Optimal. Leaf size=34 \[ \frac{x}{\sqrt{a^2-x^2}}-\tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0063195, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {288, 217, 203} \[ \frac{x}{\sqrt{a^2-x^2}}-\tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 288
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a^2-x^2\right )^{3/2}} \, dx &=\frac{x}{\sqrt{a^2-x^2}}-\int \frac{1}{\sqrt{a^2-x^2}} \, dx\\ &=\frac{x}{\sqrt{a^2-x^2}}-\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{x}{\sqrt{a^2-x^2}}\right )\\ &=\frac{x}{\sqrt{a^2-x^2}}-\tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0301574, size = 39, normalized size = 1.15 \[ \frac{x-a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left (\frac{x}{a}\right )}{\sqrt{a^2-x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 31, normalized size = 0.9 \begin{align*} -\arctan \left ({x{\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}}} \right ) +{x{\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.40408, size = 32, normalized size = 0.94 \begin{align*} \frac{x}{\sqrt{a^{2} - x^{2}}} - \arcsin \left (\frac{x}{\sqrt{a^{2}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.22168, size = 111, normalized size = 3.26 \begin{align*} \frac{2 \,{\left (a^{2} - x^{2}\right )} \arctan \left (-\frac{a - \sqrt{a^{2} - x^{2}}}{x}\right ) + \sqrt{a^{2} - x^{2}} x}{a^{2} - x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.52178, size = 51, normalized size = 1.5 \begin{align*} \begin{cases} i \operatorname{acosh}{\left (\frac{x}{a} \right )} - \frac{i x}{a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \frac{\left |{x^{2}}\right |}{\left |{a^{2}}\right |} > 1 \\- \operatorname{asin}{\left (\frac{x}{a} \right )} + \frac{x}{a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.07619, size = 32, normalized size = 0.94 \begin{align*} -\arcsin \left (\frac{x}{a}\right ) \mathrm{sgn}\left (a\right ) + \frac{x}{\sqrt{a^{2} - x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]