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

Optimal. Leaf size=90 \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^4}+\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^4}-\frac{2 x^3 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]

[Out]

(-2*x^3*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a^4 +
(Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/a^4

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Rubi [A]  time = 0.0696889, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4631, 3304, 3352} \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^4}+\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^4}-\frac{2 x^3 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x^3*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a^4 +
(Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/a^4

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

Mathematica [C]  time = 0.0458816, size = 154, normalized size = 1.71 \[ \frac{-i \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \sin ^{-1}(a x)\right )+i \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \sin ^{-1}(a x)\right )+i \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \sin ^{-1}(a x)\right )-i \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \sin ^{-1}(a x)\right )-2 \sin \left (2 \sin ^{-1}(a x)\right )+\sin \left (4 \sin ^{-1}(a x)\right )}{4 a^4 \sqrt{\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

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Maple [A]  time = 0.042, size = 83, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{a}^{4}} \left ( 2\,\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -4\,\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +2\,\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) -\sin \left ( 4\,\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^3/arcsin(a*x)^(3/2),x)

[Out]

-1/4/a^4*(2*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))-4*arcsin(a*x)^(1
/2)*Pi^(1/2)*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))+2*sin(2*arcsin(a*x))-sin(4*arcsin(a*x)))/arcsin(a*x)^(1/2)

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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^3/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^3/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^{3}}{\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**3/asin(a*x)**(3/2),x)

[Out]

Integral(x**3/asin(a*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\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^3/arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(x^3/arcsin(a*x)^(3/2), x)