Optimal. Leaf size=23 \[ -x-\frac {2 i \log (-a-b x+i)}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5093, 43} \[ -x-\frac {2 i \log (-a-b x+i)}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 5093
Rubi steps
\begin {align*} \int e^{-2 i \tan ^{-1}(a+b x)} \, dx &=\int \frac {1-i a-i b x}{1+i a+i b x} \, dx\\ &=\int \left (-1-\frac {2 i}{-i+a+b x}\right ) \, dx\\ &=-x-\frac {2 i \log (i-a-b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.39 \[ -\frac {i \log \left ((a+b x)^2+1\right )}{b}+\frac {2 \tan ^{-1}(a+b x)}{b}-x \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 22, normalized size = 0.96 \[ -\frac {b x + 2 i \, \log \left (\frac {b x + a - i}{b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 38, normalized size = 1.65 \[ \frac {{\left (b i x + a i + 1\right )} i}{b} + \frac {2 \, i \log \left (\frac {1}{\sqrt {{\left (b x + a\right )}^{2} + 1} {\left | b \right |}}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 40, normalized size = 1.74 \[ -x -\frac {i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{b}+\frac {2 \arctan \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 19, normalized size = 0.83 \[ -x - \frac {2 i \, \log \left (i \, b x + i \, a + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 21, normalized size = 0.91 \[ -x-\frac {\ln \left (x+\frac {a-\mathrm {i}}{b}\right )\,2{}\mathrm {i}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 19, normalized size = 0.83 \[ - x - \frac {2 i \log {\left (i a + i b x + 1 \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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