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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.106854, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}} \]

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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.53098, size = 34, normalized size = 0.81 \[ \frac{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}} \operatorname{asin}^{\frac{5}{2}}{\left (\frac{x}{a} \right )}}{5 \sqrt{a^{2} - x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(asin(x/a)**(3/2)/(a**2-x**2)**(1/2),x)

[Out]

2*a*sqrt(1 - x**2/a**2)*asin(x/a)**(5/2)/(5*sqrt(a**2 - x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0214433, size = 42, normalized size = 1. \[ \frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2-x^2}} \]

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])

_______________________________________________________________________________________

Maple [A]  time = 0.101, size = 38, normalized size = 0.9 \[{\frac{2\,a}{5} \left ( \arcsin \left ({\frac{x}{a}} \right ) \right ) ^{{\frac{5}{2}}}\sqrt{{\frac{{a}^{2}-{x}^{2}}{{a}^{2}}}}{\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arcsin(x/a)^(3/2)/(a^2-x^2)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 4.26498, size = 486, normalized size = 11.57 \[ \frac{2}{15} \, \sqrt{2}{\left (\arctan \left (x, \sqrt{a + x} \sqrt{a - x}\right )^{2} + \frac{1}{4} \, \log \left (a^{2}\right )^{2} - \log \left (a^{2}\right ) \log \left (a\right ) + \log \left (a\right )^{2}\right )}^{\frac{5}{4}} \cos \left (\frac{5}{2} \, \arctan \left (\arctan \left (x, \sqrt{a + x} \sqrt{a - x}\right ), -\frac{1}{2} \, \log \left (a^{2}\right ) + \log \left (a\right )\right )\right ) - \frac{1}{6} \,{\left (\arctan \left (x, \sqrt{a + x} \sqrt{a - x}\right )^{2} + \frac{1}{4} \, \log \left (a^{2}\right )^{2} - \log \left (a^{2}\right ) \log \left (a\right ) + \log \left (a\right )^{2}\right )}^{\frac{3}{4}}{\left (2 \, \sqrt{2} \arctan \left (x, \sqrt{a + x} \sqrt{a - x}\right ) - \sqrt{2} \log \left (a^{2}\right ) + 2 \, \sqrt{2} \log \left (a\right )\right )} \cos \left (\frac{3}{2} \, \arctan \left (\arctan \left (x, \sqrt{a + x} \sqrt{a - x}\right ), -\frac{1}{2} \, \log \left (a^{2}\right ) + \log \left (a\right )\right )\right ) + \frac{2}{15} \, \sqrt{2}{\left (\arctan \left (x, \sqrt{a + x} \sqrt{a - x}\right )^{2} + \frac{1}{4} \, \log \left (a^{2}\right )^{2} - \log \left (a^{2}\right ) \log \left (a\right ) + \log \left (a\right )^{2}\right )}^{\frac{5}{4}} \sin \left (\frac{5}{2} \, \arctan \left (\arctan \left (x, \sqrt{a + x} \sqrt{a - x}\right ), -\frac{1}{2} \, \log \left (a^{2}\right ) + \log \left (a\right )\right )\right ) + \frac{1}{6} \,{\left (\arctan \left (x, \sqrt{a + x} \sqrt{a - x}\right )^{2} + \frac{1}{4} \, \log \left (a^{2}\right )^{2} - \log \left (a^{2}\right ) \log \left (a\right ) + \log \left (a\right )^{2}\right )}^{\frac{3}{4}}{\left (2 \, \sqrt{2} \arctan \left (x, \sqrt{a + x} \sqrt{a - x}\right ) + \sqrt{2} \log \left (a^{2}\right ) - 2 \, \sqrt{2} \log \left (a\right )\right )} \sin \left (\frac{3}{2} \, \arctan \left (\arctan \left (x, \sqrt{a + x} \sqrt{a - x}\right ), -\frac{1}{2} \, \log \left (a^{2}\right ) + \log \left (a\right )\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*sqrt(2)*(arctan2(x, sqrt(a + x)*sqrt(a - x))^2 + 1/4*log(a^2)^2 - log(a^2)*
log(a) + log(a)^2)^(5/4)*cos(5/2*arctan2(arctan2(x, sqrt(a + x)*sqrt(a - x)), -1
/2*log(a^2) + log(a))) - 1/6*(arctan2(x, sqrt(a + x)*sqrt(a - x))^2 + 1/4*log(a^
2)^2 - log(a^2)*log(a) + log(a)^2)^(3/4)*(2*sqrt(2)*arctan2(x, sqrt(a + x)*sqrt(
a - x)) - sqrt(2)*log(a^2) + 2*sqrt(2)*log(a))*cos(3/2*arctan2(arctan2(x, sqrt(a
+ x)*sqrt(a - x)), -1/2*log(a^2) + log(a))) + 2/15*sqrt(2)*(arctan2(x, sqrt(a +
x)*sqrt(a - x))^2 + 1/4*log(a^2)^2 - log(a^2)*log(a) + log(a)^2)^(5/4)*sin(5/2*
arctan2(arctan2(x, sqrt(a + x)*sqrt(a - x)), -1/2*log(a^2) + log(a))) + 1/6*(arc
tan2(x, sqrt(a + x)*sqrt(a - x))^2 + 1/4*log(a^2)^2 - log(a^2)*log(a) + log(a)^2
)^(3/4)*(2*sqrt(2)*arctan2(x, sqrt(a + x)*sqrt(a - x)) + sqrt(2)*log(a^2) - 2*sq
rt(2)*log(a))*sin(3/2*arctan2(arctan2(x, sqrt(a + x)*sqrt(a - x)), -1/2*log(a^2)
+ log(a)))

_______________________________________________________________________________________

Fricas [A]  time = 0.218736, size = 24, normalized size = 0.57 \[ \frac{2}{5} \, \arctan \left (\frac{x}{\sqrt{a^{2} - x^{2}}}\right )^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*arctan(x/sqrt(a^2 - x^2))^(5/2)

_______________________________________________________________________________________

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

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]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.310963, size = 16, normalized size = 0.38 \[ \frac{2}{5} \, \arcsin \left (\frac{x}{a}\right )^{\frac{5}{2}}{\rm sign}\left (a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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