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

Optimal. Leaf size=55 \[ \frac{2 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^2}-\frac{2 x \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]

[Out]

(-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

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Rubi [A]  time = 0.031562, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4631, 3304, 3352} \[ \frac{2 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^2}-\frac{2 x \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-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

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]

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3352

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

Rubi steps

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

Mathematica [C]  time = 0.0326046, size = 91, normalized size = 1.65 \[ -\frac{i \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \sin ^{-1}(a x)\right )-i \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \sin ^{-1}(a x)\right )+2 \sin \left (2 \sin ^{-1}(a x)\right )}{2 a^2 \sqrt{\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(I*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-2*I)*ArcSin[a*x]] - I*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (
2*I)*ArcSin[a*x]] + 2*Sin[2*ArcSin[a*x]])/(2*a^2*Sqrt[ArcSin[a*x]])

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Maple [A]  time = 0.031, size = 43, normalized size = 0.8 \begin{align*} -{\frac{1}{{a}^{2}} \left ( -2\,\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arcsin \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^2/arcsin(a*x)^(1/2)*(-2*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))+sin(2*arcsin(a*
x)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\operatorname{asin}^{\frac{3}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/asin(a*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\arcsin \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/arcsin(a*x)^(3/2), x)