Optimal. Leaf size=18 \[ -\log (x)+i \cot (a+i \log (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3473, 8} \[ -\log (x)+i \cot (a+i \log (x)) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \frac {\cot ^2(a+i \log (x))}{x} \, dx &=\operatorname {Subst}\left (\int \cot ^2(a+i x) \, dx,x,\log (x)\right )\\ &=i \cot (a+i \log (x))-\operatorname {Subst}(\int 1 \, dx,x,\log (x))\\ &=i \cot (a+i \log (x))-\log (x)\\ \end {align*}
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Mathematica [C] time = 0.05, size = 34, normalized size = 1.89 \[ i \cot (a+i \log (x)) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(a+i \log (x))\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 34, normalized size = 1.89 \[ -\frac {{\left (x^{2} - e^{\left (2 i \, a\right )}\right )} \log \relax (x) - 2 \, e^{\left (2 i \, a\right )}}{x^{2} - e^{\left (2 i \, a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 76, normalized size = 4.22 \[ \frac {i \, {\left (\tan \left (\frac {1}{2} \, a\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, a\right )^{2} + 1\right )}}{{\left (\frac {i \, {\left (x^{2} - 1\right )} \tan \left (\frac {1}{2} \, a\right )^{2}}{x^{2} + 1} - \frac {i \, {\left (x^{2} - 1\right )}}{x^{2} + 1} - 2 \, \tan \left (\frac {1}{2} \, a\right )\right )} {\left (\tan \left (\frac {1}{2} \, a\right )^{2} - 1\right )}} - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 1.50 \[ i \cot \left (a +i \ln \relax (x )\right )-\frac {i \pi }{2}+i \left (a +i \ln \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 19, normalized size = 1.06 \[ i \, a + \frac {i}{\tan \left (a + i \, \log \relax (x)\right )} - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.49, size = 16, normalized size = 0.89 \[ -\ln \relax (x)+\mathrm {cot}\left (a+\ln \relax (x)\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 20, normalized size = 1.11 \[ - \log {\relax (x )} + \frac {2 e^{2 i a}}{x^{2} - e^{2 i a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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