3.6 \(\int \frac{\sqrt{-1+x^2}}{(-i+x)^2} \, dx\)

Optimal. Leaf size=64 \[ \frac{\sqrt{x^2-1}}{-x+i}-\frac{i \tan ^{-1}\left (\frac{1-i x}{\sqrt{2} \sqrt{x^2-1}}\right )}{\sqrt{2}}+\tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0759454, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{\sqrt{x^2-1}}{-x+i}-\frac{i \tan ^{-1}\left (\frac{1-i x}{\sqrt{2} \sqrt{x^2-1}}\right )}{\sqrt{2}}+\tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-1 + x^2]/(-I + x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.03943, size = 53, normalized size = 0.83 \[ \frac{\sqrt{2} i \operatorname{atan}{\left (\frac{\sqrt{2} \left (i x - 1\right )}{2 \sqrt{x^{2} - 1}} \right )}}{2} + \operatorname{atanh}{\left (\frac{x}{\sqrt{x^{2} - 1}} \right )} + \frac{\sqrt{x^{2} - 1}}{- x + i} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*I*atan(sqrt(2)*(I*x - 1)/(2*sqrt(x**2 - 1)))/2 + atanh(x/sqrt(x**2 - 1))
+ sqrt(x**2 - 1)/(-x + I)

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Mathematica [B]  time = 0.178963, size = 165, normalized size = 2.58 \[ \frac{1}{4} \left (-\frac{4 \sqrt{x^2-1}}{x-i}+\sqrt{2} \log \left (2 \sqrt{2} \sqrt{x^2-1}-3 x-i\right )+2 \log \left (-2 x^2-2 \sqrt{x^2-1} x+2 i \sqrt{x^2-1}+2 i x+1\right )-2 i \sqrt{2} \tan ^{-1}\left (\frac{1}{2} \left (-\sqrt{2} \sqrt{x^2-1}+x-i\right )\right )+4 \tanh ^{-1}\left (\frac{2 x}{\sqrt{x^2-1}-x+i}\right )-\sqrt{2} \log (x-i)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-4*Sqrt[-1 + x^2])/(-I + x) - (2*I)*Sqrt[2]*ArcTan[(-I + x - Sqrt[2]*Sqrt[-1 +
x^2])/2] + 4*ArcTanh[(2*x)/(I - x + Sqrt[-1 + x^2])] - Sqrt[2]*Log[-I + x] + Sq
rt[2]*Log[-I - 3*x + 2*Sqrt[2]*Sqrt[-1 + x^2]] + 2*Log[1 + (2*I)*x - 2*x^2 + (2*
I)*Sqrt[-1 + x^2] - 2*x*Sqrt[-1 + x^2]])/4

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Maple [B]  time = 0.038, size = 125, normalized size = 2. \[{\frac{1}{2\,x-2\,i} \left ( \left ( x-i \right ) ^{2}-2+2\,i \left ( x-i \right ) \right ) ^{{\frac{3}{2}}}}+\ln \left ( x+\sqrt{ \left ( x-i \right ) ^{2}-2+2\,i \left ( x-i \right ) } \right ) +{\frac{i}{2}}\sqrt{2}\arctan \left ({\frac{ \left ( -4+2\,i \left ( x-i \right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{ \left ( x-i \right ) ^{2}-2+2\,i \left ( x-i \right ) }}}} \right ) -{\frac{i}{2}}\sqrt{ \left ( x-i \right ) ^{2}-2+2\,i \left ( x-i \right ) }-{\frac{x}{2}\sqrt{ \left ( x-i \right ) ^{2}-2+2\,i \left ( x-i \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-1)^(1/2)/(x-I)^2,x)

[Out]

1/2/(x-I)*((x-I)^2-2+2*I*(x-I))^(3/2)+ln(x+((x-I)^2-2+2*I*(x-I))^(1/2))+1/2*I*2^
(1/2)*arctan(1/4*(-4+2*I*(x-I))*2^(1/2)/((x-I)^2-2+2*I*(x-I))^(1/2))-1/2*I*((x-I
)^2-2+2*I*(x-I))^(1/2)-1/2*x*((x-I)^2-2+2*I*(x-I))^(1/2)

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Maxima [A]  time = 1.54556, size = 72, normalized size = 1.12 \[ \frac{1}{2} i \, \sqrt{2} \arcsin \left (\frac{i \, x}{{\left | x - i \right |}} - \frac{1}{{\left | x - i \right |}}\right ) - \frac{\sqrt{x^{2} - 1}}{x - i} + \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*I*sqrt(2)*arcsin(I*x/abs(x - I) - 1/abs(x - I)) - sqrt(x^2 - 1)/(x - I) + lo
g(2*x + 2*sqrt(x^2 - 1))

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Fricas [A]  time = 0.231246, size = 230, normalized size = 3.59 \[ \frac{{\left (\sqrt{2} \sqrt{x^{2} - 1}{\left (x - i\right )} - \sqrt{2}{\left (x^{2} - i \, x\right )}\right )} \log \left (-x + i \, \sqrt{2} + \sqrt{x^{2} - 1} + i\right ) -{\left (\sqrt{2} \sqrt{x^{2} - 1}{\left (x - i\right )} - \sqrt{2}{\left (x^{2} - i \, x\right )}\right )} \log \left (-x - i \, \sqrt{2} + \sqrt{x^{2} - 1} + i\right ) -{\left (2 \, x^{2} - \sqrt{x^{2} - 1}{\left (2 \, x - 2 i\right )} - 2 i \, x\right )} \log \left (-x + \sqrt{x^{2} - 1}\right ) + 2 i \, x - 2 i \, \sqrt{x^{2} - 1} - 2}{2 \, x^{2} - \sqrt{x^{2} - 1}{\left (2 \, x - 2 i\right )} - 2 i \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (x - 1\right ) \left (x + 1\right )}}{\left (x - i\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((x - 1)*(x + 1))/(x - I)**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} - 1}}{{\left (x - i\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x^2 - 1)/(x - I)^2, x)