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

Optimal. Leaf size=86 \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a^3}+\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{12 a^3}+\frac{1}{3} x^3 \sqrt{\sin ^{-1}(a x)} \]

[Out]

(x^3*Sqrt[ArcSin[a*x]])/3 - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(4*a^3) + (Sqrt[Pi/6]*FresnelS
[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(12*a^3)

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Rubi [A]  time = 0.180875, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4629, 4723, 3312, 3305, 3351} \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a^3}+\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{12 a^3}+\frac{1}{3} x^3 \sqrt{\sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*Sqrt[ArcSin[a*x]])/3 - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(4*a^3) + (Sqrt[Pi/6]*FresnelS
[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(12*a^3)

Rule 4629

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcSin[c*x])^n)/(m
+ 1), x] - Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^(
m + 1), Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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]

Rubi steps

\begin{align*} \int x^2 \sqrt{\sin ^{-1}(a x)} \, dx &=\frac{1}{3} x^3 \sqrt{\sin ^{-1}(a x)}-\frac{1}{6} a \int \frac{x^3}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx\\ &=\frac{1}{3} x^3 \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sin ^3(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{6 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{3 \sin (x)}{4 \sqrt{x}}-\frac{\sin (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{6 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\sin ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{24 a^3}-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\sin ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{12 a^3}-\frac{\operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{4 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\sin ^{-1}(a x)}-\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a^3}+\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{12 a^3}\\ \end{align*}

Mathematica [C]  time = 0.0495305, size = 126, normalized size = 1.47 \[ \frac{9 \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-i \sin ^{-1}(a x)\right )+9 \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},i \sin ^{-1}(a x)\right )-\sqrt{3} \left (\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-3 i \sin ^{-1}(a x)\right )+\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},3 i \sin ^{-1}(a x)\right )\right )}{72 a^3 \sqrt{\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(9*Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (-I)*ArcSin[a*x]] + 9*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, I*ArcSin[a*x]] - Sqr
t[3]*(Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (-3*I)*ArcSin[a*x]] + Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (3*I)*ArcSin[a*x]
]))/(72*a^3*Sqrt[ArcSin[a*x]])

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Maple [A]  time = 0.044, size = 96, normalized size = 1.1 \begin{align*} -{\frac{1}{72\,{a}^{3}} \left ( -\sqrt{3}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{3}\sqrt{2}}{\sqrt{\pi }}\sqrt{\arcsin \left ( ax \right ) }} \right ) +9\,\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -18\,ax\arcsin \left ( ax \right ) +6\,\arcsin \left ( ax \right ) \sin \left ( 3\,\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^2*arcsin(a*x)^(1/2),x)

[Out]

-1/72/a^3/arcsin(a*x)^(1/2)*(-3^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arc
sin(a*x)^(1/2))+9*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))-18*a*x*arcsi
n(a*x)+6*arcsin(a*x)*sin(3*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^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 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 = 1.48785, size = 223, normalized size = 2.59 \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 )}{288 \, 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 )}{288 \, 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 )}{32 \, 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 )}{32 \, a^{3}} + \frac{i \, \sqrt{\arcsin \left (a x\right )} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{24 \, a^{3}} - \frac{i \, \sqrt{\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{8 \, a^{3}} + \frac{i \, \sqrt{\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{8 \, a^{3}} - \frac{i \, \sqrt{\arcsin \left (a x\right )} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{24 \, 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]

(1/288*I - 1/288)*sqrt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^3 - (1/288*I + 1/288)*sqrt(6
)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^3 - (1/32*I - 1/32)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1
/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^3 + (1/32*I + 1/32)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcsin(a
*x)))/a^3 + 1/24*I*sqrt(arcsin(a*x))*e^(3*I*arcsin(a*x))/a^3 - 1/8*I*sqrt(arcsin(a*x))*e^(I*arcsin(a*x))/a^3 +
1/8*I*sqrt(arcsin(a*x))*e^(-I*arcsin(a*x))/a^3 - 1/24*I*sqrt(arcsin(a*x))*e^(-3*I*arcsin(a*x))/a^3