\(\int \frac {x^2}{\arcsin (a x)^{3/2}} \, dx\) [103]

Optimal result

Integrand size = 12, antiderivative size = 96 \[ \int \frac {x^2}{\arcsin (a x)^{3/2}} \, dx=-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^3}+\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^3} \]

[Out]

-1/2*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^3+1/2*FresnelS(6^(1/2)/Pi^(1/2)*arcsin(a*
x)^(1/2))*6^(1/2)*Pi^(1/2)/a^3-2*x^2*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4727, 3386, 3432} \[ \int \frac {x^2}{\arcsin (a x)^{3/2}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^3}+\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}} \]

[In]

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

[Out]

(-2*x^2*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a^3 + (
Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/a^3

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[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 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 4727

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}+\frac {2 \text {Subst}\left (\int \left (-\frac {\sin (x)}{4 \sqrt {x}}+\frac {3 \sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arcsin (a x)\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}-\frac {\text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{2 a^3}+\frac {3 \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{2 a^3} \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}-\frac {\text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{a^3}+\frac {3 \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^3}+\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{a^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.06 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.20 \[ \int \frac {x^2}{\arcsin (a x)^{3/2}} \, dx=\frac {-\frac {e^{i \arcsin (a x)}-\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )}{4 \sqrt {\arcsin (a x)}}-\frac {e^{-i \arcsin (a x)}-\sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )}{4 \sqrt {\arcsin (a x)}}+\frac {e^{3 i \arcsin (a x)}-\sqrt {3} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-3 i \arcsin (a x)\right )}{4 \sqrt {\arcsin (a x)}}+\frac {e^{-3 i \arcsin (a x)}-\sqrt {3} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},3 i \arcsin (a x)\right )}{4 \sqrt {\arcsin (a x)}}}{a^3} \]

[In]

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

[Out]

(-1/4*(E^(I*ArcSin[a*x]) - Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-I)*ArcSin[a*x]])/Sqrt[ArcSin[a*x]] - (E^((-I)*A
rcSin[a*x]) - Sqrt[I*ArcSin[a*x]]*Gamma[1/2, I*ArcSin[a*x]])/(4*Sqrt[ArcSin[a*x]]) + (E^((3*I)*ArcSin[a*x]) -
Sqrt[3]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-3*I)*ArcSin[a*x]])/(4*Sqrt[ArcSin[a*x]]) + (E^((-3*I)*ArcSin[a*x])
- Sqrt[3]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (3*I)*ArcSin[a*x]])/(4*Sqrt[ArcSin[a*x]]))/a^3

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.99

method	result	size
default	\(-\frac {-\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+\sqrt {-a^{2} x^{2}+1}-\cos \left (3 \arcsin \left (a x \right )\right )}{2 a^{3} \sqrt {\arcsin \left (a x \right )}}\)	\(95\)

[In]

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

[Out]

-1/2/a^3*(-FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))*3^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)+Fre
snelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)+(-a^2*x^2+1)^(1/2)-cos(3*arcsin(a
*x)))/arcsin(a*x)^(1/2)

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

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

[In]

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

[Out]

Integral(x**2/asin(a*x)**(3/2), x)

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(x^2/arcsin(a*x)^(3/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2/asin(a*x)^(3/2), x)