Optimal. Leaf size=89 \[ \frac{x \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{a^2 \sqrt{a^2-x^2}}-\frac{\sqrt{1-\frac{x^2}{a^2}} \text{Unintegrable}\left (\frac{x}{\left (1-\frac{x^2}{a^2}\right ) \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}},x\right )}{2 a^3 \sqrt{a^2-x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0754712, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{\sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx &=\frac{x \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{a^2 \sqrt{a^2-x^2}}-\frac{\sqrt{1-\frac{x^2}{a^2}} \int \frac{x}{\left (1-\frac{x^2}{a^2}\right ) \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}} \, dx}{2 a^3 \sqrt{a^2-x^2}}\\ \end{align*}
Mathematica [A] time = 0.573014, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.227, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{\arcsin \left ({\frac{x}{a}} \right ) } \left ({a}^{2}-{x}^{2} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\arcsin \left (\frac{x}{a}\right )}}{{\left (a^{2} - x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{asin}{\left (\frac{x}{a} \right )}}}{\left (- \left (- a + x\right ) \left (a + x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\arcsin \left (\frac{x}{a}\right )}}{{\left (a^{2} - x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]