3.94 \(\int \frac{x^2}{\sqrt{\sin ^{-1}(a x)}} \, dx\)

Optimal. Leaf size=71 \[ \frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a^3}-\frac{\sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a^3} \]

[Out]

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

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Rubi [A]  time = 0.0892158, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4635, 4406, 3304, 3352} \[ \frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a^3}-\frac{\sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[ArcSin[a*x]],x]

[Out]

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

Rule 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

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

Mathematica [C]  time = 0.0508396, size = 128, normalized size = 1.8 \[ -\frac{i \left (3 \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )-3 \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )+\sqrt{3} \left (\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \sin ^{-1}(a x)\right )-\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \sin ^{-1}(a x)\right )\right )\right )}{24 a^3 \sqrt{\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2/Sqrt[ArcSin[a*x]],x]

[Out]

((-I/24)*(3*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-I)*ArcSin[a*x]] - 3*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, I*ArcSin[a*
x]] + Sqrt[3]*(-(Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-3*I)*ArcSin[a*x]]) + Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (3*I)
*ArcSin[a*x]])))/(a^3*Sqrt[ArcSin[a*x]])

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Maple [A]  time = 0.039, size = 51, normalized size = 0.7 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{12\,{a}^{3}} \left ( -\sqrt{3}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}}{\sqrt{\pi }}\sqrt{\arcsin \left ( ax \right ) }} \right ) +3\,{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/arcsin(a*x)^(1/2),x)

[Out]

1/12/a^3*2^(1/2)*Pi^(1/2)*(-3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))+3*FresnelC(2^(1/2)/Pi
^(1/2)*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^2/arcsin(a*x)^(1/2),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^2/arcsin(a*x)^(1/2),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^{2}}{\sqrt{\operatorname{asin}{\left (a x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/asin(a*x)**(1/2),x)

[Out]

Integral(x**2/sqrt(asin(a*x)), x)

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Giac [C]  time = 1.4345, size = 126, normalized size = 1.77 \begin{align*} \frac{\left (i + 1\right ) \, \sqrt{6} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{6} \sqrt{\arcsin \left (a x\right )}\right )}{48 \, a^{3}} - \frac{\left (i - 1\right ) \, \sqrt{6} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{6} \sqrt{\arcsin \left (a x\right )}\right )}{48 \, a^{3}} - \frac{\left (i + 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{16 \, a^{3}} + \frac{\left (i - 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{16 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arcsin(a*x)^(1/2),x, algorithm="giac")

[Out]

(1/48*I + 1/48)*sqrt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^3 - (1/48*I - 1/48)*sqrt(6)*sq
rt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^3 - (1/16*I + 1/16)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*
sqrt(2)*sqrt(arcsin(a*x)))/a^3 + (1/16*I - 1/16)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcsin(a*x))
)/a^3