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

Optimal. Leaf size=226 \[ -\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\sqrt{\pi } a^3 \sqrt{a^2-x^2} S\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{8 \sqrt{1-\frac{x^2}{a^2}}}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )} \]

[Out]

(3*a^2*x*Sqrt[a^2 - x^2]*Sqrt[ArcSin[x/a]])/8 + (x*(a^2 - x^2)^(3/2)*Sqrt[ArcSin[x/a]])/4 + (a^3*Sqrt[a^2 - x^
2]*ArcSin[x/a]^(3/2))/(4*Sqrt[1 - x^2/a^2]) - (a^3*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSi
n[x/a]]])/(64*Sqrt[1 - x^2/a^2]) - (a^3*Sqrt[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 = 0.236683, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4649, 4647, 4641, 4635, 4406, 12, 3305, 3351, 4723} \[ -\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\sqrt{\pi } a^3 \sqrt{a^2-x^2} S\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{8 \sqrt{1-\frac{x^2}{a^2}}}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2*x*Sqrt[a^2 - x^2]*Sqrt[ArcSin[x/a]])/8 + (x*(a^2 - x^2)^(3/2)*Sqrt[ArcSin[x/a]])/4 + (a^3*Sqrt[a^2 - x^
2]*ArcSin[x/a]^(3/2))/(4*Sqrt[1 - x^2/a^2]) - (a^3*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSi
n[x/a]]])/(64*Sqrt[1 - x^2/a^2]) - (a^3*Sqrt[Pi]*Sqrt[a^2 - x^2]*FresnelS[(2*Sqrt[ArcSin[x/a]])/Sqrt[Pi]])/(8*
Sqrt[1 - x^2/a^2])

Rule 4649

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

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]

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])

Rubi steps

\begin{align*} \int \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )} \, dx &=\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} \left (3 a^2\right ) \int \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )} \, dx-\frac{\left (a \sqrt{a^2-x^2}\right ) \int \frac{x \left (1-\frac{x^2}{a^2}\right )}{\sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}} \, dx}{8 \sqrt{1-\frac{x^2}{a^2}}}\\ &=\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}-\frac{\left (3 a \sqrt{a^2-x^2}\right ) \int \frac{x}{\sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}} \, dx}{16 \sqrt{1-\frac{x^2}{a^2}}}+\frac{\left (3 a^2 \sqrt{a^2-x^2}\right ) \int \frac{\sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx}{8 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (a^3 \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}\left (\frac{x}{a}\right )\right )}{8 \sqrt{1-\frac{x^2}{a^2}}}\\ &=\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (a^3 \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 \sqrt{x}}+\frac{\sin (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}\left (\frac{x}{a}\right )\right )}{8 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \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 )}{16 \sqrt{1-\frac{x^2}{a^2}}}\\ &=\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (a^3 \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (a^3 \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 )}{32 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \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 )}{16 \sqrt{1-\frac{x^2}{a^2}}}\\ &=\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (a^3 \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}\right )}{32 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (a^3 \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 )}{16 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \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 )}{32 \sqrt{1-\frac{x^2}{a^2}}}\\ &=\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1-\frac{x^2}{a^2}}}-\frac{a^3 \sqrt{\frac{\pi }{2}} \sqrt{a^2-x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{1-\frac{x^2}{a^2}}}-\frac{a^3 \sqrt{\pi } \sqrt{a^2-x^2} S\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{32 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \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 )}{16 \sqrt{1-\frac{x^2}{a^2}}}\\ &=\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{1-\frac{x^2}{a^2}}}-\frac{a^3 \sqrt{\frac{\pi }{2}} \sqrt{a^2-x^2} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{1-\frac{x^2}{a^2}}}-\frac{a^3 \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 = 0.200631, size = 183, normalized size = 0.81 \[ \frac{a^3 \sqrt{a^2-x^2} \left (8 \sqrt{2} \sqrt{-i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},-2 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+8 \sqrt{2} \sqrt{i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},2 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+\sqrt{-i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},-4 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+\sqrt{i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},4 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+32 \sin ^{-1}\left (\frac{x}{a}\right )^2\right )}{128 \sqrt{1-\frac{x^2}{a^2}} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a^2 - x^2)^(3/2)*Sqrt[ArcSin[x/a]],x]

[Out]

(a^3*Sqrt[a^2 - x^2]*(32*ArcSin[x/a]^2 + 8*Sqrt[2]*Sqrt[(-I)*ArcSin[x/a]]*Gamma[3/2, (-2*I)*ArcSin[x/a]] + 8*S
qrt[2]*Sqrt[I*ArcSin[x/a]]*Gamma[3/2, (2*I)*ArcSin[x/a]] + Sqrt[(-I)*ArcSin[x/a]]*Gamma[3/2, (-4*I)*ArcSin[x/a
]] + Sqrt[I*ArcSin[x/a]]*Gamma[3/2, (4*I)*ArcSin[x/a]]))/(128*Sqrt[1 - x^2/a^2]*Sqrt[ArcSin[x/a]])

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Maple [F]  time = 0.256, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}-{x}^{2} \right ) ^{{\frac{3}{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)^(3/2)*arcsin(x/a)^(1/2),x)

[Out]

int((a^2-x^2)^(3/2)*arcsin(x/a)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} - x^{2}\right )}^{\frac{3}{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)^(3/2)*arcsin(x/a)^(1/2),x, algorithm="maxima")

[Out]

integrate((a^2 - x^2)^(3/2)*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)^(3/2)*arcsin(x/a)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2-x**2)**(3/2)*asin(x/a)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} - x^{2}\right )}^{\frac{3}{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)^(3/2)*arcsin(x/a)^(1/2),x, algorithm="giac")

[Out]

integrate((a^2 - x^2)^(3/2)*sqrt(arcsin(x/a)), x)