Optimal. Leaf size=64 \[ \frac {\sqrt {-1+x^2}}{i-x}-\frac {i \tan ^{-1}\left (\frac {1-i x}{\sqrt {2} \sqrt {-1+x^2}}\right )}{\sqrt {2}}+\tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {747, 858, 223,
212, 739, 210} \begin {gather*} -\frac {i \text {ArcTan}\left (\frac {1-i x}{\sqrt {2} \sqrt {x^2-1}}\right )}{\sqrt {2}}+\frac {\sqrt {x^2-1}}{-x+i}+\tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 223
Rule 739
Rule 747
Rule 858
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^2}}{(-i+x)^2} \, dx &=\frac {\sqrt {-1+x^2}}{i-x}+\int \frac {x}{(-i+x) \sqrt {-1+x^2}} \, dx\\ &=\frac {\sqrt {-1+x^2}}{i-x}+i \int \frac {1}{(-i+x) \sqrt {-1+x^2}} \, dx+\int \frac {1}{\sqrt {-1+x^2}} \, dx\\ &=\frac {\sqrt {-1+x^2}}{i-x}-i \text {Subst}\left (\int \frac {1}{-2-x^2} \, dx,x,\frac {-1+i x}{\sqrt {-1+x^2}}\right )+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-1+x^2}}\right )\\ &=\frac {\sqrt {-1+x^2}}{i-x}-\frac {i \tan ^{-1}\left (\frac {1-i x}{\sqrt {2} \sqrt {-1+x^2}}\right )}{\sqrt {2}}+\tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 66, normalized size = 1.03 \begin {gather*} \frac {\sqrt {-1+x^2}}{i-x}+\tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {1+i x-i \sqrt {-1+x^2}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 149 vs. \(2 (52 ) = 104\).
time = 0.12, size = 150, normalized size = 2.34
method | result | size |
risch | \(-\frac {\sqrt {x^{2}-1}}{x -i}+\ln \left (x +\sqrt {x^{2}-1}\right )+\frac {i \sqrt {2}\, \arctan \left (\frac {\left (-4+2 i \left (x -i\right )\right ) \sqrt {2}}{4 \sqrt {\left (x -i\right )^{2}+2 i \left (x -i\right )-2}}\right )}{2}\) | \(65\) |
default | \(\frac {\left (\left (x -i\right )^{2}+2 i \left (x -i\right )-2\right )^{\frac {3}{2}}}{2 x -2 i}-\frac {i \left (\sqrt {\left (x -i\right )^{2}+2 i \left (x -i\right )-2}+i \ln \left (x +\sqrt {\left (x -i\right )^{2}+2 i \left (x -i\right )-2}\right )-\sqrt {2}\, \arctan \left (\frac {\left (-4+2 i \left (x -i\right )\right ) \sqrt {2}}{4 \sqrt {\left (x -i\right )^{2}+2 i \left (x -i\right )-2}}\right )\right )}{2}-\frac {x \sqrt {\left (x -i\right )^{2}+2 i \left (x -i\right )-2}}{2}+\frac {\ln \left (x +\sqrt {\left (x -i\right )^{2}+2 i \left (x -i\right )-2}\right )}{2}\) | \(150\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 53, normalized size = 0.83 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.86, size = 89, normalized size = 1.39 \begin {gather*} -\frac {\sqrt {2} {\left (x - i\right )} \log \left (-x + i \, \sqrt {2} + \sqrt {x^{2} - 1} + i\right ) - \sqrt {2} {\left (x - i\right )} \log \left (-x - i \, \sqrt {2} + \sqrt {x^{2} - 1} + i\right ) + 2 \, {\left (x - i\right )} \log \left (-x + \sqrt {x^{2} - 1}\right ) + 2 \, x + 2 \, \sqrt {x^{2} - 1} - 2 i}{2 \, {\left (x - i\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (x - 1\right ) \left (x + 1\right )}}{\left (x - i\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.16, size = 84, normalized size = 1.31 \begin {gather*} i \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (x - \sqrt {x^{2} - 1} - i\right )}\right ) + \frac {2 \, {\left (i \, x - i \, \sqrt {x^{2} - 1} - 1\right )}}{{\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 2 i \, x + 2 i \, \sqrt {x^{2} - 1} + 1} - \log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {x^2-1}}{{\left (x-\mathrm {i}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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