Optimal. Leaf size=19 \[ i a x+\frac {a \log (\sin (c+d x))}{d} \]
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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 = {3612, 3556}
\begin {gather*} \frac {a \log (\sin (c+d x))}{d}+i a x \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3612
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 \begin {gather*} i a x+\frac {a (\log (\cos (c+d x))+\log (\tan (c+d x)))}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 24, normalized size = 1.26
method | result | size |
derivativedivides | \(\frac {i a \left (d x +c \right )+a \ln \left (\sin \left (d x +c \right )\right )}{d}\) | \(24\) |
default | \(\frac {i a \left (d x +c \right )+a \ln \left (\sin \left (d x +c \right )\right )}{d}\) | \(24\) |
risch | \(-\frac {2 i a c}{d}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(27\) |
norman | \(i a x +\frac {a \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).
time = 0.49, size = 37, normalized size = 1.95 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.54, size = 17, normalized size = 0.89 \begin {gather*} \frac {a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 20, normalized size = 1.05 \begin {gather*} \frac {a \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 34, normalized size = 1.79 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \frac {a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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