3.1471 \(\int \frac{-2+x}{\sqrt{-3+x} \left (-8+x^2\right )} \, dx\)

Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\left (\sqrt{2}-1\right ) \sqrt{x-3}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\left (1+\sqrt{2}\right ) \sqrt{x-3}\right )}{\sqrt{2}} \]

[Out]

ArcTan[(-1 + Sqrt[2])*Sqrt[-3 + x]]/Sqrt[2] + ArcTan[(1 + Sqrt[2])*Sqrt[-3 + x]]
/Sqrt[2]

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Rubi [A]  time = 0.144215, antiderivative size = 57, normalized size of antiderivative = 1.27, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{x-3}}{\sqrt{3-2 \sqrt{2}}}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{x-3}}{\sqrt{3+2 \sqrt{2}}}\right )}{\sqrt{2}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(-2 + x)/(Sqrt[-3 + x]*(-8 + x^2)),x]

[Out]

ArcTan[Sqrt[-3 + x]/Sqrt[3 - 2*Sqrt[2]]]/Sqrt[2] + ArcTan[Sqrt[-3 + x]/Sqrt[3 +
2*Sqrt[2]]]/Sqrt[2]

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Rubi in Sympy [A]  time = 12.4693, size = 66, normalized size = 1.47 \[ - \frac{2 \left (- \frac{\sqrt{2}}{4} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{x - 3}}{-1 + \sqrt{2}} \right )}}{- \sqrt{2} + 1} + \frac{2 \left (\frac{\sqrt{2}}{4} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{x - 3}}{1 + \sqrt{2}} \right )}}{1 + \sqrt{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-2+x)/(x**2-8)/(-3+x)**(1/2),x)

[Out]

-2*(-sqrt(2)/4 + 1/2)*atan(sqrt(x - 3)/(-1 + sqrt(2)))/(-sqrt(2) + 1) + 2*(sqrt(
2)/4 + 1/2)*atan(sqrt(x - 3)/(1 + sqrt(2)))/(1 + sqrt(2))

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Mathematica [B]  time = 0.192289, size = 91, normalized size = 2.02 \[ \frac{\left (\sqrt{2}-1\right ) \tan ^{-1}\left (\frac{\sqrt{x-3}}{\sqrt{3-2 \sqrt{2}}}\right )}{\sqrt{2 \left (3-2 \sqrt{2}\right )}}+\frac{\left (1+\sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{x-3}}{\sqrt{3+2 \sqrt{2}}}\right )}{\sqrt{2 \left (3+2 \sqrt{2}\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-2 + x)/(Sqrt[-3 + x]*(-8 + x^2)),x]

[Out]

((-1 + Sqrt[2])*ArcTan[Sqrt[-3 + x]/Sqrt[3 - 2*Sqrt[2]]])/Sqrt[2*(3 - 2*Sqrt[2])
] + ((1 + Sqrt[2])*ArcTan[Sqrt[-3 + x]/Sqrt[3 + 2*Sqrt[2]]])/Sqrt[2*(3 + 2*Sqrt[
2])]

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Maple [B]  time = 0.071, size = 119, normalized size = 2.6 \[ -{\frac{\sqrt{2}}{-2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\sqrt{x-3}}{-2+2\,\sqrt{2}}} \right ) }+2\,{\frac{1}{-2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\sqrt{x-3}}{-2+2\,\sqrt{2}}} \right ) }+{\frac{\sqrt{2}}{2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\sqrt{x-3}}{2+2\,\sqrt{2}}} \right ) }+2\,{\frac{1}{2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\sqrt{x-3}}{2+2\,\sqrt{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x-2)/(x^2-8)/(x-3)^(1/2),x)

[Out]

-2^(1/2)/(-2+2*2^(1/2))*arctan(2*(x-3)^(1/2)/(-2+2*2^(1/2)))+2/(-2+2*2^(1/2))*ar
ctan(2*(x-3)^(1/2)/(-2+2*2^(1/2)))+2^(1/2)/(2+2*2^(1/2))*arctan(2*(x-3)^(1/2)/(2
+2*2^(1/2)))+2/(2+2*2^(1/2))*arctan(2*(x-3)^(1/2)/(2+2*2^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x - 2}{{\left (x^{2} - 8\right )} \sqrt{x - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 2)/((x^2 - 8)*sqrt(x - 3)),x, algorithm="maxima")

[Out]

integrate((x - 2)/((x^2 - 8)*sqrt(x - 3)), x)

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Fricas [A]  time = 0.273485, size = 26, normalized size = 0.58 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (x - 4\right )}}{4 \, \sqrt{x - 3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 2)/((x^2 - 8)*sqrt(x - 3)),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*arctan(1/4*sqrt(2)*(x - 4)/sqrt(x - 3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x - 2}{\sqrt{x - 3} \left (x^{2} - 8\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2+x)/(x**2-8)/(-3+x)**(1/2),x)

[Out]

Integral((x - 2)/(sqrt(x - 3)*(x**2 - 8)), x)

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GIAC/XCAS [A]  time = 0.270213, size = 31, normalized size = 0.69 \[ \frac{1}{4} \, \sqrt{2}{\left (\pi + 2 \, \arctan \left (\frac{\sqrt{2}{\left (x - 4\right )}}{4 \, \sqrt{x - 3}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 2)/((x^2 - 8)*sqrt(x - 3)),x, algorithm="giac")

[Out]

1/4*sqrt(2)*(pi + 2*arctan(1/4*sqrt(2)*(x - 4)/sqrt(x - 3)))