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 = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, 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} \begin {gather*} \log (x)-\left (1+\sqrt {2}\right ) \log \left (x^{\sqrt {2}-1}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 1593
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*}
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Mathematica [A] time = 0.02, size = 24, normalized size = 1.00 \begin {gather*} \log (x)-\left (1+\sqrt {2}\right ) \log \left (x^{\sqrt {2}-1}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.01, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x+x^{\sqrt {2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 24, normalized size = 1.00 \begin {gather*} -{\left (\sqrt {2} + 1\right )} \log \left (x + x^{\left (\sqrt {2}\right )}\right ) + {\left (\sqrt {2} + 2\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x + x^{\left (\sqrt {2}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 39, normalized size = 1.62 \begin {gather*} \sqrt {2}\, \ln \relax (x )+2 \ln \relax (x )-\sqrt {2}\, \ln \left (x +{\mathrm e}^{\sqrt {2}\, \ln \relax (x )}\right )-\ln \left (x +{\mathrm e}^{\sqrt {2}\, \ln \relax (x )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 31, normalized size = 1.29 \begin {gather*} \frac {\sqrt {2} \log \relax (x)}{\sqrt {2} - 1} - \frac {\log \left (x + x^{\left (\sqrt {2}\right )}\right )}{\sqrt {2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.25, size = 26, normalized size = 1.08 \begin {gather*} \ln \relax (x)\,\left (\sqrt {2}+2\right )-\frac {\ln \left (x+x^{\sqrt {2}}\right )}{\sqrt {2}-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 32, normalized size = 1.33 \begin {gather*} - \frac {2 \log {\relax (x )}}{-2 + \sqrt {2}} + \frac {\sqrt {2} \log {\left (x + x^{\sqrt {2}} \right )}}{-2 + \sqrt {2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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