Optimal. Leaf size=26 \[ \sqrt{x^2+1}+x \log \left (x-\sqrt{x^2+1}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0053162, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2534, 261} \[ \sqrt{x^2+1}+x \log \left (x-\sqrt{x^2+1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2534
Rule 261
Rubi steps
\begin{align*} \int \log \left (x-\sqrt{1+x^2}\right ) \, dx &=x \log \left (x-\sqrt{1+x^2}\right )+\int \frac{x}{\sqrt{1+x^2}} \, dx\\ &=\sqrt{1+x^2}+x \log \left (x-\sqrt{1+x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0202813, size = 26, normalized size = 1. \[ \sqrt{x^2+1}+x \log \left (x-\sqrt{x^2+1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 23, normalized size = 0.9 \begin{align*} x\ln \left ( x-\sqrt{{x}^{2}+1} \right ) +\sqrt{{x}^{2}+1} \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*} x \log \left (x - \sqrt{x^{2} + 1}\right ) - x + \arctan \left (x\right ) + \int -\frac{x}{x^{3} -{\left (x^{2} + 1\right )}^{\frac{3}{2}} + x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.36471, size = 57, normalized size = 2.19 \begin{align*} x \log \left (x - \sqrt{x^{2} + 1}\right ) + \sqrt{x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 9.29586, size = 20, normalized size = 0.77 \begin{align*} x \log{\left (x - \sqrt{x^{2} + 1} \right )} + \sqrt{x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.05946, size = 30, normalized size = 1.15 \begin{align*} x \log \left (x - \sqrt{x^{2} + 1}\right ) + \sqrt{x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]