Optimal. Leaf size=45 \[ \frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}}-\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]
[Out]
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Rubi [A] time = 0.0660179, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}}-\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(1 - x)/(Sqrt[x]*(1 + x^2)),x]
[Out]
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Rubi in Sympy [A] time = 12.3071, size = 42, normalized size = 0.93 \[ - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{2} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x)/(x**2+1)/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0230951, size = 40, normalized size = 0.89 \[ \frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )-\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x)/(Sqrt[x]*(1 + x^2)),x]
[Out]
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Maple [A] time = 0.011, size = 62, normalized size = 1.4 \[{\frac{\sqrt{2}}{4}\ln \left ({1 \left ( 1+x+\sqrt{2}\sqrt{x} \right ) \left ( 1+x-\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{4}\ln \left ({1 \left ( 1+x-\sqrt{2}\sqrt{x} \right ) \left ( 1+x+\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x)/(x^2+1)/x^(1/2),x)
[Out]
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Maxima [A] time = 0.75604, size = 46, normalized size = 1.02 \[ \frac{1}{2} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{1}{2} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - 1)/((x^2 + 1)*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.27586, size = 45, normalized size = 1. \[ \frac{1}{2} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2}{\left (x + 1\right )} \sqrt{x} + x^{2} + 4 \, x + 1}{x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - 1)/((x^2 + 1)*sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.97474, size = 94, normalized size = 2.09 \[ - \frac{\sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{4} - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{4} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{4} + \frac{\sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-x)/(x**2+1)/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.271841, size = 46, normalized size = 1.02 \[ \frac{1}{2} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{1}{2} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x - 1)/((x^2 + 1)*sqrt(x)),x, algorithm="giac")
[Out]