3.103 \(\int \frac {x^2}{\sin ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=96 \[ -\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^3}+\frac {\sqrt {\frac {3 \pi }{2}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}} \]

[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)

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Rubi [A]  time = 0.07, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4631, 3305, 3351} \[ -\frac {\sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^3}+\frac {\sqrt {\frac {3 \pi }{2}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[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 3305

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 3351

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

Rule 4631

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*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n + 1)
, Sin[x]^(m - 1)*(m - (m + 1)*Sin[x]^2), x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && G
eQ[n, -2] && LtQ[n, -1]

Rubi steps

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

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Mathematica [C]  time = 0.07, size = 211, normalized size = 2.20 \[ \frac {-\frac {e^{i \sin ^{-1}(a x)}-\sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-i \sin ^{-1}(a x)\right )}{4 \sqrt {\sin ^{-1}(a x)}}-\frac {e^{-i \sin ^{-1}(a x)}-\sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},i \sin ^{-1}(a x)\right )}{4 \sqrt {\sin ^{-1}(a x)}}+\frac {e^{3 i \sin ^{-1}(a x)}-\sqrt {3} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 i \sin ^{-1}(a x)\right )}{4 \sqrt {\sin ^{-1}(a x)}}+\frac {e^{-3 i \sin ^{-1}(a x)}-\sqrt {3} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},3 i \sin ^{-1}(a x)\right )}{4 \sqrt {\sin ^{-1}(a x)}}}{a^3} \]

Warning: Unable to verify antiderivative.

[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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\arcsin \left (a x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/arcsin(a*x)^(3/2), x)

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maple [A]  time = 0.06, size = 95, normalized size = 0.99 \[ \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-\sqrt {-a^{2} x^{2}+1}+\cos \left (3 \arcsin \left (a x \right )\right )}{2 a^{3} \sqrt {\arcsin \left (a x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\mathrm {asin}\left (a\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2/asin(a*x)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/asin(a*x)**(3/2), x)

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