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

Optimal. Leaf size=42 \[ \frac {2 a \sqrt {1-\frac {x^2}{a^2}} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4643, 4641} \[ \frac {2 a \sqrt {1-\frac {x^2}{a^2}} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*Sqrt[1 - x^2/a^2]*ArcSin[x/a]^(5/2))/(5*Sqrt[a^2 - x^2])

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 4643

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

Rubi steps

\begin {align*} \int \frac {\sin ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx &=\frac {\sqrt {1-\frac {x^2}{a^2}} \int \frac {\sin ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx}{\sqrt {a^2-x^2}}\\ &=\frac {2 a \sqrt {1-\frac {x^2}{a^2}} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 42, normalized size = 1.00 \[ \frac {2 a \sqrt {1-\frac {x^2}{a^2}} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*Sqrt[1 - x^2/a^2]*ArcSin[x/a]^(5/2))/(5*Sqrt[a^2 - x^2])

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 38, normalized size = 0.90 \[ \frac {2}{5} \, \sqrt {-\arctan \left (-\frac {x}{\sqrt {a^{2} - x^{2}}}\right )} \arctan \left (-\frac {x}{\sqrt {a^{2} - x^{2}}}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*sqrt(-arctan(-x/sqrt(a^2 - x^2)))*arctan(-x/sqrt(a^2 - x^2))^2

________________________________________________________________________________________

giac [A]  time = 1.08, size = 15, normalized size = 0.36 \[ \frac {2 \, {\left | a \right |} \arcsin \left (\frac {x}{a}\right )^{\frac {5}{2}}}{5 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*abs(a)*arcsin(x/a)^(5/2)/a

________________________________________________________________________________________

maple [A]  time = 0.25, size = 38, normalized size = 0.90

method	result	size
default	\(\frac {2 \arcsin \left (\frac {x}{a}\right )^{\frac {5}{2}} a \sqrt {\frac {a^{2}-x^{2}}{a^{2}}}}{5 \sqrt {a^{2}-x^{2}}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x/a)^(3/2)/(a^2-x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/5*arcsin(x/a)^(5/2)*a/(a^2-x^2)^(1/2)*((a^2-x^2)/a^2)^(1/2)

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {asin}\left (\frac {x}{a}\right )}^{3/2}}{\sqrt {a^2-x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________