Optimal. Leaf size=24 \[ \log (x)-\left (1+\sqrt{2}\right ) \log \left (x^{\sqrt{2}-1}+1\right ) \]
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Rubi [A] time = 0.0163587, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {1593, 266, 36, 29, 31} \[ \log (x)-\left (1+\sqrt{2}\right ) \log \left (x^{\sqrt{2}-1}+1\right ) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x+x^{\sqrt{2}}} \, dx &=\int \frac{1}{x \left (1+x^{-1+\sqrt{2}}\right )} \, dx\\ &=\left (1+\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{x (1+x)} \, dx,x,x^{-1+\sqrt{2}}\right )\\ &=\left (-1-\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^{-1+\sqrt{2}}\right )+\left (1+\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^{-1+\sqrt{2}}\right )\\ &=\log (x)-\left (1+\sqrt{2}\right ) \log \left (1+x^{-1+\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0208126, size = 24, normalized size = 1. \[ \log (x)-\left (1+\sqrt{2}\right ) \log \left (x^{\sqrt{2}-1}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 39, normalized size = 1.6 \begin{align*} \sqrt{2}\ln \left ( x \right ) +2\,\ln \left ( x \right ) -\ln \left ( x+{{\rm e}^{\sqrt{2}\ln \left ( x \right ) }} \right ) \sqrt{2}-\ln \left ( x+{{\rm e}^{\sqrt{2}\ln \left ( x \right ) }} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61263, size = 42, normalized size = 1.75 \begin{align*} \frac{\sqrt{2} \log \left (x\right )}{\sqrt{2} - 1} - \frac{\log \left (x + x^{\left (\sqrt{2}\right )}\right )}{\sqrt{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.774054, size = 78, normalized size = 3.25 \begin{align*} -{\left (\sqrt{2} + 1\right )} \log \left (x + x^{\left (\sqrt{2}\right )}\right ) +{\left (\sqrt{2} + 2\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.680891, size = 76, normalized size = 3.17 \begin{align*} \frac{140 \sqrt{2} \log{\left (x \right )}}{-338 + 239 \sqrt{2}} - \frac{198 \log{\left (x \right )}}{-338 + 239 \sqrt{2}} + \frac{99 \sqrt{2} \log{\left (x + x^{\sqrt{2}} \right )}}{-338 + 239 \sqrt{2}} - \frac{140 \log{\left (x + x^{\sqrt{2}} \right )}}{-338 + 239 \sqrt{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x + x^{\left (\sqrt{2}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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