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

Optimal. Leaf size=126 \[ -\frac{\sqrt{\pi } a \sqrt{a^2-x^2} S\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{8 \sqrt{1-\frac{x^2}{a^2}}}+\frac{a \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )} \]

[Out]

(x*Sqrt[a^2 - x^2]*Sqrt[ArcSin[x/a]])/2 + (a*Sqrt[a^2 - x^2]*ArcSin[x/a]^(3/2))/(3*Sqrt[1 - x^2/a^2]) - (a*Sqr
t[Pi]*Sqrt[a^2 - x^2]*FresnelS[(2*Sqrt[ArcSin[x/a]])/Sqrt[Pi]])/(8*Sqrt[1 - x^2/a^2])

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Rubi [A]  time = 0.105709, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {4647, 4641, 4635, 4406, 12, 3305, 3351} \[ -\frac{\sqrt{\pi } a \sqrt{a^2-x^2} S\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{8 \sqrt{1-\frac{x^2}{a^2}}}+\frac{a \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[a^2 - x^2]*Sqrt[ArcSin[x/a]])/2 + (a*Sqrt[a^2 - x^2]*ArcSin[x/a]^(3/2))/(3*Sqrt[1 - x^2/a^2]) - (a*Sqr
t[Pi]*Sqrt[a^2 - x^2]*FresnelS[(2*Sqrt[ArcSin[x/a]])/Sqrt[Pi]])/(8*Sqrt[1 - x^2/a^2])

Rule 4647

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(x*Sqrt[d + e*x^2]*(
a + b*ArcSin[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcSin[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 - c^2*x^2]), Int[x*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

Mathematica [C]  time = 0.0719123, size = 148, normalized size = 1.17 \[ \frac{\sqrt{a^2-x^2} \left (3 \sqrt{2} a \sqrt{-i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{1}{2},-2 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+3 \sqrt{2} a \sqrt{i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{1}{2},2 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+48 x \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )+32 a \sin ^{-1}\left (\frac{x}{a}\right )^2\right )}{96 \sqrt{1-\frac{x^2}{a^2}} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[a^2 - x^2]*(48*x*Sqrt[1 - x^2/a^2]*ArcSin[x/a] + 32*a*ArcSin[x/a]^2 + 3*Sqrt[2]*a*Sqrt[(-I)*ArcSin[x/a]]
*Gamma[1/2, (-2*I)*ArcSin[x/a]] + 3*Sqrt[2]*a*Sqrt[I*ArcSin[x/a]]*Gamma[1/2, (2*I)*ArcSin[x/a]]))/(96*Sqrt[1 -
x^2/a^2]*Sqrt[ArcSin[x/a]])

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Maple [F]  time = 0.267, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{a}^{2}-{x}^{2}}\sqrt{\arcsin \left ({\frac{x}{a}} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a^2 - x^2)*sqrt(arcsin(x/a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-(-a + x)*(a + x))*sqrt(asin(x/a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a^2 - x^2)*sqrt(arcsin(x/a)), x)