Optimal. Leaf size=25 \[ \frac{2 \sqrt{1-x^2}}{1-x}-\sin ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0068515, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {663, 216} \[ \frac{2 \sqrt{1-x^2}}{1-x}-\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 663
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x^2}}{(1-x)^2} \, dx &=\frac{2 \sqrt{1-x^2}}{1-x}-\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{2 \sqrt{1-x^2}}{1-x}-\sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0548329, size = 50, normalized size = 2. \[ 2 \sqrt{1-x^2} \left (\frac{1}{1-x}+\frac{\sinh ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )}{\sqrt{x-1} \sqrt{x+1}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 40, normalized size = 1.6 \begin{align*}{\frac{1}{ \left ( -1+x \right ) ^{2}} \left ( - \left ( -1+x \right ) ^{2}+2-2\,x \right ) ^{{\frac{3}{2}}}}+\sqrt{- \left ( -1+x \right ) ^{2}+2-2\,x}-\arcsin \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.53006, size = 28, normalized size = 1.12 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}}{x - 1} - \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.14285, size = 104, normalized size = 4.16 \begin{align*} \frac{2 \,{\left ({\left (x - 1\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) + x - \sqrt{-x^{2} + 1} - 1\right )}}{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{\left (x - 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]