Optimal. Leaf size=18 \[ -\log (x)-i \tan (a+i \log (x)) \]
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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} \[ -\log (x)-i \tan (a+i \log (x)) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \frac {\tan ^2(a+i \log (x))}{x} \, dx &=\operatorname {Subst}\left (\int \tan ^2(a+i x) \, dx,x,\log (x)\right )\\ &=-i \tan (a+i \log (x))-\operatorname {Subst}(\int 1 \, dx,x,\log (x))\\ &=-\log (x)-i \tan (a+i \log (x))\\ \end {align*}
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Mathematica [A] time = 0.04, size = 28, normalized size = 1.56 \[ i \tan ^{-1}(\tan (a+i \log (x)))-i \tan (a+i \log (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 30, normalized size = 1.67 \[ -\frac {{\left (x^{2} + e^{\left (2 i \, a\right )}\right )} \log \relax (x) + 2 \, e^{\left (2 i \, a\right )}}{x^{2} + e^{\left (2 i \, a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 38, normalized size = 2.11 \[ \frac {i \, \tan \relax (a)^{2} + i}{{\left (\frac {i \, {\left (x^{2} - 1\right )} \tan \relax (a)}{x^{2} + 1} - 1\right )} \tan \relax (a)} - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 23, normalized size = 1.28 \[ -i \tan \left (a +i \ln \relax (x )\right )+i \left (a +i \ln \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 17, normalized size = 0.94 \[ i \, a - \log \relax (x) - i \, \tan \left (a + i \, \log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.38, size = 16, normalized size = 0.89 \[ -\ln \relax (x)-\mathrm {tan}\left (a+\ln \relax (x)\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 22, normalized size = 1.22 \[ - \log {\relax (x )} - \frac {2 e^{2 i a}}{x^{2} + e^{2 i a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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