Optimal. Leaf size=26 \[ x \log \left (\sqrt {x^2-1}+x\right )-\sqrt {x^2-1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2534, 261} \[ x \log \left (\sqrt {x^2-1}+x\right )-\sqrt {x^2-1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 2534
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 = 0.02, size = 26, normalized size = 1.00 \[ x \log \left (\sqrt {x^2-1}+x\right )-\sqrt {x^2-1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 22, normalized size = 0.85 \[ x \log \left (x + \sqrt {x^{2} - 1}\right ) - \sqrt {x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 22, normalized size = 0.85 \[ x \log \left (x + \sqrt {x^{2} - 1}\right ) - \sqrt {x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 23, normalized size = 0.88 \[ x \ln \left (x +\sqrt {x^{2}-1}\right )-\sqrt {x^{2}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ x \log \left (\sqrt {x + 1} \sqrt {x - 1} + x\right ) - x + \int \frac {x}{x^{3} + {\left (x^{2} - 1\right )} e^{\left (\frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right )\right )} - x}\,{d x} + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.55, size = 22, normalized size = 0.85 \[ x\,\ln \left (x+\sqrt {x^2-1}\right )-\sqrt {x^2-1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.78, size = 20, normalized size = 0.77 \[ x \log {\left (x + \sqrt {x^{2} - 1} \right )} - \sqrt {x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________