Optimal. Leaf size=32 \[ -a x-\frac {a \cot (c+d x)}{d}+\frac {i a \log (\sin (c+d x))}{d} \]
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Rubi [A]
time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3610, 3612,
3556} \begin {gather*} -\frac {a \cot (c+d x)}{d}+\frac {i a \log (\sin (c+d x))}{d}-a x \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3610
Rule 3612
Rubi steps
\begin {align*} \int \cot ^2(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac {a \cot (c+d x)}{d}+\int \cot (c+d x) (i a-a \tan (c+d x)) \, dx\\ &=-a x-\frac {a \cot (c+d x)}{d}+(i a) \int \cot (c+d x) \, dx\\ &=-a x-\frac {a \cot (c+d x)}{d}+\frac {i a \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.13, size = 54, normalized size = 1.69 \begin {gather*} -\frac {a \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{d}+\frac {i 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 = 35, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {i a \ln \left (\sin \left (d x +c \right )\right )+a \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(35\) |
default | \(\frac {i a \ln \left (\sin \left (d x +c \right )\right )+a \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(35\) |
risch | \(\frac {2 a c}{d}-\frac {2 i a}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {i a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(48\) |
norman | \(\frac {-\frac {a}{d}-a x \tan \left (d x +c \right )}{\tan \left (d x +c \right )}+\frac {i a \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {i a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 49, normalized size = 1.53 \begin {gather*} -\frac {2 \, {\left (d x + c\right )} a + i \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 i \, a \log \left (\tan \left (d x + c\right )\right ) + \frac {2 \, a}{\tan \left (d x + c\right )}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 51, normalized size = 1.59 \begin {gather*} \frac {{\left (i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 2 i \, a}{d e^{\left (2 i \, d x + 2 i \, c\right )} - d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 46, normalized size = 1.44 \begin {gather*} - \frac {2 i a}{d e^{2 i c} e^{2 i d x} - d} + \frac {i 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 [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 75 vs. \(2 (30) = 60\).
time = 0.61, size = 75, normalized size = 2.34 \begin {gather*} -\frac {4 i \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 2 i \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {-2 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.73, size = 27, normalized size = 0.84 \begin {gather*} -\frac {a\,\left (\mathrm {cot}\left (c+d\,x\right )+2\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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