\(\int \frac {x^3}{\arcsin (a x)^{7/2}} \, dx\) [136]

Optimal result

Integrand size = 12, antiderivative size = 190 \[ \int \frac {x^3}{\arcsin (a x)^{7/2}} \, dx=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4 x^2}{5 a^2 \arcsin (a x)^{3/2}}+\frac {16 x^4}{15 \arcsin (a x)^{3/2}}-\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arcsin (a x)}}+\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arcsin (a x)}}+\frac {32 \sqrt {2 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{15 a^4}-\frac {16 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{15 a^4} \] Output:

-2/5*x^3*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(5/2)-4/5*x^2/a^2/arcsin(a*x)^(3 
/2)+16/15*x^4/arcsin(a*x)^(3/2)-16/5*x*(-a^2*x^2+1)^(1/2)/a^3/arcsin(a*x)^ 
(1/2)+128/15*x^3*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(1/2)+32/15*2^(1/2)*Pi^( 
1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^4-16/15*Pi^(1/2)*Fre 
snelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))/a^4
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.76 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.43 \[ \int \frac {x^3}{\arcsin (a x)^{7/2}} \, dx=\frac {4 \arcsin (a x) \left (i e^{2 i \arcsin (a x)} (i-4 \arcsin (a x))-4 \sqrt {2} (-i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},-2 i \arcsin (a x)\right )+e^{-2 i \arcsin (a x)} \left (-1+4 i \arcsin (a x)-4 \sqrt {2} e^{2 i \arcsin (a x)} (i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},2 i \arcsin (a x)\right )\right )\right )-4 \arcsin (a x) \left (i e^{4 i \arcsin (a x)} (i-8 \arcsin (a x))-16 (-i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},-4 i \arcsin (a x)\right )+e^{-4 i \arcsin (a x)} \left (-1+8 i \arcsin (a x)-16 e^{4 i \arcsin (a x)} (i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},4 i \arcsin (a x)\right )\right )\right )-6 \sin (2 \arcsin (a x))+3 \sin (4 \arcsin (a x))}{60 a^4 \arcsin (a x)^{5/2}} \] Input:

Integrate[x^3/ArcSin[a*x]^(7/2),x]
 

Output:

(4*ArcSin[a*x]*(I*E^((2*I)*ArcSin[a*x])*(I - 4*ArcSin[a*x]) - 4*Sqrt[2]*(( 
-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-2*I)*ArcSin[a*x]] + (-1 + (4*I)*ArcSin 
[a*x] - 4*Sqrt[2]*E^((2*I)*ArcSin[a*x])*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, ( 
2*I)*ArcSin[a*x]])/E^((2*I)*ArcSin[a*x])) - 4*ArcSin[a*x]*(I*E^((4*I)*ArcS 
in[a*x])*(I - 8*ArcSin[a*x]) - 16*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-4* 
I)*ArcSin[a*x]] + (-1 + (8*I)*ArcSin[a*x] - 16*E^((4*I)*ArcSin[a*x])*(I*Ar 
cSin[a*x])^(3/2)*Gamma[1/2, (4*I)*ArcSin[a*x]])/E^((4*I)*ArcSin[a*x])) - 6 
*Sin[2*ArcSin[a*x]] + 3*Sin[4*ArcSin[a*x]])/(60*a^4*ArcSin[a*x]^(5/2))
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.30, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5144, 5222, 5142, 2009, 3042, 3785, 3833}

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.

\(\displaystyle \int \frac {x^3}{\arcsin (a x)^{7/2}} \, dx\)

\(\Big \downarrow \) 5144

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5142

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3833

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

Input:

Int[x^3/ArcSin[a*x]^(7/2),x]
 

Output:

(-2*x^3*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) + (6*((-2*x^2)/(3*a*Arc 
Sin[a*x]^(3/2)) + (4*((-2*x*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) + (2* 
Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/a^2))/(3*a)))/(5*a) - ( 
8*a*((-2*x^4)/(3*a*ArcSin[a*x]^(3/2)) + (8*((-2*x^3*Sqrt[1 - a^2*x^2])/(a* 
Sqrt[ArcSin[a*x]]) + (2*(-1/2*(Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSi 
n[a*x]]]) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/2))/a^4))/ 
(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 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 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]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.73

method	result	size
default	\(\frac {128 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-64 \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}+32 \sin \left (2 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )^{2}-64 \sin \left (4 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )^{2}-8 \arcsin \left (a x \right ) \cos \left (2 \arcsin \left (a x \right )\right )+8 \arcsin \left (a x \right ) \cos \left (4 \arcsin \left (a x \right )\right )-6 \sin \left (2 \arcsin \left (a x \right )\right )+3 \sin \left (4 \arcsin \left (a x \right )\right )}{60 a^{4} \arcsin \left (a x \right )^{\frac {5}{2}}}\)	\(139\)

Input:

int(x^3/arcsin(a*x)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/60/a^4*(128*2^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/ 
2))*arcsin(a*x)^(5/2)-64*Pi^(1/2)*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))*a 
rcsin(a*x)^(5/2)+32*sin(2*arcsin(a*x))*arcsin(a*x)^2-64*sin(4*arcsin(a*x)) 
*arcsin(a*x)^2-8*arcsin(a*x)*cos(2*arcsin(a*x))+8*arcsin(a*x)*cos(4*arcsin 
(a*x))-6*sin(2*arcsin(a*x))+3*sin(4*arcsin(a*x)))/arcsin(a*x)^(5/2)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3/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]

\[ \int \frac {x^3}{\arcsin (a x)^{7/2}} \, dx=\int \frac {x^{3}}{\operatorname {asin}^{\frac {7}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**3/asin(a*x)**(7/2),x)
 

Output:

Integral(x**3/asin(a*x)**(7/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^3/arcsin(a*x)^(7/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^3/arcsin(a*x)^(7/2),x, algorithm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\arcsin (a x)^{7/2}} \, dx=\int \frac {x^3}{{\mathrm {asin}\left (a\,x\right )}^{7/2}} \,d x \] Input:

int(x^3/asin(a*x)^(7/2),x)
 

Output:

int(x^3/asin(a*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {x^3}{\arcsin (a x)^{7/2}} \, dx=\frac {-15 \mathit {asin} \left (a x \right )^{3} \left (\int \frac {\sqrt {\mathit {asin} \left (a x \right )}\, x^{3}}{\mathit {asin} \left (a x \right )^{4} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{4}}d x \right ) a^{4}+15 \mathit {asin} \left (a x \right )^{3} \left (\int \frac {\sqrt {\mathit {asin} \left (a x \right )}\, x}{\mathit {asin} \left (a x \right )^{4} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{4}}d x \right ) a^{2}+16 \mathit {asin} \left (a x \right )^{3} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x^{4}}{\mathit {asin} \left (a x \right )^{3} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{3}}d x \right ) a^{5}-4 \sqrt {\mathit {asin} \left (a x \right )}\, \mathit {asin} \left (a x \right )-4 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, a^{3} x^{3}-6 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, a x}{10 \mathit {asin} \left (a x \right )^{3} a^{4}} \] Input:

int(x^3/asin(a*x)^(7/2),x)
 

Output:

( - 15*asin(a*x)**3*int((sqrt(asin(a*x))*x**3)/(asin(a*x)**4*a**2*x**2 - a 
sin(a*x)**4),x)*a**4 + 15*asin(a*x)**3*int((sqrt(asin(a*x))*x)/(asin(a*x)* 
*4*a**2*x**2 - asin(a*x)**4),x)*a**2 + 16*asin(a*x)**3*int((sqrt( - a**2*x 
**2 + 1)*sqrt(asin(a*x))*x**4)/(asin(a*x)**3*a**2*x**2 - asin(a*x)**3),x)* 
a**5 - 4*sqrt(asin(a*x))*asin(a*x) - 4*sqrt( - a**2*x**2 + 1)*sqrt(asin(a* 
x))*a**3*x**3 - 6*sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))*a*x)/(10*asin(a*x 
)**3*a**4)