3.318 \(\int \frac{1}{x^{3/2} \left (1+x^2\right )} \, dx\)

Optimal. Leaf size=99 \[ -\frac{2}{\sqrt{x}}-\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]

[Out]

-2/Sqrt[x] + ArcTan[1 - Sqrt[2]*Sqrt[x]]/Sqrt[2] - ArcTan[1 + Sqrt[2]*Sqrt[x]]/S
qrt[2] - Log[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 = 0.133273, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{2}{\sqrt{x}}-\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(1 + x^2)),x]

[Out]

-2/Sqrt[x] + ArcTan[1 - Sqrt[2]*Sqrt[x]]/Sqrt[2] - ArcTan[1 + Sqrt[2]*Sqrt[x]]/S
qrt[2] - Log[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 = 18.9603, size = 90, normalized size = 0.91 \[ - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{4} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{2} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{2} - \frac{2}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(3/2)/(x**2+1),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 -
2/sqrt(x)

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Mathematica [A]  time = 0.0554511, size = 99, normalized size = 1. \[ \frac{1}{4} \left (-\frac{8}{\sqrt{x}}-\sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )+\sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )+2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(3/2)*(1 + x^2)),x]

[Out]

(-8/Sqrt[x] + 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[
2]*Sqrt[x]] - Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] + Sqrt[2]*Log[1 + Sqrt[2]*Sqr
t[x] + x])/4

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Maple [A]  time = 0.01, size = 67, normalized size = 0.7 \[ -{\frac{\sqrt{2}}{2}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }-{\frac{\sqrt{2}}{2}\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) }-{\frac{\sqrt{2}}{4}\ln \left ({1 \left ( 1+x-\sqrt{2}\sqrt{x} \right ) \left ( 1+x+\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) }-2\,{\frac{1}{\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(x^2+1),x)

[Out]

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

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Maxima [A]  time = 1.51559, size = 107, normalized size = 1.08 \[ -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{2}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)*x^(3/2)),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) - 2/sqrt(x)

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Fricas [A]  time = 0.247709, size = 157, normalized size = 1.59 \[ \frac{4 \, \sqrt{2} x \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} + 1}\right ) + 4 \, \sqrt{2} x \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} - 1}\right ) + \sqrt{2} x \log \left (2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - \sqrt{2} x \log \left (-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 8 \, \sqrt{x}}{4 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)*x^(3/2)),x, algorithm="fricas")

[Out]

1/4*(4*sqrt(2)*x*arctan(1/(sqrt(2)*sqrt(x) + sqrt(2*sqrt(2)*sqrt(x) + 2*x + 2) +
1)) + 4*sqrt(2)*x*arctan(1/(sqrt(2)*sqrt(x) + sqrt(-2*sqrt(2)*sqrt(x) + 2*x + 2
) - 1)) + sqrt(2)*x*log(2*sqrt(2)*sqrt(x) + 2*x + 2) - sqrt(2)*x*log(-2*sqrt(2)*
sqrt(x) + 2*x + 2) - 8*sqrt(x))/x

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Sympy [A]  time = 6.21941, size = 97, normalized size = 0.98 \[ - \frac{\sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{4} + \frac{\sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{2} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{2} - \frac{2}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(3/2)/(x**2+1),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 - 2/sqrt(x)

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GIAC/XCAS [A]  time = 0.211222, size = 107, normalized size = 1.08 \[ -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{1}{4} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{1}{4} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{2}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 1)*x^(3/2)),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*
sqrt(2)*ln(-sqrt(2)*sqrt(x) + x + 1) - 2/sqrt(x)