\(\int \frac {\arcsin (\frac {x}{a})^{3/2}}{(a^2-x^2)^{3/2}} \, dx\) [463]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx=\int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

(x*ArcSin[x/a]^(3/2))/(a^2*Sqrt[a^2 - x^2]) - (3*Sqrt[1 - x^2/a^2]*Defer[Int][(x*Sqrt[ArcSin[x/a]])/(1 - x^2/a
^2), x])/(2*a^3*Sqrt[a^2 - x^2])

Rubi steps \begin{align*} \text {integral}& = \frac {x \arcsin \left (\frac {x}{a}\right )^{3/2}}{a^2 \sqrt {a^2-x^2}}-\frac {\left (3 \sqrt {1-\frac {x^2}{a^2}}\right ) \int \frac {x \sqrt {\arcsin \left (\frac {x}{a}\right )}}{1-\frac {x^2}{a^2}} \, dx}{2 a^3 \sqrt {a^2-x^2}} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx=\int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

\[\int \frac {\arcsin \left (\frac {x}{a}\right )^{\frac {3}{2}}}{\left (a^{2}-x^{2}\right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 15.88 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {asin}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}}{\left (- \left (- a + x\right ) \left (a + x\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.90 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx=\int { \frac {\arcsin \left (\frac {x}{a}\right )^{\frac {3}{2}}}{{\left (a^{2} - x^{2}\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {asin}\left (\frac {x}{a}\right )}^{3/2}}{{\left (a^2-x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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