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

Optimal. Leaf size=59 \[ \frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a^2}-\frac{\sqrt{\sin ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\sin ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.150785, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4629, 4723, 3312, 3304, 3352} \[ \frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a^2}-\frac{\sqrt{\sin ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 x \sqrt{\sin ^{-1}(a x)} \, dx &=\frac{1}{2} x^2 \sqrt{\sin ^{-1}(a x)}-\frac{1}{4} a \int \frac{x^2}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx\\ &=\frac{1}{2} x^2 \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sin ^2(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^2}\\ &=\frac{1}{2} x^2 \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}-\frac{\cos (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a^2}\\ &=-\frac{\sqrt{\sin ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\sin ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^2}\\ &=-\frac{\sqrt{\sin ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\sin ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{4 a^2}\\ &=-\frac{\sqrt{\sin ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\sin ^{-1}(a x)}+\frac{\sqrt{\pi } C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{8 a^2}\\ \end{align*}

Mathematica [C]  time = 0.021174, size = 81, normalized size = 1.37 \[ -\frac{\sqrt{\sin ^{-1}(a x)} \left (\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-2 i \sin ^{-1}(a x)\right )+\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},2 i \sin ^{-1}(a x)\right )\right )}{8 \sqrt{2} a^2 \sqrt{\sin ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(Sqrt[ArcSin[a*x]]*(Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (-2*I)*ArcSin[a*x]] + Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (2
*I)*ArcSin[a*x]]))/(8*Sqrt[2]*a^2*Sqrt[ArcSin[a*x]^2])

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Maple [A]  time = 0.033, size = 42, normalized size = 0.7 \begin{align*}{\frac{1}{8\,{a}^{2}\sqrt{\pi }} \left ( -2\,\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }\cos \left ( 2\,\arcsin \left ( ax \right ) \right ) +\pi \,{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/a^2/Pi^(1/2)*(-2*arcsin(a*x)^(1/2)*Pi^(1/2)*cos(2*arcsin(a*x))+Pi*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(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*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*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 \sqrt{\operatorname{asin}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(asin(a*x)), x)

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Giac [C]  time = 1.37763, size = 96, normalized size = 1.63 \begin{align*} -\frac{\left (i + 1\right ) \, \sqrt{\pi } \operatorname{erf}\left (\left (i - 1\right ) \, \sqrt{\arcsin \left (a x\right )}\right )}{32 \, a^{2}} + \frac{\left (i - 1\right ) \, \sqrt{\pi } \operatorname{erf}\left (-\left (i + 1\right ) \, \sqrt{\arcsin \left (a x\right )}\right )}{32 \, a^{2}} - \frac{\sqrt{\arcsin \left (a x\right )} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} - \frac{\sqrt{\arcsin \left (a x\right )} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1/32*I + 1/32)*sqrt(pi)*erf((I - 1)*sqrt(arcsin(a*x)))/a^2 + (1/32*I - 1/32)*sqrt(pi)*erf(-(I + 1)*sqrt(arcs
in(a*x)))/a^2 - 1/8*sqrt(arcsin(a*x))*e^(2*I*arcsin(a*x))/a^2 - 1/8*sqrt(arcsin(a*x))*e^(-2*I*arcsin(a*x))/a^2