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

Optimal. Leaf size=113 \[ \frac{x^{3/2}}{2 \left (x^2+1\right )}+\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{4 \sqrt{2}} \]

[Out]

x^(3/2)/(2*(1 + x^2)) - ArcTan[1 - Sqrt[2]*Sqrt[x]]/(4*Sqrt[2]) + ArcTan[1 + Sqr
t[2]*Sqrt[x]]/(4*Sqrt[2]) + Log[1 - Sqrt[2]*Sqrt[x] + x]/(8*Sqrt[2]) - Log[1 + S
qrt[2]*Sqrt[x] + x]/(8*Sqrt[2])

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Rubi [A]  time = 0.145266, antiderivative size = 113, 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 \[ \frac{x^{3/2}}{2 \left (x^2+1\right )}+\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(1 + x^2)^2,x]

[Out]

x^(3/2)/(2*(1 + x^2)) - ArcTan[1 - Sqrt[2]*Sqrt[x]]/(4*Sqrt[2]) + ArcTan[1 + Sqr
t[2]*Sqrt[x]]/(4*Sqrt[2]) + Log[1 - Sqrt[2]*Sqrt[x] + x]/(8*Sqrt[2]) - Log[1 + S
qrt[2]*Sqrt[x] + x]/(8*Sqrt[2])

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Rubi in Sympy [A]  time = 19.3458, size = 95, normalized size = 0.84 \[ \frac{x^{\frac{3}{2}}}{2 \left (x^{2} + 1\right )} + \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{16} - \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{16} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(3/2)/(2*(x**2 + 1)) + sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1)/16 - sqrt(2)*log
(sqrt(2)*sqrt(x) + x + 1)/16 + sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/8 + sqrt(2)*ata
n(sqrt(2)*sqrt(x) + 1)/8

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Mathematica [A]  time = 0.0991947, size = 106, normalized size = 0.94 \[ \frac{1}{16} \left (\frac{8 x^{3/2}}{x^2+1}+\sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )-\sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )-2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((8*x^(3/2))/(1 + x^2) - 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 2*Sqrt[2]*ArcTa
n[1 + Sqrt[2]*Sqrt[x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] - Sqrt[2]*Log[1 +
Sqrt[2]*Sqrt[x] + x])/16

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Maple [A]  time = 0.011, size = 74, normalized size = 0.7 \[{\frac{1}{2\,{x}^{2}+2}{x}^{{\frac{3}{2}}}}+{\frac{\sqrt{2}}{8}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }+{\frac{\sqrt{2}}{8}\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) }+{\frac{\sqrt{2}}{16}\ln \left ({1 \left ( 1+x-\sqrt{2}\sqrt{x} \right ) \left ( 1+x+\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(x^2+1)^2,x)

[Out]

1/2*x^(3/2)/(x^2+1)+1/8*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)+1/8*arctan(2^(1/2)*x^(
1/2)-1)*2^(1/2)+1/16*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 = 1.50476, size = 116, normalized size = 1.03 \[ \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{1}{8} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{1}{16} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{1}{16} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{x^{\frac{3}{2}}}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 1/8*sqrt(2)*arctan(-1/2*
sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 1/16*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) + 1/1
6*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/2*x^(3/2)/(x^2 + 1)

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Fricas [A]  time = 0.248485, size = 184, normalized size = 1.63 \[ -\frac{4 \, \sqrt{2}{\left (x^{2} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} + 1}\right ) + 4 \, \sqrt{2}{\left (x^{2} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} - 1}\right ) + \sqrt{2}{\left (x^{2} + 1\right )} \log \left (2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - \sqrt{2}{\left (x^{2} + 1\right )} \log \left (-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 8 \, x^{\frac{3}{2}}}{16 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(4*sqrt(2)*(x^2 + 1)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(2*sqrt(2)*sqrt(x) +
2*x + 2) + 1)) + 4*sqrt(2)*(x^2 + 1)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(-2*sqrt(2)
*sqrt(x) + 2*x + 2) - 1)) + sqrt(2)*(x^2 + 1)*log(2*sqrt(2)*sqrt(x) + 2*x + 2) -
sqrt(2)*(x^2 + 1)*log(-2*sqrt(2)*sqrt(x) + 2*x + 2) - 8*x^(3/2))/(x^2 + 1)

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Sympy [A]  time = 12.1386, size = 257, normalized size = 2.27 \[ \frac{8 x^{\frac{3}{2}}}{16 x^{2} + 16} + \frac{\sqrt{2} x^{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{16 x^{2} + 16} - \frac{\sqrt{2} x^{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac{2 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{16 x^{2} + 16} + \frac{2 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{16 x^{2} + 16} + \frac{\sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{16 x^{2} + 16} - \frac{\sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac{2 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{16 x^{2} + 16} + \frac{2 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{16 x^{2} + 16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*x**(3/2)/(16*x**2 + 16) + sqrt(2)*x**2*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(16*x
**2 + 16) - sqrt(2)*x**2*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(16*x**2 + 16) + 2*sqr
t(2)*x**2*atan(sqrt(2)*sqrt(x) - 1)/(16*x**2 + 16) + 2*sqrt(2)*x**2*atan(sqrt(2)
*sqrt(x) + 1)/(16*x**2 + 16) + sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(16*x**
2 + 16) - sqrt(2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(16*x**2 + 16) + 2*sqrt(2)*at
an(sqrt(2)*sqrt(x) - 1)/(16*x**2 + 16) + 2*sqrt(2)*atan(sqrt(2)*sqrt(x) + 1)/(16
*x**2 + 16)

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GIAC/XCAS [A]  time = 0.21371, size = 116, normalized size = 1.03 \[ \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{1}{8} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{1}{16} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{1}{16} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{x^{\frac{3}{2}}}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 1/8*sqrt(2)*arctan(-1/2*
sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 1/16*sqrt(2)*ln(sqrt(2)*sqrt(x) + x + 1) + 1/16
*sqrt(2)*ln(-sqrt(2)*sqrt(x) + x + 1) + 1/2*x^(3/2)/(x^2 + 1)