\(\int \frac {x}{\arcsin (a x)^{7/2}} \, dx\) [138]

Optimal result

Integrand size = 10, antiderivative size = 119 \[ \int \frac {x}{\arcsin (a x)^{7/2}} \, dx=-\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4}{15 a^2 \arcsin (a x)^{3/2}}+\frac {8 x^2}{15 \arcsin (a x)^{3/2}}+\frac {32 x \sqrt {1-a^2 x^2}}{15 a \sqrt {\arcsin (a x)}}-\frac {32 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{15 a^2} \] Output:

-2/5*x*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(5/2)-4/15/a^2/arcsin(a*x)^(3/2)+8 
/15*x^2/arcsin(a*x)^(3/2)+32/15*x*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(1/2)-3 
2/15*Pi^(1/2)*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))/a^2
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.23 \[ \int \frac {x}{\arcsin (a x)^{7/2}} \, dx=-\frac {\arcsin (a x) \left (2 e^{2 i \arcsin (a x)} (1+4 i \arcsin (a x))+8 \sqrt {2} (-i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},-2 i \arcsin (a x)\right )+e^{-2 i \arcsin (a x)} \left (2-8 i \arcsin (a x)+8 \sqrt {2} e^{2 i \arcsin (a x)} (i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},2 i \arcsin (a x)\right )\right )\right )+3 \sin (2 \arcsin (a x))}{15 a^2 \arcsin (a x)^{5/2}} \] Input:

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

Output:

-1/15*(ArcSin[a*x]*(2*E^((2*I)*ArcSin[a*x])*(1 + (4*I)*ArcSin[a*x]) + 8*Sq 
rt[2]*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-2*I)*ArcSin[a*x]] + (2 - (8*I) 
*ArcSin[a*x] + 8*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])) + 3*Sin[2*ArcSin[a*x]])/( 
a^2*ArcSin[a*x]^(5/2))
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5144, 5152, 5222, 5142, 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.

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

\(\Big \downarrow \) 5144

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

\(\Big \downarrow \) 5152

\(\displaystyle -\frac {4}{5} a \int \frac {x^2}{\sqrt {1-a^2 x^2} \arcsin (a x)^{5/2}}dx-\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4}{15 a^2 \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 5222

\(\displaystyle -\frac {4}{5} a \left (\frac {4 \int \frac {x}{\arcsin (a x)^{3/2}}dx}{3 a}-\frac {2 x^2}{3 a \arcsin (a x)^{3/2}}\right )-\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4}{15 a^2 \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 5142

\(\displaystyle -\frac {4}{5} a \left (\frac {4 \left (\frac {2 \int \frac {\cos (2 \arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a^2}-\frac {2 x \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \arcsin (a x)^{3/2}}\right )-\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4}{15 a^2 \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4}{5} a \left (\frac {4 \left (\frac {2 \int \frac {\sin \left (2 \arcsin (a x)+\frac {\pi }{2}\right )}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a^2}-\frac {2 x \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \arcsin (a x)^{3/2}}\right )-\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4}{15 a^2 \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 3785

\(\displaystyle -\frac {4}{5} a \left (\frac {4 \left (\frac {4 \int \cos (2 \arcsin (a x))d\sqrt {\arcsin (a x)}}{a^2}-\frac {2 x \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \arcsin (a x)^{3/2}}\right )-\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4}{15 a^2 \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 3833

\(\displaystyle -\frac {4}{5} a \left (\frac {4 \left (\frac {2 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{a^2}-\frac {2 x \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \arcsin (a x)^{3/2}}\right )-\frac {2 x \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}-\frac {4}{15 a^2 \arcsin (a x)^{3/2}}\)

Input:

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

Output:

(-2*x*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) - 4/(15*a^2*ArcSin[a*x]^( 
3/2)) - (4*a*((-2*x^2)/(3*a*ArcSin[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]])/Sqr 
t[Pi]])/a^2))/(3*a)))/5
 

Defintions of rubi rules used

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 5152

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a 
 + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d 
+ e, 0] && NeQ[n, -1]
 

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 = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.61

method	result	size
default	\(\frac {-32 \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}+16 \sin \left (2 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )^{2}-4 \arcsin \left (a x \right ) \cos \left (2 \arcsin \left (a x \right )\right )-3 \sin \left (2 \arcsin \left (a x \right )\right )}{15 a^{2} \arcsin \left (a x \right )^{\frac {5}{2}}}\)	\(73\)

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate(x/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]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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