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

Optimal. Leaf size=55 \[ -\frac{\log \left (x^2-\sqrt{2} x+1\right )}{2 \sqrt{2}}-x \tan ^{-1}\left (1-\sqrt{2} x\right )+\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0666246, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\log \left (x^2-\sqrt{2} x+1\right )}{2 \sqrt{2}}-x \tan ^{-1}\left (1-\sqrt{2} x\right )+\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[ArcTan[(-Sqrt[2] + 2*x)/Sqrt[2]],x]

[Out]

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

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Rubi in Sympy [A]  time = 3.58589, size = 56, normalized size = 1.02 \[ x \operatorname{atan}{\left (\sqrt{2} \left (x - \frac{\sqrt{2}}{2}\right ) \right )} - \frac{\sqrt{2} \log{\left (4 x^{2} - 4 \sqrt{2} x + 4 \right )}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.063924, size = 48, normalized size = 0.87 \[ \frac{1}{4} \left (2 \left (\sqrt{2}-2 x\right ) \tan ^{-1}\left (1-\sqrt{2} x\right )-\sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[ArcTan[(-Sqrt[2] + 2*x)/Sqrt[2]],x]

[Out]

(2*(Sqrt[2] - 2*x)*ArcTan[1 - Sqrt[2]*x] - Sqrt[2]*Log[1 - Sqrt[2]*x + x^2])/4

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Maple [A]  time = 0.006, size = 42, normalized size = 0.8 \[ x\arctan \left ( x\sqrt{2}-1 \right ) -{\frac{\arctan \left ( x\sqrt{2}-1 \right ) \sqrt{2}}{2}}-{\frac{\sqrt{2}\ln \left ( \left ( x\sqrt{2}-1 \right ) ^{2}+1 \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arctan(1/2*(2*x-2^(1/2))*2^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.55182, size = 70, normalized size = 1.27 \[ \frac{1}{4} \, \sqrt{2}{\left (\sqrt{2}{\left (2 \, x - \sqrt{2}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \log \left (\frac{1}{2} \,{\left (2 \, x - \sqrt{2}\right )}^{2} + 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.237161, size = 50, normalized size = 0.91 \[ \frac{1}{4} \, \sqrt{2}{\left (2 \,{\left (\sqrt{2} x - 1\right )} \arctan \left (\sqrt{2} x - 1\right ) - \log \left (x^{2} - \sqrt{2} x + 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.10909, size = 230, normalized size = 4.18 \[ \frac{4 x^{3} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{4 x^{2} - 4 \sqrt{2} x + 4} - \frac{\sqrt{2} x^{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{4 x^{2} - 4 \sqrt{2} x + 4} - \frac{6 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{4 x^{2} - 4 \sqrt{2} x + 4} + \frac{2 x \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{4 x^{2} - 4 \sqrt{2} x + 4} + \frac{8 x \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{4 x^{2} - 4 \sqrt{2} x + 4} - \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{4 x^{2} - 4 \sqrt{2} x + 4} - \frac{2 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{4 x^{2} - 4 \sqrt{2} x + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*x**3*atan(sqrt(2)*x - 1)/(4*x**2 - 4*sqrt(2)*x + 4) - sqrt(2)*x**2*log(x**2 -
sqrt(2)*x + 1)/(4*x**2 - 4*sqrt(2)*x + 4) - 6*sqrt(2)*x**2*atan(sqrt(2)*x - 1)/(
4*x**2 - 4*sqrt(2)*x + 4) + 2*x*log(x**2 - sqrt(2)*x + 1)/(4*x**2 - 4*sqrt(2)*x
+ 4) + 8*x*atan(sqrt(2)*x - 1)/(4*x**2 - 4*sqrt(2)*x + 4) - sqrt(2)*log(x**2 - s
qrt(2)*x + 1)/(4*x**2 - 4*sqrt(2)*x + 4) - 2*sqrt(2)*atan(sqrt(2)*x - 1)/(4*x**2
- 4*sqrt(2)*x + 4)

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GIAC/XCAS [A]  time = 0.20187, size = 70, normalized size = 1.27 \[ \frac{1}{4} \, \sqrt{2}{\left (\sqrt{2}{\left (2 \, x - \sqrt{2}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) -{\rm ln}\left (\frac{1}{2} \,{\left (2 \, x - \sqrt{2}\right )}^{2} + 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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