\(\int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx\) [135]

Optimal result

Integrand size = 12, antiderivative size = 264 \[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arcsin (a x)^{3/2}}+\frac {4 x^5}{3 \arcsin (a x)^{3/2}}-\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arcsin (a x)}}+\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arcsin (a x)}}+\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{15 a^5}-\frac {5 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^5}+\frac {8 \sqrt {6 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{3 a^5} \] Output:

-2/5*x^4*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(5/2)-16/15*x^3/a^2/arcsin(a*x)^ 
(3/2)+4/3*x^5/arcsin(a*x)^(3/2)-32/5*x^2*(-a^2*x^2+1)^(1/2)/a^3/arcsin(a*x 
)^(1/2)+40/3*x^4*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(1/2)+1/15*2^(1/2)*Pi^(1 
/2)*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^5-9/10*6^(1/2)*Pi^(1/2) 
*FresnelS(6^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^5+5/6*10^(1/2)*Pi^(1/2)*Fr 
esnelS(10^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^5
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.58 \[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\frac {9 e^{3 i \arcsin (a x)} \left (1+2 i \arcsin (a x)-12 \arcsin (a x)^2\right )+2 e^{i \arcsin (a x)} \left (-3-2 i \arcsin (a x)+4 \arcsin (a x)^2\right )+e^{5 i \arcsin (a x)} \left (-3-10 i \arcsin (a x)+100 \arcsin (a x)^2\right )-8 \sqrt {-i \arcsin (a x)} \arcsin (a x)^2 \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )+e^{-i \arcsin (a x)} \left (-6+4 i \arcsin (a x)+8 \arcsin (a x)^2+8 e^{i \arcsin (a x)} (i \arcsin (a x))^{5/2} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )\right )+108 \sqrt {3} \sqrt {-i \arcsin (a x)} \arcsin (a x)^2 \Gamma \left (\frac {1}{2},-3 i \arcsin (a x)\right )-9 e^{-3 i \arcsin (a x)} \left (-1+2 i \arcsin (a x)+12 \arcsin (a x)^2+12 \sqrt {3} e^{3 i \arcsin (a x)} (i \arcsin (a x))^{5/2} \Gamma \left (\frac {1}{2},3 i \arcsin (a x)\right )\right )-100 \sqrt {5} \sqrt {-i \arcsin (a x)} \arcsin (a x)^2 \Gamma \left (\frac {1}{2},-5 i \arcsin (a x)\right )+e^{-5 i \arcsin (a x)} \left (-3+10 i \arcsin (a x)+100 \arcsin (a x)^2+100 \sqrt {5} e^{5 i \arcsin (a x)} (i \arcsin (a x))^{5/2} \Gamma \left (\frac {1}{2},5 i \arcsin (a x)\right )\right )}{240 a^5 \arcsin (a x)^{5/2}} \] Input:

Integrate[x^4/ArcSin[a*x]^(7/2),x]
 

Output:

(9*E^((3*I)*ArcSin[a*x])*(1 + (2*I)*ArcSin[a*x] - 12*ArcSin[a*x]^2) + 2*E^ 
(I*ArcSin[a*x])*(-3 - (2*I)*ArcSin[a*x] + 4*ArcSin[a*x]^2) + E^((5*I)*ArcS 
in[a*x])*(-3 - (10*I)*ArcSin[a*x] + 100*ArcSin[a*x]^2) - 8*Sqrt[(-I)*ArcSi 
n[a*x]]*ArcSin[a*x]^2*Gamma[1/2, (-I)*ArcSin[a*x]] + (-6 + (4*I)*ArcSin[a* 
x] + 8*ArcSin[a*x]^2 + 8*E^(I*ArcSin[a*x])*(I*ArcSin[a*x])^(5/2)*Gamma[1/2 
, I*ArcSin[a*x]])/E^(I*ArcSin[a*x]) + 108*Sqrt[3]*Sqrt[(-I)*ArcSin[a*x]]*A 
rcSin[a*x]^2*Gamma[1/2, (-3*I)*ArcSin[a*x]] - (9*(-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]) - 100*Sqrt[5]*Sqrt[ 
(-I)*ArcSin[a*x]]*ArcSin[a*x]^2*Gamma[1/2, (-5*I)*ArcSin[a*x]] + (-3 + (10 
*I)*ArcSin[a*x] + 100*ArcSin[a*x]^2 + 100*Sqrt[5]*E^((5*I)*ArcSin[a*x])*(I 
*ArcSin[a*x])^(5/2)*Gamma[1/2, (5*I)*ArcSin[a*x]])/E^((5*I)*ArcSin[a*x]))/ 
(240*a^5*ArcSin[a*x]^(5/2))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5144, 5222, 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.

\(\displaystyle \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx\)

\(\Big \downarrow \) 5144

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5142

\(\displaystyle -2 a \left (\frac {10 \left (\frac {2 \int \left (-\frac {a x}{8 \sqrt {\arcsin (a x)}}+\frac {9 \sin (3 \arcsin (a x))}{16 \sqrt {\arcsin (a x)}}-\frac {5 \sin (5 \arcsin (a x))}{16 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{a^5}-\frac {2 x^4 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x^5}{3 a \arcsin (a x)^{3/2}}\right )+\frac {8 \left (\frac {2 \left (\frac {2 \int \left (\frac {3 \sin (3 \arcsin (a x))}{4 \sqrt {\arcsin (a x)}}-\frac {a x}{4 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{a}-\frac {2 x^3}{3 a \arcsin (a x)^{3/2}}\right )}{5 a}-\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}\)

\(\Big \downarrow \) 2009

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

Input:

Int[x^4/ArcSin[a*x]^(7/2),x]
 

Output:

(-2*x^4*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) + (8*((-2*x^3)/(3*a*Arc 
Sin[a*x]^(3/2)) + (2*((-2*x^2*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[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*a) - 2*a*((-2 
*x^5)/(3*a*ArcSin[a*x]^(3/2)) + (10*((-2*x^4*Sqrt[1 - a^2*x^2])/(a*Sqrt[Ar 
cSin[a*x]]) + (2*(-1/4*(Sqrt[Pi/2]*FresnelS[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[ArcSin[a*x]]])/8))/a^5))/(3*a))
 

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]
 

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 = 0.14 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.85

method	result	size
default	\(-\frac {108 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-100 \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-8 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-8 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+108 \arcsin \left (a x \right )^{2} \cos \left (3 \arcsin \left (a x \right )\right )-100 \arcsin \left (a x \right )^{2} \cos \left (5 \arcsin \left (a x \right )\right )-4 \arcsin \left (a x \right ) a x +18 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )-10 \arcsin \left (a x \right ) \sin \left (5 \arcsin \left (a x \right )\right )+6 \sqrt {-a^{2} x^{2}+1}-9 \cos \left (3 \arcsin \left (a x \right )\right )+3 \cos \left (5 \arcsin \left (a x \right )\right )}{120 a^{5} \arcsin \left (a x \right )^{\frac {5}{2}}}\)	\(225\)

Input:

int(x^4/arcsin(a*x)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/120/a^5*(108*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2) 
*arcsin(a*x)^(1/2))*arcsin(a*x)^(5/2)-100*2^(1/2)*Pi^(1/2)*5^(1/2)*Fresnel 
S(2^(1/2)/Pi^(1/2)*5^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(5/2)-8*2^(1/2)* 
Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(5/2)-8* 
arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)+108*arcsin(a*x)^2*cos(3*arcsin(a*x))-100* 
arcsin(a*x)^2*cos(5*arcsin(a*x))-4*arcsin(a*x)*a*x+18*arcsin(a*x)*sin(3*ar 
csin(a*x))-10*arcsin(a*x)*sin(5*arcsin(a*x))+6*(-a^2*x^2+1)^(1/2)-9*cos(3* 
arcsin(a*x))+3*cos(5*arcsin(a*x)))/arcsin(a*x)^(5/2)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4/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]

\[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\int \frac {x^{4}}{\operatorname {asin}^{\frac {7}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**4/asin(a*x)**(7/2),x)
 

Output:

Integral(x**4/asin(a*x)**(7/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4/arcsin(a*x)^(7/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\int { \frac {x^{4}}{\arcsin \left (a x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(x^4/arcsin(a*x)^(7/2),x, algorithm="giac")
 

Output:

integrate(x^4/arcsin(a*x)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\int \frac {x^4}{{\mathrm {asin}\left (a\,x\right )}^{7/2}} \,d x \] Input:

int(x^4/asin(a*x)^(7/2),x)
 

Output:

int(x^4/asin(a*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {x^4}{\arcsin (a x)^{7/2}} \, dx=\frac {-\frac {4 \mathit {asin} \left (a x \right )^{3} \left (\int \frac {\sqrt {\mathit {asin} \left (a x \right )}}{\mathit {asin} \left (a x \right )^{4} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{4}}d x \right ) a}{3}+\frac {4 \mathit {asin} \left (a x \right )^{3} \left (\int \frac {\sqrt {\mathit {asin} \left (a x \right )}\, x^{2}}{\mathit {asin} \left (a x \right )^{4} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{4}}d x \right ) a^{3}}{3}+2 \mathit {asin} \left (a x \right )^{3} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x^{5}}{\mathit {asin} \left (a x \right )^{3} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{3}}d x \right ) a^{6}-\frac {8 \mathit {asin} \left (a x \right )^{3} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x^{3}}{\mathit {asin} \left (a x \right )^{3} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{3}}d x \right ) a^{4}}{5}-\frac {8 \mathit {asin} \left (a x \right )^{3} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x}{\mathit {asin} \left (a x \right )^{3} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{3}}d x \right ) a^{2}}{15}-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, a^{4} x^{4}}{5}+\frac {8 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}}{15}}{\mathit {asin} \left (a x \right )^{3} a^{5}} \] Input:

int(x^4/asin(a*x)^(7/2),x)
 

Output:

(2*( - 10*asin(a*x)**3*int(sqrt(asin(a*x))/(asin(a*x)**4*a**2*x**2 - asin( 
a*x)**4),x)*a + 10*asin(a*x)**3*int((sqrt(asin(a*x))*x**2)/(asin(a*x)**4*a 
**2*x**2 - asin(a*x)**4),x)*a**3 + 15*asin(a*x)**3*int((sqrt( - a**2*x**2 
+ 1)*sqrt(asin(a*x))*x**5)/(asin(a*x)**3*a**2*x**2 - asin(a*x)**3),x)*a**6 
 - 12*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**4 - 4*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 
)*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))))/(15*asin(a*x)**3*a**5)