Optimal. Leaf size=40 \[ -\frac {2 i x}{b}-\frac {x^2}{2}+\frac {2 (1+i a) \log (i-a-b x)}{b^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5203, 78}
\begin {gather*} \frac {2 (1+i a) \log (-a-b x+i)}{b^2}-\frac {2 i x}{b}-\frac {x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 5203
Rubi steps
\begin {align*} \int e^{-2 i \tan ^{-1}(a+b x)} x \, dx &=\int \frac {x (1-i a-i b x)}{1+i a+i b x} \, dx\\ &=\int \left (-\frac {2 i}{b}-x+\frac {2 (1+i a)}{b (-i+a+b x)}\right ) \, dx\\ &=-\frac {2 i x}{b}-\frac {x^2}{2}+\frac {2 (1+i a) \log (i-a-b x)}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 40, normalized size = 1.00 \begin {gather*} -\frac {2 i x}{b}-\frac {x^2}{2}+\frac {2 (1+i a) \log (i-a-b x)}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 39, normalized size = 0.98
method | result | size |
default | \(-\frac {\frac {1}{2} x^{2} b +2 i x}{b}+\frac {\left (2 i a +2\right ) \ln \left (-b x -a +i\right )}{b^{2}}\) | \(39\) |
risch | \(-\frac {x^{2}}{2}-\frac {2 i x}{b}+\frac {\ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{b^{2}}+\frac {2 i \arctan \left (b x +a \right )}{b^{2}}+\frac {i a \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{b^{2}}-\frac {2 a \arctan \left (b x +a \right )}{b^{2}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 36, normalized size = 0.90 \begin {gather*} \frac {i \, {\left (i \, b x^{2} - 4 \, x\right )}}{2 \, b} - \frac {2 \, {\left (-i \, a - 1\right )} \log \left (i \, b x + i \, a + 1\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.04, size = 35, normalized size = 0.88 \begin {gather*} -\frac {b^{2} x^{2} + 4 i \, b x + 4 \, {\left (-i \, a - 1\right )} \log \left (\frac {b x + a - i}{b}\right )}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 29, normalized size = 0.72 \begin {gather*} - \frac {x^{2}}{2} - \frac {2 i x}{b} + \frac {2 i \left (a - i\right ) \log {\left (a + b x - i \right )}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 72 vs. \(2 (32) = 64\).
time = 0.42, size = 72, normalized size = 1.80 \begin {gather*} -\frac {i \, {\left (\frac {{\left (i \, b x + i \, a + 1\right )}^{2} {\left (-\frac {2 i \, {\left (i \, a b + 3 \, b\right )}}{{\left (i \, b x + i \, a + 1\right )} b} + i\right )}}{b} + \frac {4 \, {\left (a - i\right )} \log \left (\frac {1}{\sqrt {{\left (b x + a\right )}^{2} + 1} {\left | b \right |}}\right )}{b}\right )}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.50, size = 51, normalized size = 1.28 \begin {gather*} \ln \left (x+\frac {a-\mathrm {i}}{b}\right )\,\left (\frac {2}{b^2}+\frac {a\,2{}\mathrm {i}}{b^2}\right )-\frac {x^2}{2}+x\,\left (\frac {a-\mathrm {i}}{b}-\frac {a+1{}\mathrm {i}}{b}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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