Optimal. Leaf size=20 \[ -x+\frac {2 i \log (a+b x+i)}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 20, 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 = 1.60 \[ \frac {i \log \left ((a+b x)^2+1\right )}{b}+\frac {2 \tan ^{-1}(a+b x)}{b}-x \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.58, size = 22, normalized size = 1.10 \[ -\frac {b x - 2 i \, \log \left (\frac {b x + a + i}{b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 17, normalized size = 0.85 \[ -x + \frac {2 \, i \log \left (b x + a + i\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 51, normalized size = 2.55 \[ -x +\frac {i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{b}+\frac {2 \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 46, normalized size = 2.30 \[ -x + \frac {2 \, \arctan \left (\frac {b^{2} x + a b}{b}\right )}{b} + \frac {i \, \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 21, normalized size = 1.05 \[ -x+\frac {\ln \left (x+\frac {a+1{}\mathrm {i}}{b}\right )\,2{}\mathrm {i}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 17, normalized size = 0.85 \[ - x + \frac {2 i \log {\left (i a + i b x - 1 \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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