\(\int \frac {\arcsin (\frac {x}{a})^{3/2}}{\sqrt {a^2-x^2}} \, dx\) [69]

Optimal result

Integrand size = 24, antiderivative size = 42 \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 a \sqrt {1-\frac {x^2}{a^2}} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {4737} \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 a \sqrt {1-\frac {x^2}{a^2}} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]

[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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 a \sqrt {1-\frac {x^2}{a^2}} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 a \sqrt {1-\frac {x^2}{a^2}} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]

[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])

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90

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

[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^2-x^2)^(1/2)*((a^2-x^2)/a^2)^(1/2)*a

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90 \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2}{5} \, \sqrt {-\arctan \left (-\frac {x}{\sqrt {a^{2} - x^{2}}}\right )} \arctan \left (-\frac {x}{\sqrt {a^{2} - x^{2}}}\right )^{2} \]

[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

Sympy [F]

\[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\int \frac {\operatorname {asin}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}}{\sqrt {- \left (- a + x\right ) \left (a + x\right )}}\, dx \]

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.36 \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 \, {\left | a \right |} \arcsin \left (\frac {x}{a}\right )^{\frac {5}{2}}}{5 \, a} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\int \frac {{\mathrm {asin}\left (\frac {x}{a}\right )}^{3/2}}{\sqrt {a^2-x^2}} \,d x \]

[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)