Optimal. Leaf size=19 \[ \frac {a \log (\sin (c+d x))}{d}+i a x \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3531, 3475} \[ \frac {a \log (\sin (c+d x))}{d}+i a x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3531
Rubi steps
\begin {align*} \int \cot (c+d x) (a+i a \tan (c+d x)) \, dx &=i a x+a \int \cot (c+d x) \, dx\\ &=i a x+\frac {a \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 1.42 \[ \frac {a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+i a x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 17, normalized size = 0.89 \[ \frac {a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.36, size = 34, normalized size = 1.79 \[ -\frac {2 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 27, normalized size = 1.42 \[ i a x +\frac {i a c}{d}+\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 37, normalized size = 1.95 \[ -\frac {-2 i \, {\left (d x + c\right )} a + a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, a \log \left (\tan \left (d x + c\right )\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.78, size = 19, normalized size = 1.00 \[ \frac {a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 20, normalized size = 1.05 \[ \frac {a \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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