Optimal. Leaf size=26 \[ \sqrt{x^2-1} \log \left (\sqrt{x^2-1}+x\right )-x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0360943, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {261, 2554, 8} \[ \sqrt{x^2-1} \log \left (\sqrt{x^2-1}+x\right )-x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 2554
Rule 8
Rubi steps
\begin{align*} \int \frac{x \log \left (x+\sqrt{-1+x^2}\right )}{\sqrt{-1+x^2}} \, dx &=\sqrt{-1+x^2} \log \left (x+\sqrt{-1+x^2}\right )-\int 1 \, dx\\ &=-x+\sqrt{-1+x^2} \log \left (x+\sqrt{-1+x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0214885, size = 26, normalized size = 1. \[ \sqrt{x^2-1} \log \left (\sqrt{x^2-1}+x\right )-x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{x\ln \left ( x+\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0574, size = 30, normalized size = 1.15 \begin{align*} \sqrt{x^{2} - 1} \log \left (x + \sqrt{x^{2} - 1}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.99764, size = 57, normalized size = 2.19 \begin{align*} \sqrt{x^{2} - 1} \log \left (x + \sqrt{x^{2} - 1}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 23.102, size = 20, normalized size = 0.77 \begin{align*} - x + \sqrt{x^{2} - 1} \log{\left (x + \sqrt{x^{2} - 1} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09971, size = 30, normalized size = 1.15 \begin{align*} \sqrt{x^{2} - 1} \log \left (x + \sqrt{x^{2} - 1}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]