Sqrt[7]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-7*I) 
*ArcSin[a*x]])/(64*Sqrt[ArcSin[a*x]]) + (E^((-7*I)*ArcSin[a*x]) - Sqrt[7]* 
Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (7*I)*ArcSin[a*x]])/(64*Sqrt[ArcSin[a*x]])) 
/a^7
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.96, 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.

\(\displaystyle \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx\)

\(\Big \downarrow \) 5142

\(\displaystyle \frac {2 \int \left (-\frac {5 a x}{64 \sqrt {\arcsin (a x)}}+\frac {27 \sin (3 \arcsin (a x))}{64 \sqrt {\arcsin (a x)}}-\frac {25 \sin (5 \arcsin (a x))}{64 \sqrt {\arcsin (a x)}}+\frac {7 \sin (7 \arcsin (a x))}{64 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{a^7}-\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {5}{32} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )+\frac {9}{32} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {5}{32} \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )+\frac {1}{32} \sqrt {\frac {7 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {14}{\pi }} \sqrt {\arcsin (a x)}\right )\right )}{a^7}-\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\)

Input:

Int[x^6/ArcSin[a*x]^(3/2),x]
 

Output:

(-2*x^6*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) + (2*((-5*Sqrt[Pi/2]*Fres 
nelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/32 + (9*Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6 
/Pi]*Sqrt[ArcSin[a*x]]])/32 - (5*Sqrt[(5*Pi)/2]*FresnelS[Sqrt[10/Pi]*Sqrt[ 
ArcSin[a*x]]])/32 + (Sqrt[(7*Pi)/2]*FresnelS[Sqrt[14/Pi]*Sqrt[ArcSin[a*x]] 
])/32))/a^7
 

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 = 0.14 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.08

method	result	size
default	\(-\frac {5 \sqrt {5}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-9 \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-\sqrt {2}\, \sqrt {\pi }\, \sqrt {7}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {7}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arcsin \left (a x \right )}+5 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+5 \sqrt {-a^{2} x^{2}+1}-9 \cos \left (3 \arcsin \left (a x \right )\right )+5 \cos \left (5 \arcsin \left (a x \right )\right )-\cos \left (7 \arcsin \left (a x \right )\right )}{32 a^{7} \sqrt {\arcsin \left (a x \right )}}\)	\(184\)

Input:

int(x^6/arcsin(a*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/32/a^7*(5*5^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/P 
i^(1/2)*5^(1/2)*arcsin(a*x)^(1/2))-9*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^(1/2)*Pi^(1/2 
)*7^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*7^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x) 
^(1/2)+5*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arcs 
in(a*x)^(1/2))+5*(-a^2*x^2+1)^(1/2)-9*cos(3*arcsin(a*x))+5*cos(5*arcsin(a* 
x))-cos(7*arcsin(a*x)))/arcsin(a*x)^(1/2)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^6/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]

\[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=\int \frac {x^{6}}{\operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**6/asin(a*x)**(3/2),x)
 

Output:

Integral(x**6/asin(a*x)**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^6/arcsin(a*x)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=\int { \frac {x^{6}}{\arcsin \left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^6/arcsin(a*x)^(3/2),x, algorithm="giac")
 

Output:

integrate(x^6/arcsin(a*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=\int \frac {x^6}{{\mathrm {asin}\left (a\,x\right )}^{3/2}} \,d x \] Input:

int(x^6/asin(a*x)^(3/2),x)
 

Output:

int(x^6/asin(a*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {x^6}{\arcsin (a x)^{3/2}} \, dx=\frac {-\frac {12 \mathit {asin} \left (a x \right ) \left (\int \frac {\sqrt {\mathit {asin} \left (a x \right )}}{\mathit {asin} \left (a x \right )^{2} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{2}}d x \right ) a}{5}+\frac {2 \mathit {asin} \left (a x \right ) \left (\int \frac {\sqrt {\mathit {asin} \left (a x \right )}\, x^{4}}{\mathit {asin} \left (a x \right )^{2} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{2}}d x \right ) a^{5}}{5}+2 \mathit {asin} \left (a x \right ) \left (\int \frac {\sqrt {\mathit {asin} \left (a x \right )}\, x^{2}}{\mathit {asin} \left (a x \right )^{2} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{2}}d x \right ) a^{3}+14 \mathit {asin} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x^{7}}{\mathit {asin} \left (a x \right ) a^{2} x^{2}-\mathit {asin} \left (a x \right )}d x \right ) a^{8}-12 \mathit {asin} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x^{5}}{\mathit {asin} \left (a x \right ) a^{2} x^{2}-\mathit {asin} \left (a x \right )}d x \right ) a^{6}-\frac {12 \mathit {asin} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x^{3}}{\mathit {asin} \left (a x \right ) a^{2} x^{2}-\mathit {asin} \left (a x \right )}d x \right ) a^{4}}{5}-\frac {16 \mathit {asin} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x}{\mathit {asin} \left (a x \right ) a^{2} x^{2}-\mathit {asin} \left (a x \right )}d x \right ) a^{2}}{5}-2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, a^{6} x^{6}+\frac {4 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, a^{2} x^{2}}{5}+\frac {24 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}}{5}}{\mathit {asin} \left (a x \right ) a^{7}} \] Input:

int(x^6/asin(a*x)^(3/2),x)
 

Output:

(2*( - 6*asin(a*x)*int(sqrt(asin(a*x))/(asin(a*x)**2*a**2*x**2 - asin(a*x) 
**2),x)*a + asin(a*x)*int((sqrt(asin(a*x))*x**4)/(asin(a*x)**2*a**2*x**2 - 
 asin(a*x)**2),x)*a**5 + 5*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 + 35*asin(a*x)*int((sqrt( - a**2*x** 
2 + 1)*sqrt(asin(a*x))*x**7)/(asin(a*x)*a**2*x**2 - asin(a*x)),x)*a**8 - 3 
0*asin(a*x)*int((sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))*x**5)/(asin(a*x)*a 
**2*x**2 - asin(a*x)),x)*a**6 - 6*asin(a*x)*int((sqrt( - a**2*x**2 + 1)*sq 
rt(asin(a*x))*x**3)/(asin(a*x)*a**2*x**2 - asin(a*x)),x)*a**4 - 8*asin(a*x 
)*int((sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))*x)/(asin(a*x)*a**2*x**2 - as 
in(a*x)),x)*a**2 - 5*sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))*a**6*x**6 + 2* 
sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))*a**2*x**2 + 12*sqrt( - a**2*x**2 + 
1)*sqrt(asin(a*x))))/(5*asin(a*x)*a**7)