Optimal. Leaf size=40 \[ -\frac {1}{2} \log \left (x^2+1\right )+\tanh ^{-1}\left (\frac {x}{\sqrt {2 x^2+1}}\right )-\sqrt {2} \sinh ^{-1}\left (\sqrt {2} x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {6742, 260, 402, 215, 377, 206} \[ -\frac {1}{2} \log \left (x^2+1\right )+\tanh ^{-1}\left (\frac {x}{\sqrt {2 x^2+1}}\right )-\sqrt {2} \sinh ^{-1}\left (\sqrt {2} x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 215
Rule 260
Rule 377
Rule 402
Rule 6742
Rubi steps
\begin {align*} \int \frac {1}{x-\sqrt {1+2 x^2}} \, dx &=\int \left (-\frac {x}{1+x^2}-\frac {\sqrt {1+2 x^2}}{1+x^2}\right ) \, dx\\ &=-\int \frac {x}{1+x^2} \, dx-\int \frac {\sqrt {1+2 x^2}}{1+x^2} \, dx\\ &=-\frac {1}{2} \log \left (1+x^2\right )-2 \int \frac {1}{\sqrt {1+2 x^2}} \, dx+\int \frac {1}{\left (1+x^2\right ) \sqrt {1+2 x^2}} \, dx\\ &=-\sqrt {2} \sinh ^{-1}\left (\sqrt {2} x\right )-\frac {1}{2} \log \left (1+x^2\right )+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {1+2 x^2}}\right )\\ &=-\sqrt {2} \sinh ^{-1}\left (\sqrt {2} x\right )+\tanh ^{-1}\left (\frac {x}{\sqrt {1+2 x^2}}\right )-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 74, normalized size = 1.85 \[ \frac {1}{4} \left (-2 \log \left (x^2+1\right )-\log \left (3 x^2-2 \sqrt {2 x^2+1} x+1\right )+\log \left (3 x^2+2 \sqrt {2 x^2+1} x+1\right )-4 \sqrt {2} \sinh ^{-1}\left (\sqrt {2} x\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.44, size = 90, normalized size = 2.25 \[ \sqrt {2} \log \left (\sqrt {2} x - \sqrt {2 \, x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \log \left (\frac {2 \, x^{2} - \sqrt {2 \, x^{2} + 1} {\left (x + 1\right )} + x + 1}{x^{2}}\right ) + \frac {1}{2} \, \log \left (\frac {2 \, x^{2} + \sqrt {2 \, x^{2} + 1} {\left (x - 1\right )} - x + 1}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.41, size = 88, normalized size = 2.20 \[ \sqrt {2} \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 1}\right ) + \frac {1}{2} \, \log \left ({\left (\sqrt {2} x - \sqrt {2 \, x^{2} + 1}\right )}^{2} + 2 \, \sqrt {2} + 3\right ) - \frac {1}{2} \, \log \left ({\left (\sqrt {2} x - \sqrt {2 \, x^{2} + 1}\right )}^{2} - 2 \, \sqrt {2} + 3\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 33, normalized size = 0.82 \[ -\sqrt {2}\, \arcsinh \left (\sqrt {2}\, x \right )+\arctanh \left (\frac {x}{\sqrt {2 x^{2}+1}}\right )-\frac {\ln \left (x^{2}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x - \sqrt {2 \, x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.50, size = 57, normalized size = 1.42 \[ -\ln \left (x-\mathrm {i}\right )-\frac {\ln \left (x-\frac {\sqrt {2}\,\sqrt {x^2+\frac {1}{2}}}{2}+\frac {1}{2}{}\mathrm {i}\right )}{2}+\frac {\ln \left (x+\frac {\sqrt {2}\,\sqrt {x^2+\frac {1}{2}}}{2}-\frac {1}{2}{}\mathrm {i}\right )}{2}-\sqrt {2}\,\mathrm {asinh}\left (\sqrt {2}\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.21, size = 27, normalized size = 0.68 \[ - \log {\left (x - \sqrt {2 x^{2} + 1} \right )} - \sqrt {2} \operatorname {asinh}{\left (\sqrt {2} x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________