3.5.55 \(\int (a^2-x^2)^{3/2} \sqrt {\text {ArcSin}(\frac {x}{a})} \, dx\) [455]

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

[Out]

1/4*a^3*arcsin(x/a)^(3/2)*(a^2-x^2)^(1/2)/(1-x^2/a^2)^(1/2)-1/128*a^3*FresnelS(2*2^(1/2)/Pi^(1/2)*arcsin(x/a)^
(1/2))*2^(1/2)*Pi^(1/2)*(a^2-x^2)^(1/2)/(1-x^2/a^2)^(1/2)-1/8*a^3*FresnelS(2*arcsin(x/a)^(1/2)/Pi^(1/2))*Pi^(1
/2)*(a^2-x^2)^(1/2)/(1-x^2/a^2)^(1/2)+1/4*x*(a^2-x^2)^(3/2)*arcsin(x/a)^(1/2)+3/8*a^2*x*(a^2-x^2)^(1/2)*arcsin
(x/a)^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 226, normalized size of antiderivative = 1.00, 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 = {4743, 4741, 4737, 4731, 4491, 12, 3386, 3432, 4809} \begin {gather*} \frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}-\frac {\sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2-x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}\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 {\text {ArcSin}\left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}+\frac {a^3 \sqrt {a^2-x^2} \text {ArcSin}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1-\frac {x^2}{a^2}}} \end {gather*}

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 12

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

Rule 3386

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 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 4491

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 4731

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

Rule 4737

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

Rule 4741

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[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^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 4743

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/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcS
in[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]

Rule 4809

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.15, size = 183, normalized size = 0.81 \begin {gather*} \frac {a^3 \sqrt {a^2-x^2} \left (32 \text {ArcSin}\left (\frac {x}{a}\right )^2+8 \sqrt {2} \sqrt {-i \text {ArcSin}\left (\frac {x}{a}\right )} \text {Gamma}\left (\frac {3}{2},-2 i \text {ArcSin}\left (\frac {x}{a}\right )\right )+8 \sqrt {2} \sqrt {i \text {ArcSin}\left (\frac {x}{a}\right )} \text {Gamma}\left (\frac {3}{2},2 i \text {ArcSin}\left (\frac {x}{a}\right )\right )+\sqrt {-i \text {ArcSin}\left (\frac {x}{a}\right )} \text {Gamma}\left (\frac {3}{2},-4 i \text {ArcSin}\left (\frac {x}{a}\right )\right )+\sqrt {i \text {ArcSin}\left (\frac {x}{a}\right )} \text {Gamma}\left (\frac {3}{2},4 i \text {ArcSin}\left (\frac {x}{a}\right )\right )\right )}{128 \sqrt {1-\frac {x^2}{a^2}} \sqrt {\text {ArcSin}\left (\frac {x}{a}\right )}} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.32, size = 0, normalized size = 0.00 \[\int \left (a^{2}-x^{2}\right )^{\frac {3}{2}} \sqrt {\arcsin \left (\frac {x}{a}\right )}\, dx\]

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- \left (- a + x\right ) \left (a + x\right )\right )^{\frac {3}{2}} \sqrt {\operatorname {asin}{\left (\frac {x}{a} \right )}}\, dx \end {gather*}

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]

Integral((-(-a + x)*(a + x))**(3/2)*sqrt(asin(x/a)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {\mathrm {asin}\left (\frac {x}{a}\right )}\,{\left (a^2-x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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