Optimal. Leaf size=36 \[ \frac{2 x \sqrt{\frac{\left (1-x^2\right )^2}{x \left (x^2+1\right )}}}{1-x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.139978, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {6718, 449} \[ \frac{2 x \sqrt{\frac{\left (1-x^2\right )^2}{x \left (x^2+1\right )}}}{1-x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6718
Rule 449
Rubi steps
\begin{align*} \int \frac{\sqrt{\frac{\left (-1+x^2\right )^2}{x \left (1+x^2\right )}}}{1+x^2} \, dx &=\frac{\left (\sqrt{x} \sqrt{\frac{\left (-1+x^2\right )^2}{x \left (1+x^2\right )}} \sqrt{1+x^2}\right ) \int \frac{-1+x^2}{\sqrt{x} \left (1+x^2\right )^{3/2}} \, dx}{-1+x^2}\\ &=\frac{2 x \sqrt{\frac{\left (1-x^2\right )^2}{x \left (1+x^2\right )}}}{1-x^2}\\ \end{align*}
Mathematica [A] time = 0.0196528, size = 29, normalized size = 0.81 \[ -\frac{2 x \sqrt{\frac{\left (x^2-1\right )^2}{x^3+x}}}{x^2-1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 34, normalized size = 0.9 \begin{align*} -2\,{\frac{x}{ \left ( x-1 \right ) \left ( 1+x \right ) }\sqrt{{\frac{ \left ({x}^{2}-1 \right ) ^{2}}{x \left ({x}^{2}+1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{{\left (x^{2} - 1\right )}^{2}}{{\left (x^{2} + 1\right )} x}}}{x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.45234, size = 68, normalized size = 1.89 \begin{align*} -\frac{2 \, x \sqrt{\frac{x^{4} - 2 \, x^{2} + 1}{x^{3} + x}}}{x^{2} - 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{\frac{\left (x - 1\right )^{2} \left (x + 1\right )^{2}}{x^{3} + x}}}{x^{2} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{{\left (x^{2} - 1\right )}^{2}}{{\left (x^{2} + 1\right )} x}}}{x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]