3.432 \(\int \frac{1-x}{\sqrt{x} \left (1+x^2\right )} \, dx\)

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]

-(Log[1 - Sqrt[2]*Sqrt[x] + x]/Sqrt[2]) + Log[1 + Sqrt[2]*Sqrt[x] + x]/Sqrt[2]

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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]

-(Log[1 - Sqrt[2]*Sqrt[x] + x]/Sqrt[2]) + Log[1 + Sqrt[2]*Sqrt[x] + x]/Sqrt[2]

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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]

-sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1)/2 + sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1)/
2

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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]

(-Log[1 - Sqrt[2]*Sqrt[x] + x] + Log[1 + Sqrt[2]*Sqrt[x] + x])/Sqrt[2]

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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]

1/4*2^(1/2)*ln((1+x+2^(1/2)*x^(1/2))/(1+x-2^(1/2)*x^(1/2)))-1/4*2^(1/2)*ln((1+x-
2^(1/2)*x^(1/2))/(1+x+2^(1/2)*x^(1/2)))

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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]

1/2*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) - 1/2*sqrt(2)*log(-sqrt(2)*sqrt(x) + x
+ 1)

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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]

1/2*sqrt(2)*log((2*sqrt(2)*(x + 1)*sqrt(x) + x^2 + 4*x + 1)/(x^2 + 1))

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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]

-sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/4 - sqrt(2)*log(-sqrt(2)*sqrt(x) + x
+ 1)/4 + sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1)/4 + sqrt(2)*log(4*sqrt(2)*sqrt(x)
+ 4*x + 4)/4

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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]

1/2*sqrt(2)*ln(sqrt(2)*sqrt(x) + x + 1) - 1/2*sqrt(2)*ln(-sqrt(2)*sqrt(x) + x +
1)