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

Optimal. Leaf size=92 \[ -\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.123013, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*(1 + x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.2786, size = 83, normalized size = 0.9 \[ - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{4} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2+1)/x**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0220481, size = 76, normalized size = 0.83 \[ \frac{-\log \left (x-\sqrt{2} \sqrt{x}+1\right )+\log \left (x+\sqrt{2} \sqrt{x}+1\right )-2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+2 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[x]*(1 + x^2)),x]

[Out]

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

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Maple [A]  time = 0.007, size = 62, normalized size = 0.7 \[{\frac{\sqrt{2}}{2}\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) }+{\frac{\sqrt{2}}{4}\ln \left ({1 \left ( 1+x+\sqrt{2}\sqrt{x} \right ) \left ( 1+x-\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{2}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+1)/x^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.48905, size = 100, normalized size = 1.09 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)*sqrt(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.250424, size = 139, normalized size = 1.51 \[ -\sqrt{2} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} + 1}\right ) - \sqrt{2} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} - 1}\right ) + \frac{1}{4} \, \sqrt{2} \log \left (2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - \frac{1}{4} \, \sqrt{2} \log \left (-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)*sqrt(x)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.85345, size = 90, normalized size = 0.98 \[ - \frac{\sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{4} + \frac{\sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2+1)/x**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.208833, size = 100, normalized size = 1.09 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{1}{4} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{1}{4} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)*sqrt(x)),x, algorithm="giac")

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 1/2*sqrt(2)*arctan(-1/2*
sqrt(2)*(sqrt(2) - 2*sqrt(x))) + 1/4*sqrt(2)*ln(sqrt(2)*sqrt(x) + x + 1) - 1/4*s
qrt(2)*ln(-sqrt(2)*sqrt(x) + x + 1)